structured/0709.1201.structured.json

{"paper_meta":{"paper_id":"arxiv:0709.1201","title":"0709.1201","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0709.1201v3 [cs.CC] 19 Apr 2009\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nAbstract. We obtain two results about the proof complexity of deep inference: 1) deep-\ninference proof systems are as powerful as Frege ones, even when both are extended with\nthe Tseitin extension rule or with the substitution rule; 2) there are analytic deep-inference\nproof systems that exhibit an exponential speedup over analytic Gentzen proof systems\nthat they polynomially simulate.\n1. Introduction\nDeep inference is a relatively new methodology in proof theory, consisting in deal-\ning with proof systems whose inference rules are applicable at any depth inside formulae\n[Gug07b]. We obtain two results about the proof complexity of deep inference:\n• deep-inference proof systems are as powerful as Frege ones, even when both are\nextended with the Tseitin extension rule or with the substitution rule;\n• there are analytic deep-inference proof systems that exhibit an exponential speed-\nup over analytic Gentzen proof systems that they polynomially simulate.\nThese results are established for the calculus of structures, or CoS, the simplest formal-\nism in deep inference [Gug07b], and in particular for its proof system SKS, introduced\nby Br ̈unnler in [Br ̈u04] and then extensively studied [Br ̈u03a, Br ̈u03b, Br ̈u06a, Br ̈u06d,\nBG04, BT01].\nOur contributions fit in the following picture.\nCoS +\nextension\nCoS +\nsubstitution\nFrege +\nextension\nFrege +\nsubstitution\n⋆\n4\n3\nKraj ́ıˇcek-Pudl ́ak ’89\n⋆\n5\nCook-Reckhow ’79\nFrege\nCoS\nGentzen\nopen\n2\nCook-\nReckhow ’74\nanalytic\nCoS\nanalytic\nGentzen\nBr ̈unnler\n’04\n1 ×\nStatman ’78\n×\nopen\nThe notation F\nG indicates that formalism F polynomially simulates formalism G ;\nthe notation F\nG\n×\nindicates that it is known that this does not happen.\nThe left side of the picture represents, in part, the following. Analytic Gentzen systems,\ni.e., Gentzen proof systems without the cut rule, can only prove certain formulae, which\nwe call ‘Statman tautologies’, with proofs that grow exponentially in the size of the for-\nmulae. On the contrary, Gentzen systems with the cut rule can prove Statman tautologies\nby polynomially growing proofs. So, Gentzen systems p-simulate analytic Gentzen ones,\nDate: October 24, 2018.\nThis research was partially supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in\nDeep Inference.\nc⃝ACM, 2009. This is the authors’ version of the work. It is posted here by permission of ACM for your\npersonal use. Not for redistribution. The definitive version was published in ACM Transactions on Computational\nLogic 10 (2:14) 2009, pp. 1–34, http://doi.acm.org/10.1145/1462179.1462186.\n1\n\n2\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nbut not vice versa [Sta78]. Cook and Reckhow proved that Frege and Gentzen systems\nare polynomially equivalent, i.e., each Frege system polynomially simulates any Gentzen\nsystem and vice versa [CR74].\nIn the box at the right of the figure, ‘extension’ refers to the Tseitin extension rule,\nand ‘substitution’ to the substitution rule. The works of Cook and Reckhow [CR79] and\nKraj ́ıˇcek and Pudl ́ak [KP89] established that Frege + extension and Frege + substitution\nare polynomially equivalent. It is immediate to see that these formalisms polynomially\nsimulate Frege and Gentzen, which, in turn, polynomially simulate analytic Gentzen. It is\na major open problem to establish whether Frege polynomially simulates Frege + exten-\nsion/substitution.\nIn this work, we establish the following results (numbered as in the previous figure):\n(1) Analytic Gentzen does not polynomially simulate analytic CoS (essentially in the\nform of system SKS without cut); in fact, Statman tautologies admit polynomially\ngrowing proofs in analytic CoS (Theorem 3.12).\n(2) CoS and Frege are polynomially equivalent (Theorems 4.7 and 4.12).\n(3) There is a natural notion of (Tseitin) extension for CoS, and CoS + extension is\npolynomially equivalent to Frege + extension (Theorem 5.6).\n(4) There is a natural notion of substitution for CoS, and CoS + substitution polyno-\nmially simulates CoS + extension (Theorem 5.12).\n(5) Frege + substitution polynomially simulates CoS + substitution; this way, we\nknow that all extended formalisms are polynomially equivalent (Theorem 5.14).\nThe polynomial simulations indicated by ⋆arcs in the picture follow from the others.\nEstablishing whether analytic CoS polynomially simulates CoS is an open problem.\nAfter the necessary preliminaries, in Section 2, we see how CoS expresses Gentzen\nsystems, including their properties, like analyticity, and then in Section 3 how it provides\nfor exponentially more compact proofs. The relation between CoS and Frege systems is\nexplored in Section 4 and the extensions are studied in Section 5. We conclude the article\nwith a list of open problems, in Section 6.\n2. Preliminaries\nIn this section, we quickly introduce the necessary deep-inference notions. A more\nextensive treatment of much of this material is in Br ̈unnler’s [Br ̈u04].\nWe only need the following, basic proof complexity notions (see [CR79]).\nDefinition 2.1. A (propositional) proof system is a binary relation S between formulae\nα and proofs Π such that S is computable in polynomial time, and the formula α is a\ntautology if and only if there is a proof Π such that S(α, Π); in this case we say that Π is\na proof of α in S. We say that proof system S p-simulates proof system S′ if there is a\npolynomial-time computable algorithm that transforms every proof in S′ into a proof in S\nof the same tautology. Two proof systems are p-equivalent if each p-simulates the other.\nRemark 2.2. In the following, we state theorems on the existence of proofs in one proof\nsystem when proofs exist in another proof system, such that their size is polynomially\nrelated. Implicitly, we always mean that the new proofs are obtained by transforming the\nold ones by way of a polynomial-time computable algorithm.\nDeep inference is a relatively recent development in proof theory. Its main idea is\nto provide a finer analysis of inference than possible with traditional methods, and one\nof the main objectives is to obtain a geometric semantics for proofs, inspired by linear\nlogic’s proof nets [Gir87]. Another objective is to provide a uniform and useful syntactic\ntreatment of several logics, especially modal ones, for which no satisfactory proof theory\nexisted before.\nIn deep inference, several formalisms can be defined with excellent structural proper-\nties, like locality for all the inference rules. The calculus of structures [Gug07b] is one\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n3\nof them and is now well developed for classical [Br ̈u03a, Br ̈u06a, Br ̈u06d, BT01], intu-\nitionistic [Tiu06a], linear [Str02, Str03b], modal [Br ̈u06c, GT07, Sto07] and commuta-\ntive/noncommutative logics [Gug07b, Tiu06b, Str03a, Bru02, DG04, GS01, GS02, GS09,\nKah06b, Kah07]; for all these logics, quantification can be defined at any order. We em-\nphasise that deep inference is developing the first reasonable proof theory for modal logics;\nthe large number of different modal logic systems can be studied in simple and modular\ndeep-inference systems, which are similar to their propositional logic counterparts and en-\njoy the same locality properties. The calculus of structures promoted the discovery of a\nnew class of proof nets for classical and linear logic [LS05a, LS05b, LS06, SL04] (see\nalso [Gui06]). Moreover, there exist implementations in Maude of deep-inference proof\nsystems [Kah08].\nIn this article, we focus on the calculus of structures because it is well developed and is\nprobably the simplest formalism definable in deep inference. The complexity results that\nwe present here are not dependent on the choice of formalism; rather, they only depend on\nthe deep-inference methodology and the finer granularity of inference rules that it yields.\nAdopting deep inference basically means that it is possible to replace subformulae inside\nformulae by other, implied subformulae, and that there is no limit to the nesting depth of\nsubformulae. Formalisms like Gentzen’s sequent calculus differ because they only rewrite\nformulae, or sequents, around their root connectives, and (we argue) they suffer excessive\nrigidity in the syntax and they do not sufficiently support geometric semantics.\nBecause of its geometric nature, it is important, in deep inference, to control whether\npropositional variables can be instantiated by formulae. In particular, normalisation (cut\nelimination) in deep inference crucially depends on the availability of ‘atomic’ inference\nrules, which are rules related to some topological invariants (see, for example, [GG08]\nfor normalisation in propositional logic). In practice, we need two kinds of propositional\nvariables: the atoms, only subject to renaming, and the formula variables, subject to (unre-\nstricted) substitution. This distinction does not bear dramatic effects on proof complexity,\nbut it does allow for some finer measures than otherwise possible.\nDefinition 2.3. Formulae of the calculus of structures, or CoS, are denoted by α, β, γ, δ\nand are freely built from: units, like f (false) and t (true); atoms a, b, c, d and ̄a, ̄b, ̄c, ̄d;\n(formula) variables A, B, C, D and ̄A, ̄B, ̄C, ̄D; logical relations, like disjunction [α ∨β]\nand conjunction (α ∧β). A formula is ground if it contains no variables. We usually omit\nexternal brackets of formulae, and sometimes we omit dispensable brackets under associa-\ntivity. We use ≡to denote literal equality of formulae. The size |α| of a formula α is the\nnumber of unit, atom, and variable occurrences appearing in it. On the set of atoms, there\nis an involution ̄·, called negation (i.e., ̄· is a bijection from the set of atoms to itself such\nthat ̄ ̄a ≡a); we require that ̄a . a for every a; when both a and ̄a appear in a formula,\nwe mean that atom a is mapped to ̄a by ̄·. An analogous involution is defined on the set of\nformula variables. The (De Morgan) dual of a formula is obtained by exchanging disjunc-\ntion and conjunction and applying negation to all atoms and variables; we denote duals by\nusing ̄·; for example, the De Morgan dual of α ≡t ∨(a ∧[ ̄B ∨c]) is ̄α ≡f ∧[ ̄a ∨(B ∧ ̄c)]. A\ncontext is a formula where one hole { } appears in the place of a subformula; for example,\nA ∨(b ∧{ }) is a context; the generic context is denoted by ξ{ }. The hole can be filled with\nformulae; for example, if ξ{ } ≡b ∧[{ } ∨c], then ξ{a} ≡b ∧[a ∨c], ξ{b} ≡b ∧[b ∨c] and\nξ{a ∧B} ≡b ∧[(a ∧B) ∨c]. The size of ξ{ } is defined as |ξ{ }| = |ξ{a}| −1.\nRemark 2.4. We do not say that a is positive and ̄a is negative. It only matters that,\nwhen a and ̄a appear in the same formula, if one is negative the other is positive. In\nabsence of disambiguating information, there are two ways in which ξ{b} might correspond\nto b ∧[b ∨c]: one such that ξ{a} ≡a ∧[b ∨c] and another such that ξ{a} ≡b ∧[a ∨c].\nThe language of formulae is redundant because we can choose whether to use atoms\nor formula variables whenever a propositional variable is needed. The distinction between\n\n4\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\natoms and formula variables only plays a role in the choice of applicable inference rules,\nand this aspect is controlled by renamings and substitutions.\nDefinition 2.5. A renaming is a map from the set of atoms to itself, and is denoted by\n{a1/b1, a2/b2, . . .}; we use ρ for renamings. A renaming of α by ρ = {a1/b1, a2/b2, . . .}\nis indicated by αρ and is obtained by simultaneously substituting every occurrence of ai\nin α by bi and every occurrence of ̄ai by ̄bi; for example, if α ≡a ∧[b ∨(a ∧[ ̄a ∨C])]\nthen α{a/ ̄b, ̄b/c} ≡ ̄b ∧[ ̄c ∨( ̄b ∧[b ∨C])]. A substitution is a map from the set of for-\nmula variables to formulae, denoted by {A1/β1, A2/β2, . . .}; we use σ for substitutions.\nAn instance of α by σ = {A1/β1, A2/β2, . . . } is indicated by ασ and is obtained by si-\nmultaneously substituting every occurrence of variable Ai in α by formula βi and every\noccurrence of ̄Ai by the De Morgan dual of βi; for example, if α ≡(A ∧[A ∨ ̄A]) ∨b then\nα{A/(c ∧ ̄B)} ≡((c ∧ ̄B) ∧[(c ∧ ̄B) ∨[ ̄c ∨B]]) ∨b.\nDefinition 2.6. A CoS (inference) rule ν is an expression\nα\nν β , where formulae α and β\nare called premiss and conclusion, respectively. A (rule) instance\nγ\nν δ of\nα\nν β is such that\nγ ≡αρσ and δ ≡βρσ, for some renaming ρ and substitution σ. For some context ξ{ }, a\nCoS (inference) step, generated by rule\nα\nν β via its instance\nγ\nν δ , is the expression\nξ{γ}\nν ξ{δ}.\nExample 2.7. Given\nt\ni↓A ∨ ̄A and\nt\nai↓a ∨ ̄a, then\nt\ni↓(A ∧b) ∨[ ̄A ∨ ̄b] is an instance of i↓and\nt\nν b ∨ ̄b is an instance of both i↓and ai↓. The rule\nA ∨A\nc↓\nA\ngenerates the inference step\n[a ∨b] ∧[(c ∧D) ∨(c ∧D)]\nc↓\n[a ∨b] ∧(c ∧D)\n.\nWe usually classify deep-inference rules in three classes. For this, we rely on the notion\nof linearity, which, in this context, essentially means the same as in term-rewriting: a\nrewriting rule is linear if variables appear once in both sides of the rule. In other words, a\nlinear rule does not create or destroy anything. These are the three classes of rules:\n(1) Atomic rules. They usually correspond to structural rules in Gentzen systems; in\nnormalisation and in semantics of proofs, they play a crucial role because they\nexpress the causality relations between atoms, so shaping the geometry of proofs.\nTheir instances are obtained by renaming.\n(2) Noninvertible linear rules. They usually correspond to logical rules in Gentzen\nsystems. Since they are noninvertible, they express proper inference choices, but\nsince they are linear, they do not alter the geometry of causality between atoms.\nTheir instances are obtained by substitution.\n(3) Invertible linear rules. These rules are equivalences between formulae that do not\ncorrespond to proper inference choices and have no impact on the geometry of\nproofs. For this reason, they are usually gathered into one big equivalence relation\nbetween formulae, corresponding to just one rule, defined via substitution.\nThe success of deep inference is due to its ability to separate rules into classes 1 and\n2, which is only possible by adopting deep inference. The references to the ‘geometry of\nproofs’ can be understood by reading [GG08, LS05b]. Class 3 allows us to greatly simplify\nproofs and to hide, so to speak, a great deal of logical complexity (in the sense of size of\nproofs). We start by defining our ‘class 3’ rule, the others being dependent on specific\nproof systems.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n5\nCommutativity\nα ∨β = β ∨α\nα ∧β = β ∧α\nAssociativity\n[α ∨β] ∨γ = α ∨[β ∨γ]\n(α ∧β) ∧γ = α ∧(β ∧γ)\nUnits\nα ∨f = α\nα ∧t = α\nt ∨t = t\nf ∧f = f\nContext closure\nif α = β then ξ{α} = ξ{β}\nFigure 1. Equality = on formulae.\nDefinition 2.8. The equality relation = on formulae is defined by closing the equations in\nFigure 1 by reflexivity, symmetry, transitivity and by applying context closure. We define\nthe inference rule = as\nα\n= β , where α = β.\nThe following remark helps in assessing how much complexity is hidden in =.\nRemark 2.9. It is possible to decide α = β in polynomial time by reducing α and β to\nsome canonical form and comparing the canonical forms. A canonical form under = of\nany given formula can be obtained, for example, by removing as many units as possible\nand ordering units, atoms, and variables according to an arbitrary order; the canonical\nform is normal for associativity and units equations, when these are orientated from left\nto right. Let us assume a total order on the set of units, atoms, and variables; we now\nsee in detail how to find an equivalent canonical formula in the case of a formula only\ncontaining one logical relation. On the formula, use commutativity until the minimal unit,\natom, or variable appears in the leftmost position, then use associativity, orientated from\nleft to right, until normality is reached. For example, on [b ∨d] ∨[c ∨a], we perform the\nsteps\n[b ∨d] ∨[c ∨a] ❀[c ∨a] ∨[b ∨d] ❀[a ∨c] ∨[b ∨d] ❀a ∨[c ∨[b ∨d]]\n.\nThis phase requires O(n) steps, where n is the size of the formula. We proceed the same\nway on the subformula immediately following the first element, and so on recursively; for\nexample,\na ∨[c ∨[b ∨d]] ❀a ∨[[b ∨d] ∨c] ❀a ∨[b ∨[d ∨c]] ❀a ∨[b ∨[c ∨d]]\n.\nThe number of steps of the algorithm for a formula only containing one logical relation is\nthen O(n2). On a generic formula, the same algorithm can be used, with the same number-\nof-steps complexity O(n2) on the size n of the given formula, by adopting the lexicographic\norder induced by the given total order. This is an example, also involving an initial O(n)\nphase of simplification of units:\n(t ∧t) ∧[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f]\n❀t ∧[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f]\n❀[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f] ∧t\n❀[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f\n❀a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])\n❀⋆a ∨([a ∨[b ∨[c ∨d]]] ∧[[B ∨c] ∨[t ∨a]])\n❀⋆a ∨([a ∨[b ∨[c ∨d]]] ∧[t ∨[a ∨[c ∨B]]])\n❀a ∨([t ∨[a ∨[c ∨B]]] ∧[a ∨[b ∨[c ∨d]]])\n\n6\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\n❀([t ∨[a ∨[c ∨B]]] ∧[a ∨[b ∨[c ∨d]]]) ∨a\n.\nThis way we obtain a (unique, of course) canonical formula in O(n2) steps, given any\nformula of size n, so we can decide the equivalence of two formulae α and β in O(n2)\nsteps, where n = |α| + |β|. Notice that at each step the size of the formula stays the same or\ndiminishes.\nDefinition 2.10. A CoS (proof) system is a finite set of inference rules. A CoS derivation\nΦ of length k in proof system S, whose premiss is α0 and conclusion is αk, is a chain of\ninference steps\nΦ =\nα0\nν1 α1\nν2 ...\nνk−1 αk−1\nνk αk\n,\nsuch that ν1, . . ., νk is a sequence of inference rules that alternate between the = rule and\nany rule of system S, where k ≥0. The same derivation can be indicated by\nα0\nΦ ∥\n∥S\nαk\n, when\nthe details are known or irrelevant; a proof is a derivation whose premiss is t. A derivation\nis ground if it contains no variables. Sometimes, we omit to indicate the inference steps\ngenerated by =. The size |Φ| of derivation Φ is the number of unit, atom, and variable\noccurrences appearing in it. We denote by ξ{Φ} the result of including every formula of\nΦ into the context ξ{ }. We denote by Φρ and Φσ the expression obtained from Φ by\napplying renaming ρ and substitution σ to every formula in Φ. A CoS proof system that,\nfor every valid implication α →β, contains a derivation with premiss α and conclusion β, is\nsaid to be implicationally complete.\nRemark 2.11. If Φ is a derivation, then ξ{Φ}, Φρ, and Φσ are derivations, for every context\nξ{ }, renaming ρ, and substitution σ.\nWe use the notion of groundness to relate the complexity of deep-inference proof sys-\ntems with atomic rules to proof systems without atomic rules, including those outside of\ndeep inference. Due to the aforementioned redundancy in the language, groundness is not\nreally a restriction.\nRemark 2.12. Every nonground derivation can be transformed into an equivalent,\nground one, by replacing variables with atoms in such a way that newly introduced atoms\nare different from the already present one.\nWe can now define some deep-inference proof systems. System SKS is the most impor-\ntant for the proof theory of classical logic, because of its atomic structural rules. System\nSKSg relates SKS to proof systems in other formalisms, like Frege.\nDefinition 2.13. CoS proof systems KSg = {i↓, w↓, c↓, s}, SKSg = KSg ∪{i↑, w↑,\nc↑}, KS = {ai↓, aw↓, ac↓, s, m} and SKS = KS ∪{ai↑, aw↑, ac↑} are defined in Figures 2\nand 3, for a language containing f, t, disjunction, and conjunction. Proof systems where\nnone of the rules i↑, ai↑, w↑, and aw↑appear are said to be analytic.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n7\nStructural rules\nLogical rule\nSKSg\n \nA ∧ ̄A\ni↑\nf\nA\nw↑t\nA\nc↑A ∧A\ncointeraction\ncoweakening\ncocontraction\nor cut\nt\ni↓A ∨ ̄A\nf\nw↓A\nA ∨A\nc↓\nA\nA ∧[B ∨C]\ns (A ∧B) ∨C\n \nKSg\ninteraction\nweakening\ncontraction\nswitch\nor identity\nFigure 2. Systems SKSg and KSg.\nAtomic structural rules\nLogical rules\nSKS\n \na ∧ ̄a\nai↑\nf\na\naw↑t\na\nac↑a ∧a\ncointeraction\ncoweakening\ncocontraction\nor cut\nt\nai↓a ∨ ̄a\nf\naw↓a\na ∨a\nac↓\na\nA ∧[B ∨C]\ns (A ∧B) ∨C\n(A ∧B) ∨(C ∧D)\nm [A ∨C] ∧[B ∨D]\n \nKS\ninteraction\nweakening\ncontraction\nswitch\nmedial\nor identity\nFigure 3. Systems SKS and KS.\nExample 2.14. This is a valid derivation in all CoS proof systems defined previously (and\nit plays a role in the proof of Lemma 3.11):\nγ ∨[(([ ̄α ∨α] ∧c) ∧(α ∧d)) ∨δ]\n= γ ∨[(((α ∧d) ∧c) ∧[α ∨ ̄α]) ∨δ]\ns γ ∨[[(((α ∧d) ∧c) ∧α) ∨ ̄α] ∨δ]\n= [ ̄α ∨γ] ∨[((α ∧c) ∧(α ∧d)) ∨δ]\n.\nNote that SKSg, KSg, SKS, and KS are closed under renaming and substitution (see\nRemark 2.11). This is so because of the distinction between atoms and formula variables.\nObtaining the closure of these and other systems under renaming and substitution is one of\nthe main technical reasons for distinguishing between atoms and variables.\nThe following theorem is proved in [Br ̈u04], and follows immediately from Section 3.1,\nwhere we prove that CoS systems p-simulate Gentzen systems.\nTheorem 2.15. (Br ̈unnler)\nSystems SKSg, KSg, SKS, and KS are complete; systems\nSKSg and SKS are implicationally complete.\nThe theorem holds also when restricting the language to ground derivations, since sys-\ntems SKS and KS apply to them.\nIn the presence of cut, the coweakening and cocontraction rules do not play a major role\nin terms of proof complexity:\nTheorem 2.16. Systems SKSg and KSg∪i↑are p-equivalent, and systems SKS and KS∪\nai↑are p-equivalent.\n\n8\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nProof. Observe that the rules w↑and c↑can be derived in KSg ∪i↑:\nA\n= A ∧[f ∨t]\ns (A ∧f) ∨t\nw↓(A ∧ ̄A) ∨t\ni↑\nf ∨t\n=\nt\nand\nA\n= A ∧t\ni↓A ∧[[ ̄A ∨ ̄A] ∨(A ∧A)]\ns (A ∧[ ̄A ∨ ̄A]) ∨(A ∧A)\nc↓\n(A ∧ ̄A) ∨(A ∧A)\ni↑\nf ∨(A ∧A)\n=\nA ∧A\n.\nSimilar constructions hold in KS ∪i↑for aw↑and ac↑.\n□\nIt turns out that all the systems mentioned in the previous theorem are p-equivalent, as\na consequence of Corollary 2.23.\nAnalytic systems are formally defined in Definition 2.13 for CoS, and 3.2 for Gentzen.\nThose definitions are specific to different systems in different formalisms, which is not\nnecessarily satisfactory. Defining a general, syntax-independent concept of analyticity is a\nsubject of ongoing research (see Problem 6.4). We briefly discuss now the notion of ana-\nlyticity and its connections with the proof complexity of deep inference, as an introduction\nto our result on Statman tautologies.\nA Gentzen system is said to be analytic when it does not contain the cut rule. Analytic\nGentzen systems enjoy the ‘subformula property’, i.e., proofs in these systems only contain\nsubformulae of their conclusions. In fact, we might stipulate that enjoying the subformula\nproperty is a primitive notion of analyticity, which we can use to exclude the cut rule, as\ndesired. In analytic Gentzen proofs, all formulae have lower or equal complexity than\nthat of the conclusion, when complexity is measured, for example, as the and/or depth of a\nformula (i.e., the number of alternations of conjunction and disjunction; see Definition 6.3).\nThere is another property, of interest to us, that analytic Gentzen systems enjoy: given an\ninference rule and its conclusion, there are only finitely many premisses to choose from;\nwe call such rules ‘finitely generating’. The cut rule in Gentzen does not possess the\nsubformula property nor is it finitely generating.\nThe primitive notion of analyticity that we are currently adopting for CoS is different\nfrom the one for Gentzen. We stipulate that a rule is analytic if its premiss is a formula\nobtained from a formula scheme by instantiating it with subformulae of the conclusion\n(so, the premiss is not just a subformula of the conclusion). This means that no atom or\nvariable can appear in the premiss of an analytic rule that does not appear in its conclusion.\nIt is, of course, a weaker condition than asking for the subformula property of Gentzen\nsystems, but doing so is necessary if we want to adopt deep inference and obtain linear\nrules. Like the subformula property does for Gentzen, this weaker notion for CoS excludes\nthe cut rule, but also the coweakening one. However, this is not an important difference\nwith the sequent calculus because coweakening is irrelevant for the proof complexity of\na CoS system (see, for example [GG08]). The reason for dealing with coweakening is\nthat, given the potential importance of cocontraction for proof complexity, we preferred to\nintroduce top-down-symmetric CoS systems (so, closed by duality), even if coweakening\nand cocontraction are not required for completeness.\nSo, the two notions of analyticity, for Gentzen and for CoS, are such that the only\nimportant rules that are not analytic are the respective cut rules. Note that in both cases,\nanalytic systems are made of finitely generating rules. However, there is an important\ndifference: in CoS, the complexity of formulae in an analytic proof can be unboundedly\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n9\ngreater than the complexity of the conclusion. Consider, for example, the derivation\nc ∨(a ∧[b ∨[c ∨(a ∧b)])\ns c ∨[(a ∧b) ∨[c ∨(a ∧b)]]\n= [c ∨(a ∧b)] ∨[c ∨(a ∧b)]\nc↓\nc ∨(a ∧b)\n.\nThe and/or depth of the conclusion is 1, while that of the premiss is 3. We could repeat the\nconstruction on top of itself and further increase the and/or depth of the premiss at will.\nDeep-inference systems can be top-down symmetric in the sense that a derivation can\nbe flipped upside-down and negated and still be a valid derivation (we say that two such\nderivations are dual). Accordingly, some forms of analyticity can be defined in a symmetric\nway. Then, typically asymmetric theorems that depend on the notion of analyticity, like\ncut elimination, can be generalised to symmetric statements that imply cut elimination.\nThis is not the place to be detailed about this aspect; suffice to say that we can obtain for\nCoS systems much stronger normalisation (and cut elimination) results than for Gentzen\nsystems (see [Br ̈u06b, GG08]).\nAs we see in Section 3.1, analyticity in CoS faithfully captures analyticity in Gentzen,\nin the sense that analytic CoS can produce isomorphic proofs to Gentzen ones (almost\namounting to a change of notation). However, analytic CoS admits more proofs than an-\nalytic Gentzen, and, among CoS proofs, we can find some remarkably small ones, which\nanalytic Gentzen cannot express; this is the subject of Section 3.2 on Statman tautologies.\nRemark 2.17. The rules of SKS are local, in the sense that, for any language with a\nfinite number of atoms, checking that a given expression is an instance of any of these\nrules requires time bounded by a constant (adopting a tree representation of formulae,\nfor example). This property is peculiar to deep inference; it cannot be obtained in other\nformalisms. For example, a traditional, nonatomic contraction rule is not local because\nit requires checking the identity of two unbounded formulae. Contrary to other nonlocal\nrules, like identity in a Gentzen system, contraction cannot be replaced by its local, atomic\ncounterpart without losing completeness. A counterexample showing this is in [Br ̈u03b].\nLocality can possibly lead to a new, general, productive notion of analyticity, as argued in\nProblem 6.4.\nWe conclude the section by showing the p-equivalence of systems with atomic rules to\nsystems without atomic rules. We start by proving the result on ground derivations.\nLemma 2.18. For every ground instance\nt\ni↓α ∨ ̄α there is a derivation\nt\nΦ ∥\n∥{ai↓,s}\nα ∨ ̄α\nand for\nevery ground instance\nα ∧ ̄α\ni↑\nf\nthere is a derivation\nα ∧ ̄α\nΦ ∥\n∥{ai↑,s}\nf\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. Let us see the case for i↑, the other being its dual. We make an induction on the\nstructure of α. The cases when α is a unit or an atom are trivial: in the former case Φ\nconsists of an instance of = and in the latter the instance of i↑is also an instance of ai↑. We\nonly have to consider the case when α ≡β ∨γ: we apply the induction hypothesis on the\nderivation\n[β ∨γ] ∧( ̄β ∧ ̄γ)\n= ( ̄β ∧[β ∨γ]) ∧ ̄γ\ns [( ̄β ∧β) ∨γ] ∧ ̄γ\ni↑\n[f ∨γ] ∧ ̄γ\n=\nγ ∧ ̄γ\ni↑\nf\n,\n\n10\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nand we obtain a derivation whose length is O(n), and so its size is O(n2), where n = |α|.\n□\nLemma 2.19. For every ground instance\nf\nw↓α there is a derivation\nf\nΦ ∥\n∥{aw↓,s}\nα\nand for\nevery ground instance\nα\nw↑t there is a derivation\nα\nΦ ∥\n∥{aw↑,s}\nt\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. The proof is similar to the one for Lemma 2.18. In case α ≡t an instance of w↓\nyields\nf\n= f ∧[f ∨t]\ns (f ∧f) ∨t\n=\nt\n;\nwe can do similarly if α ≡f in an instance of w↑. These are the derivations for the inductive\ncases about w↓(those about w↑are dual):\nf\nw↓γ\n= f ∨γ\nw↓β ∨γ\nand\nf\n= f ∧f\nw↓f ∧γ\nw↓β ∧γ\n.\nFrom these we obtain a derivation whose length is O(n), and so its size is O(n2), where\nn = |α|.\n□\nRemark 2.20. In the statement of Lemma 2.19, instead of {aw↓, s} and {aw↑, s} we could\nhave used {aw↓, m} and {aw↑, m}, respectively.\nLemma 2.21. For every ground instance\nα ∨α\nc↓\nα\nthere is a derivation\nα ∨α\nΦ ∥\n∥{ac↓,m}\nα\nand for\nevery ground instance\nα\nc↑α ∧α there is a derivation\nα\nΦ ∥\n∥{ac↑,m}\nα ∧α\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. The proof is similar to the one for Lemma 2.18. These are the derivations for the\ninductive cases about c↓(those about c↑are dual):\n[β ∨γ] ∨[β ∨γ]\n= [β ∨β] ∨[γ ∨γ]\nc↓\nβ ∨[γ ∨γ]\nc↓\nβ ∨γ\nand\n(β ∧γ) ∨(β ∧γ)\nm [β ∨β] ∧[γ ∨γ]\nc↓\nβ ∧[γ ∨γ]\nc↓\nβ ∧γ\n.\nFrom these we obtain a derivation whose length is O(n), and so its size is O(n2), where\nn = |α|.\n□\nTheorem 2.22. For every ground SKSg derivation Φ there is a ground SKS derivation\nΦ′ with the same premiss and conclusion of Φ; if n is the size of Φ then the size of Φ′ is\nO(n2); moreover, if Φ is in KSg then Φ′ is in KS.\nProof. The theorem follows immediately from Lemmas 2.18, 2.19, and 2.21.\n□\nBy Remark 2.12, every derivation can be ‘grounded’, so:\nCorollary 2.23. KS and KSg are p-equivalent and SKS and SKSg are p-equivalent.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n11\nφ, A\n ̄A, ψ\ncut\nφ, ψ\ncut\nid A, ̄A\nt t\nφ\nw φ, A\nφ, A, A\nc\nφ, A\nφ, A, B\n∨\nφ, A ∨B\nφ, A\nB, ψ\n∧\nφ, A ∧B, ψ\nidentity\ntrue\nweakening\ncontraction\ndisjunction\nconjunction\nFigure 4. System Gentzen.\nRemark 2.24. Sometimes, we use nonatomic structural rule instances in SKS and KS\nderivations: those instances actually stand for the SKS and KS derivations that would\nbe obtained according to the proofs of Lemmas 2.18, 2.19, and 2.21. In this sense, we\nsay that i↓, i↑, w↓, w↑, c↓, and c↑are ‘macro’ rules for SKS and KS. The reason we\nmight want to work with macro rules in SKS and KS instead of working in SKSg and\nKSg and then appealing to Theorem 2.22 is to obtain finer upper bounds. This is because\nthe size of formulae over which nonatomic structural rules operate can be much smaller\nthan the square root of the size of a derivation, which is the pessimistic assumption of\nTheorem 2.22.\nRemark 2.25. All implicationally complete CoS proof systems are p-equivalent. This can\nbe proved analogously to, or resorting to, a similar ‘robustness’ result for Frege systems\n(Theorem 4.2), as argued in Remark 4.13. This means that studying proof complexity for\nSKSg and SKS has universal value for all CoS systems for propositional logic.\n3. Calculus of Structures, Gentzen Proof Systems and Statman Tautologies\nThere are two parts in this section. In the first part, we show how CoS naturally p-\nsimulates Gentzen systems, and in particular how it realizes Gentzen’s notion of analyticity.\nIn the second part, we show that analytic CoS admits polynomial proofs when analytic\nGentzen only has exponential ones, in the case of Statman tautologies.\n3.1. Calculus of Structures and Gentzen Proof Systems. In this section, we adopt a\nspecific one-sided (Gentzen-Sch ̈utte) sequent system that we call Gentzen (and that is\ncalled GS1p in [TS96]). We could have adopted any other style of presentation without\naffecting our results. In fact, for Gentzen systems an analogous ‘robustness’ theorem to\nthat for Frege systems (Theorem 4.2) can be established. This means that studying the\nproof complexity of Gentzen has universal value for the class of Gentzen systems.\nDefinition 3.1. Over the language of SKS formulae, the sequent-calculus proof system\nGentzen is defined by the inference rules in Figure 4, where φ and ψ stand for multisets\nof formulae and the symbol ‘,’ represents multiset union. We interpret multisets of for-\nmulae as their disjunction (where associativity is irrelevant). Derivations, denoted by ∆,\nare trees obtained by composing instances of inference rules; the leaves of a derivation are\nits premisses and the root is its conclusion; a derivation ∆with premisses φ1, ..., φh and\nconclusion ψ is denoted by\nφ1 . . . φh\n∆\nψ\n.\nA derivation with no premisses is a proof. The size |∆| of derivation ∆is the number of\nunit, atom, and variable occurrences appearing in it. In the following, every SKS formula\nis translated into a Gentzen formula in the obvious way, and vice versa; in particular, we\ntranslate a Gentzen multiset φ = α1, . . ., αh into α1 ∨· · · ∨αh.\n\n12\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nIn the language, we keep the distinction between atoms and variables because, thanks\nto atoms, we obtain a better upper bound for the size of Statman tautologies proofs, in\nthe Section 3.2. As we said in the case of CoS, the redundancy in the language has no\nconsequences outside of the possibility of using certain CoS rules instead of others.\nDefinition 3.2. The proof system analytic Gentzen is proof system Gentzen without the\ncut rule; analytic derivations and proofs are those derivations and proofs in Gentzen where\nno instances of the cut rule appear.\nWe know, of course, that both Gentzen and analytic Gentzen are complete, and that\nGentzen proofs can be transformed into analytic Gentzen proofs by a cut-elimination\nprocedure, which, in general, blows-up a given proof exponentially.\nEvery Gentzen derivation has natural counterparts in CoS: the idea is to (arbitrarily)\nsequentialise its tree structure. This is possible because the natural logical relation between\ntree branches is conjunction, which CoS can represent, of course. In doing so, we pay\nin terms of complexity because the tree structure is less redundant than CoS contexts:\nthe size of derivations grows quadratically. Other deep-inference formalisms (currently\nunder development, see [BL05, Gug04, Gug05]) are more efficient than CoS and Gentzen\nformalisms in dealing with this so-called ‘bureaucracy’.\nRemark 3.3. In the following, we assume that an empty conjunction can be represented\nby a nonempty conjunction of t units.\nTheorem 3.4. For every Gentzen derivation ∆with premisses φ1, ..., φh and conclusion\nψ there is a derivation\nφ1 ∧· · · ∧φh\nΦ ∥\n∥SKSg\nψ\n; if n is the size of ∆, the size of Φ is O(n2); moreover,\nif ∆is analytic then Φ is in KSg.\nProof. We proceed by induction on the tree structure of ∆. The base cases id A, ̄A and t t\nare, respectively, translated into\nt\ni↓A, ̄A and t. The derivations\nφ1 . . . φh\n∆1\nφ\nw φ, A\n,\nφ1 . . . φh\n∆1\nφ, A, A\nc\nφ, A\n,\nand\nφ1 . . . φh\n∆1\nφ, A, B\n∨\nφ, A ∨B\nare, respectively, translated into\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ\n= φ ∨f\nw↓φ ∨A\n,\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ ∨[A ∨A]\nc↓\nφ ∨A\n,\nand\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ ∨[A ∨B]\n,\nwhere Φ1 is obtained by induction from ∆1, and some possibly necessary instances of the\n= rule have been omitted (they depend on the exact translation of Gentzen multisets into\nSKSg formulae). The derivations\nφ1 . . . φh\n∆1\nφ, A\nφh+1 . . . φk\n∆2\nB, ψ\n∧\nφ, A ∧B, ψ\nand\nφ1 . . . φh\n∆1\nφ, A\nφh+1 . . . φk\n∆2\n ̄A, ψ\ncut\nφ, ψ\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n13\nare, respectively, translated into\nφ1 ∧· · · ∧φk\nΦ1∧Φ2 ∥\n∥\n[φ ∨A] ∧[B ∨ψ]\n= [B ∨ψ] ∧[A ∨φ]\ns ([B ∨ψ] ∧A) ∨φ\n= φ ∨(A ∧[B ∨ψ])\ns φ ∨[(A ∧B) ∨ψ]\nand\nφ1 ∧· · · ∧φk\nΦ1∧Φ2 ∥\n∥\n[φ ∨A] ∧[ ̄A ∨ψ]\n= [ ̄A ∨ψ] ∧[A ∨φ]\ns ([ ̄A ∨ψ] ∧A) ∨φ\n= φ ∨(A ∧[ ̄A ∨ψ])\ns φ ∨[(A ∧ ̄A) ∨ψ]\ni↑\nφ ∨[f ∨ψ]\n=\nφ ∨ψ\n,\nwhere Φ1 and Φ2 are obtained by induction from ∆1 and ∆2, some possibly necessary\ninstances of the = rule have been omitted, and Φ1 ∧Φ2 stands for the derivation\nφ1 ∧· · · ∧φk\nΦ1∧(φh+1∧···∧φk) ∥\n∥\n[φ ∨A] ∧(φh+1 ∧· · · ∧φk)\n[φ∨A]∧Φ2 ∥\n∥\n[φ ∨A] ∧[B ∨ψ]\n,\nwhere B is possibly instantiated by ̄A; the length of this derivation and the size of the largest\nformula appearing in it are both O(n). The resulting O(n2) measure of these last two cases\ndominates the others.\n□\nCorollary 3.5. SKSg p-simulates Gentzen and KSg p-simulates analytic Gentzen.\nAlthough it does not explicitly address complexity, [Br ̈u04] is more exhaustive than the\naforesaid on the two-way translation between SKSg and Gentzen. Translating from SKSg\nto Gentzen crucially employs the cut rule: for every inference step in SKSg, a cut instance\nis used in Gentzen. So, while it is very natural and easy to show that Gentzen p-simulates\nSKSg (see [Br ̈u04]), we are left with the question: does analytic Gentzen p-simulate KSg?\n3.2. Analytic Calculus of Structures on Statman Tautologies. We prove here that an-\nalytic Gentzen does not p-simulate KSg. In fact, the CoS (polynomial) inefficiency in\ndealing with context bureaucracy is compensated by its freedom in applying inference\nrules, which leads to exponential speedups on certain classes of tautologies. Here, we\nstudy Statman tautologies, which have been used to provide the classic lower bound for\nanalytic Gentzen systems: no cut-free proofs of Statman tautologies are possible in an-\nalytic Gentzen without the proofs growing exponentially over the size of the tautologies\nthey prove [Sta78]. We show that, on the contrary, KSg and KS prove Statman tautologies\nwith polynomially growing analytic proofs.\nRemark 3.6. The subset of SKSg only containing analytic rules is equal to KSg plus\nthe cocontraction rule. We do not know whether cocontraction provides for exponential\nspeedups, so separating, proof-complexity-wise, the class of KSg from that of ‘analytic\nCoS’; about this, see Problem 6.2. In our opinion, the very notion of analyticity would\nbenefit from some further analysis; about this, see Problem 6.4.\nDefinition 3.7. For n ≥1, consider the following formulae:\nαi ≡ ̄ci ∨ ̄di\nfor i ≥1,\nβn\nk ≡Vk\ni=n αi ≡αn ∧βn−1\nk\nfor n ≥k > 1,\nγn\nk ≡βn\nk+1 ∧ck\nfor n > k ≥1,\nδn\nk ≡βn\nk+1 ∧dk\nfor n > k ≥1.\n\n14\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nStatman tautologies are, for n ≥1, the formulae:\nSn ≡ ̄αn ∨[(γn\nn−1 ∧δn\nn−1) ∨[· · · ∨[(γn\n1 ∧δn\n1) ∨α1] · · ·]]\n.\nExample 3.8. These are the first three Statman tautologies:\nS1 ≡(c1 ∧d1) ∨[ ̄c1 ∨ ̄d1]\n,\nS2 ≡(c2 ∧d2) ∨[(([ ̄c2 ∨ ̄d2] ∧c1) ∧([ ̄c2 ∨ ̄d2] ∧d1)) ∨[ ̄c1 ∨ ̄d1]]\n,\nS3 ≡(c3 ∧d3) ∨[(([ ̄c3 ∨ ̄d3] ∧c2) ∧([ ̄c3 ∨ ̄d3] ∧d2)) ∨\n[((([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) ∧c1) ∧(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) ∧d1)) ∨\n[ ̄c1 ∨ ̄d1]]]\n.\nIt is perhaps easier to understand their meaning by using implication, as in\nS′\n3 ≡[ ̄c3 ∨ ̄d3] →([([ ̄c3 ∨ ̄d3] → ̄c2) ∨([ ̄c3 ∨ ̄d3] → ̄d2)] →\n([(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) → ̄c1) ∨(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) → ̄d1)] →\n[ ̄c1 ∨ ̄d1]))\n.\nRemark 3.9. |Sn| = 2 + 2 Pn−1\nk=1 |γn\nk| + 2 = 2 Pn\nk=2(|βn\nk| + 1) + 4 = 2n2 + 2.\nIt is not difficult to see why analytic Gentzen proofs of Statman tautologies grow expo-\nnentially (this is a classic argument that can be found in many textbooks; see, for example,\n[CK02]). Basically, all what analytic Gentzen can do while building a proof of Sn is to\ngenerate a proof tree with O(2n) branches. The next lemma shows the crucial advantage\nof deep inference over Gentzen systems: Statman tautologies can be proved ‘from the in-\nside out’, which is precisely what Gentzen systems can only do by resorting to convoluted\nproofs involving cuts (so, nonanalytic proofs).\nRemark 3.10. In the following, for brevity, we label inference steps with expressions like\nn · ν, to denote n inference steps involving rule ν.\nLemma 3.11. For Statman tautologies Sn and Sn+1 there exists a derivation\nSn\n∥\n∥KS\nSn+1\nwhose\nlength is O(n) and size is O(n3).\nProof. We refer to Definition 3.7. The requested derivation is\nΦ =\n(cn ∧dn) ∨\n[((βn\nn ∧cn−1) ∧(βn\nn ∧dn−1)) ∨[· · · ∨\n[((βn\n2 ∧c1) ∧(βn\n2 ∧d1)) ∨α1] · · ·]]\n2n · i↓(([αn+1 ∨ ̄αn+1] ∧cn) ∧([αn+1 ∨ ̄αn+1] ∧dn)) ∨\n[((([αn+1 ∨ ̄αn+1] ∧βn\nn) ∧cn−1) ∧(([αn+1 ∨ ̄αn+1] ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[((([αn+1 ∨ ̄αn+1] ∧βn\n2) ∧c1) ∧(([αn+1 ∨ ̄αn+1] ∧βn\n2) ∧d1)) ∨α1] · · ·]]\n2n · s\n2n\nz }| {\n[ ̄αn+1 ∨[· · · ∨[ ̄αn+1 ∨ ̄αn+1] · · ·]] ∨\n[((αn+1 ∧cn) ∧(αn+1 ∧dn)) ∨\n[(((αn+1 ∧βn\nn) ∧cn−1) ∧((αn+1 ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[(((αn+1 ∧βn\n2) ∧c1) ∧((αn+1 ∧βn\n2) ∧d1)) ∨α1] · · ·]]]\n(2n −1) · c↓ ̄αn+1 ∨\n[((αn+1 ∧cn) ∧(αn+1 ∧dn)) ∨\n[(((αn+1 ∧βn\nn) ∧cn−1) ∧((αn+1 ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[(((αn+1 ∧βn\n2) ∧c1) ∧((αn+1 ∧βn\n2) ∧d1)) ∨α1] · · ·]]]\n,\nwhere we use macro inference rules as explained in Remark 2.22. (Example 2.14 explains\nthe central step in the preceding derivation.) The formulae appearing in the middle of the\nprevious derivation are the largest. Since |αn+1| = 2 and |βn\nk| = 2(n −k + 1), their size is\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n15\n2n·2+6+2(3n−3+Pn\nk=2 |βn\nk|)+2 = 2n2+8n+2. Each i↓macro inference step involves one s\nand two ai↓steps in KS, for a total of six steps, including = ones; each c↓macro inference\nstep involves one m and two ac↓steps in KS, for a total of six steps. So, the length of Φ is\n2n · 6 + 2n · 2 + (2n −1) · 6 = 28n −6, and so |Φ| ≤(28n −6)(2n2 + 8n + 2) ∈O(n3).\n□\nNote that in the previous proof, by working with macro inference rules, we get a better\nupper bound for KS than if we worked in KSg and then applied Theorem 2.22.\nTheorem 3.12. There are KS proofs of Statman tautologies whose size is quadratic in the\nsize of the tautologies they prove.\nProof. Tautology S1 is trivially provable by an instance of the i↓macro rule. By repeatedly\napplying the previous lemma, we obtain proofs of all Statman tautologies Sn, whose size\nis O(n4). Since |Sn| ∈O(n2) (see Remark 3.9), the statement follows.\n□\nThis is enough to conclude that analytic Gentzen does not polynomially simulate KSg\nand KS. Some could argue that Statman tautologies are artificial in their forcing exponen-\ntial Gentzen proofs into ‘wildly’ branching. However, notice that both notions of proof\nand analyticity in Gentzen systems ‘get into tautologies’ from the outside inwards. In\nother words, the restricted notion of analyticity in Gentzen systems is strongly correlated\nto the restricted notion of proof that leads to exponential-size proofs. In CoS, both notions\nare more liberal, to the advantage of proof complexity. We pay a price for this in terms\nof proof-search complexity: there is research aimed at improving the situation, with very\npromising results; see [Kah06a].\nWe note that polynomial proofs on Statman tautologies are obtained by a very small\ndose of deep inference. In fact, the trick is done by the switch and interaction instances in\nthe proof in Lemma 3.11: they all operate just below the ‘surface’ of a formula. This leads\nus to state a currently open problem, in Section 6.6.\n4. Calculus of Structures and Frege Systems\nIn this section, we prove that CoS and Frege systems are p-equivalent.\nIn the following definitions about Frege systems, we do not assume that the language of\nformulae coincides with the CoS one, but, as always, this is not a very important issue.\nDefinition 4.1. Given a language of propositional logic formulae built over a complete\nbase of connectives, a Frege (proof) system is a finite collection of sound inference rules,\neach of which is a tuple of n > 0 formulae such that from n −1 premisses one conclusion\nis derived; inference rules with 0 premisses are called axioms. Given a Frege system, a\nFrege derivation of length l with premisses α1, ..., αh and conclusion βl is a sequence of\nformulae β1, . . ., βl, such that each βi either belongs to {α1, . . ., αh} or is the conclusion of\nan instance of an inference rule whose premisses belong to β1, . . ., βi−1, where 1 ≤i ≤l;\na Frege proof of β is a Frege derivation with no premisses and conclusion β; we use Υ for\nderivations. We require of each Frege system to be implicationally complete, i.e., whenever\n(α1 ∧· · · ∧αh) →β is valid there is a derivation with premisses α1, ..., αh and conclusion\nβ in the proof system. The size of a Frege derivation Υ is the number of unit, atom, and\nvariable occurrences that it contains, and is indicated by |Υ|.\nThe following ‘robustness’ theorem can easily be proved.\nTheorem 4.2. (Robustness, Cook-Reckhow, [CR79])\nAll Frege systems in the same lan-\nguage are p-equivalent.\nThe theorem has been generalised by Reckhow to Frege systems in any language (under\n‘natural translations’) [Rec76], but we do not need this level of generality in our article.\nThe robustness theorem allows us to work with just one Frege system, and we arbitrarily\nchoose the following, taken from [Bus87] and modified by adding axioms F14, F15, F16,\nand F17 in order to deal with units.\n\n16\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nAxioms:\nF1 ≡A →(B →(A ∧B))\nF2 ≡(A ∧B) →A\nF3 ≡(A ∧B) →B\nF4 ≡A →[A ∨B]\nF5 ≡B →[A ∨B]\nF6 ≡¬¬A →A\nF7 ≡A →¬¬A\nF8 ≡A →(B →A)\nF9 ≡¬A →(A →B)\nF10 ≡(A →(B →C)) →((A →B) →(A →C))\nF11 ≡(A →C) →((B →C) →([A ∨B] →C))\nF12 ≡(A →(B →C)) →(B →(A →C))\nF13 ≡(A →B) →(¬B →¬A)\nF14 ≡f →(A ∧¬A)\nF15 ≡(A ∧¬A) →f\nF16 ≡t →[A ∨¬A]\nF17 ≡[A ∨¬A] →t\nInference rule:\nA\nA →B\nmp\nB\nFigure 5. System Frege.\nDefinition 4.3. Frege system Frege, over the language of formulae freely generated by\nunits, non-negated formula variables, and the connectives ∨, ∧, →, and ¬, has inference\nrules as shown in Figure 5, where the formulae F1, ..., F17 are axioms and the inference\nrule mp is called modus ponens.\nRemark 4.4. In the following, every SKS formula is implicitly translated into a Frege\nformula in the obvious way, and vice versa; in particular, we translate Frege’s formulae of\nthe kind α →β into SKS formulae ̄α ∨β.\nRemark 4.5. In Frege systems, distinguishing atoms from formula variables is unnec-\nessary because we only instantiate rules by general substitution (as opposed to renam-\ning). So, from now on, we assume that CoS atoms correspond to Frege formula vari-\nables. Since in system Frege we have a connective for negation, we can also assume that\nwhen dual atoms and formula variables appear in an SKSg formula, their Frege translation\nonly uses ¬; for example, the SKSg formula [A ∨ ̄A] ∧[a ∨ ̄a] is translated into Frege for-\nmula [A ∨¬A] ∧[B ∨¬B] or [A ∨¬A] ∧[¬B ∨B] or [¬A ∨A] ∧[B ∨¬B] or [¬A ∨A] ∧[¬B ∨B].\nConversely, Frege formula [A ∨¬A] ∧[B ∨¬B] is translated into SKSg or SKS formula\n[a ∨ ̄a] ∧[b ∨ ̄b] or [a ∨ ̄a] ∧[ ̄b ∨b] or [ ̄a ∨a] ∧[b ∨ ̄b] or [ ̄a ∨a] ∧[ ̄b ∨b] or one such for-\nmula with formula variables in the place of some of the atoms. As always, we use atoms\nwhen we need to use SKS atomic structural rules, we use formula variables when we need\nto instantiate formulae and derivations, and otherwise we can choose both.\nTranslating Frege into SKSg derivations is straightforward, given that the cut rule of\nSKSg can easily simulate modus ponens.\nTheorem 4.6. For every Frege derivation Υ with premisses α1, ..., αh, where h ≥0, and\nconclusion β, there is a derivation\nα1 ∧· · · ∧αh\nΦ ∥\n∥SKSg\nβ\n; if l and n are, respectively, the length and\nsize of Υ, then the length and size of Φ are, respectively, O(l) and O(n2).\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n17\nProof. The axioms Fi of Frege are tautologies, so each one has a proof Φi in SKSg, for\n1 ≤i ≤17; for example F1 and F10 are, respectively, proved by\nΦ1 =\nt\ni↓[ ̄A ∨ ̄B] ∨(A ∧B)\n= ̄A ∨[ ̄B ∨(A ∧B)]\nand\nΦ10 =\nt\ni↓(A ∧(B ∧ ̄C)) ∨[ ̄A ∨[ ̄B ∨C]]\n= (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧t) ∨C]]\ni↓(A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧[A ∨ ̄A]) ∨C]]\ns (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[[( ̄B ∧A) ∨ ̄A] ∨C]]\n= (A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[[ ̄A ∨ ̄A] ∨C]]\nc↓\n(A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[ ̄A ∨C]]\n.\nWe proceed by induction on the length of Υ = β1, . . ., βk, β and we prove the existence\nof a derivation\nα1 ∧· · · ∧αh\nΦ′ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n. The base case k = 0 is as follows: 1) if β is a premiss,\nthen Φ′ = β; 2) if β ≡Fiσ, for some i and σ, then Φ′ = Φiσ. For the inductive step,\ngiven Υk = β1, . . . , βk and\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk\n, where γk is the conjunction of premisses of Υk, we\nconsider the following cases:\n• if β is a premiss, then Φ′ =\nγk ∧β\nΦ′\nk∧β ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n;\n• if β ≡Fiσ, for some i and substitution σ, then Φ′ =\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk\n= (β1 ∧· · · ∧βk) ∧t\n(β1∧···∧βk)∧Φiσ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n;\n• if β is the conclusion of an instance\nβk′\nβk′ →β\nmp\nβ\n, where βk′′ ≡βk′ →β and\n1 ≤k′, k′′ ≤k, then\nΦ′ =\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk′′ ∧· · · ∧βk\nc↑β1 ∧· · · ∧βk′ ∧· · · ∧(βk′′ ∧βk′′) ∧· · · ∧βk\nc↑β1 ∧· · · ∧(βk′ ∧βk′) ∧· · · ∧(βk′′ ∧βk′′) ∧· · · ∧βk\n=\n(β1 ∧· · · ∧βk) ∧(βk′ ∧[ ̄βk′ ∨β])\ns (β1 ∧· · · ∧βk) ∧[(βk′ ∧ ̄βk′) ∨β]\ni↑\n(β1 ∧· · · ∧βk) ∧[f ∨β]\n=\n(β1 ∧· · · ∧βk) ∧β\n,\nwhere, without loss of generality, we assumed k′ < k′′.\nAt every inductive step the length of the SKSg derivation is only increased by an O(1)\nnumber of inference steps. From\nα1 ∧· · · ∧αh\nΦ′ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\nwe can obtain the desired derivation\nα1 ∧· · · ∧αh\nΦ ∥\n∥SKSg\nβ\nby applying once the w↑rule. So, the length of Φ is O(k). From this, and after\ninspecting the aforesaid derivations, it follows that |Φ| ∈O(k2m), where m is the maximum\nsize of a formula appearing in Υ, and so |Φ| ∈O(n2), where |Υ| = n.\n□\nCorollary 4.7. SKSg and SKS p-simulate Frege.\n\n18\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nProof. The statement for SKSg follows from Theorem 4.6, and that for SKS from this and\nCorollary 2.23.\n□\nTranslating derivations from SKSg to Frege requires more effort than the converse,\npartly because of the need to simulate deep inference, and partly because of the large\n‘amount of inference’ of =-rule instances. The next two lemmas take care of these two\nissues.\nLemma 4.8. For every SKS context ξ{ } and formulae α and β, there is a Frege derivation\nwith premiss α→β and conclusion ξ{α}→ξ{β} whose length is O(m) and size is O(n2), where\nm = |ξ{ }| and n = |ξ{α} →ξ{β}|.\nProof. Consider four Frege proofs Υ′, Υ′′, Υ′′′, and Υ′′′′, respectively of the four tautolo-\ngies\n(A →B) →([A ∨C] →[B ∨C])\n,\n(A →B) →([C ∨A] →[C ∨B])\n,\n(A →B) →((A ∧C) →(B ∧C))\n,\n(A →B) →((C ∧A) →(C ∧B))\n.\nWe proceed by induction on the structure of ξ{ }. If ξ{ } ≡ξ1{{ } ∨γ1}, we build Frege\nderivation\nΥ1 = α →β, Υ′{A/α, B/β,C/γ1}, α ∨γ1 →β ∨γ1\n;\nwe build Υ1 similarly if ξ{ } ≡ξ1{{ } ∧γ1} or ξ{ } ≡ξ1{γ1 ∨{ }} or ξ{ } ≡ξ1{γ1 ∧{ }}.\nGiven ξ1{ } ≡ξ2{{ } ∨γ2} or ξ1{ } ≡ξ2{{ } ∧γ2} or ξ1{ } ≡ξ2{γ2 ∨{ }} or ξ1{ } ≡\nξ2{γ2 ∧{ }} we build Υ2 analogously to Υ1, and the premiss of Υ2 is the conclusion of\nΥ1. We proceed this way until we build Υl, whose conclusion is ξ{α} →ξ{β}, where l ≤m.\nWe obtain the desired derivation Υ by concatenating Υ1, ..., Υl. Since the length and size\nof Υ′, Υ′′, Υ′′′, and Υ′′′′ are independent of ξ{ }, α, and β, the length of Υ is O(m) and its\nsize is O(mn), and so O(n2).\n□\nLemma 4.9. For every SKS formulae α and β such that α = β there is a Frege derivation\nwith premiss α, conclusion β, length O(n3), and size O(n4), where n = |α| + |β|.\nProof. Consider the following tautologies, derived from the equations in Figure 1:\n(1)\n[A ∨B] ↔[B ∨A]\n,\n[A ∨f] ↔A\n,\n(A ∧B) ↔(B ∧A)\n,\n(A ∧t) ↔A\n,\n[[A ∨B] ∨C] ↔[A ∨[B ∨C]]\n,\n[t ∨t] ↔t\n,\n((A ∧B) ∧C) ↔(A ∧(B ∧C))\n,\n(f ∧f) ↔f\n,\nwhere each expression corresponds to the two tautologies obtained by orientating each\ndouble implication. Every such tautology can be proved in Frege with a constant-size\nproof, so every instance γ →γ′ of any of these tautologies has a Frege proof of length O(1)\nand size O(m′), where m′ = |γ| + |γ′|. By Lemma 4.8, for every ξ{ } there is a derivation\nwith premiss γ →γ′ and conclusion ξ{γ} →ξ{γ′} whose length is O(m) and size is O(m2),\nwhere m = |ξ{γ} →ξ{γ′}|. By concatenating the proof and derivation so obtained, we can\nbuild a proof of ξ{γ} →ξ{γ′} whose length is O(m) and size is O(m2). By Remark 2.9, we\ncan build a chain of implications\nα ≡α1 →· · · →αh ≡δ ≡βk →· · · →β1 ≡β\n,\nwhere δ is a canonical form for α and β, h + k is O(n2), and each implication αi →αi+1 and\nβi+1 →βi is a tautology of the form ξ{γ} →ξ{γ′}, such that γ →γ′ is an instance of one of\nthe tautologies 1. By concatenating the proofs of every ξ{γ} →ξ{γ′} by mp, we obtain a\nderivation with premiss α, conclusion β, length O(n3), and size O(n4).\n□\nLemma 4.10. For every inference step\nα\nν β , where ν is a rule of SKSg, there is a Frege\nderivation with premiss α, conclusion β, length O(n), and size O(n2), where n = |α| + |β|.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n19\nProof. Each of the following tautologies, corresponding to the inference rules in Figure 2,\ncan be proved in Frege with a constant-size proof:\n(2)\n(A ∧¬A) →f\n,\nA →t\n,\nA →(A ∧A)\n,\nf →[A ∨¬A]\n,\nf →A\n,\n[A ∨A] →A\n,\n(A ∧[B ∨C]) →[(A ∧B) ∨C]\n.\nLet\nα\nν β =\nξ{γ}\nν ξ{δ}, where\nγ\nν δ is an instance of ν. There is a Frege proof Υ of γ →δ,\nwhose length is O(1) and size is O(n), obtained by instantiating the corresponding proof to\nν among those in 2. By Lemma 4.8, there exists a Frege derivation Υ′ with premiss γ →δ,\nconclusion ξ{γ} →ξ{δ}, length O(n), and size O(n2). By concatenating Υ and Υ′ we obtain\na proof Υ′′ of ξ{γ} →ξ{δ}. From Υ′′, by using mp, we obtain the desired derivation with\npremiss α ≡ξ{γ} and conclusion β ≡ξ{δ}.\n□\nTheorem 4.11. For every derivation\nα\nΦ ∥\n∥SKSg\nβ\nthere is a Frege derivation Υ with premiss\nα and conclusion β; if n is the size of Φ, then the length and size of Υ are, respectively,\nO(n4) and O(n5).\nProof. The statement immediately follows from Lemmas 4.9 and 4.10, after assuming that\nthe length of Φ is O(n).\n□\nCorollary 4.12. Frege p-simulates SKSg and SKS.\nRemark 4.13. As evidenced by the proofs of Theorems 4.6 and 4.11, it does not really\nmatter, for establishing the p-simulations, precisely which inference rules are adopted by\nthe CoS and Frege systems. In fact, the simulations work because the simulating systems\nare implicationally complete and their set of proofs is closed under substitution. This way,\nthe constant-size proofs in one system, simulating the rules of the other system, can be\ninstantiated at a linear cost in order to simulate instances of rules. We can then use a ro-\nbustness theorem (see Theorem 4.2) for Frege in order to establish a robustness theorem\nfor CoS, possibly also for systems on mutually different languages: given two implication-\nally complete CoS systems, we p-simulate each in two appropriate Frege systems and use\nFrege robustness.\n5. Extension and Substitution\nIn this section, we show how CoS systems can be extended with the Tseitin exten-\nsion rule and with the substitution rule, analogously to Frege systems. We also show the\np-equivalence of all these systems, as described in the box of the diagram in the Introduc-\ntion. As always, we operate under robustness theorems (relying on the mentioned one,\nTheorem 4.2) that ensure that the proof complexity properties we establish for the specific\nsystems actually hold for the formalisms they belong to.\nDefinition 5.1. An extended Frege (proof) system is a Frege system augmented with the\n(Tseitin) extension rule, which is a rule with no premisses and whose instances A ↔β are\nsuch that the variable A does not appear before in the derivation, nor appears in β or in\nthe conclusion of the proof. We write A < α to state that variables A and ̄A do not appear\nin formula α. The symbol ↔stands for logical equivalence, and the specific syntax of the\nexpressions A ↔β depends on the language of the Frege system in use. In the following,\nwe consider A ↔β a shortcut for (A →β) ∧(β →A). We denote by xFrege the proof system\nwhere a proof is a derivation with no premisses, conclusion αk, and shape\nα1, . . ., αi1−1,\nαi1 ≡\nz }| {\nA1 ↔β1 , αi1+1, . . ., αih−1,\nαih ≡\nz }| {\nAh ↔βh , αih+1, . . . , αk\n,\n\n20\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nwhere all the conclusions of extension instances αi1, ..., αih are singled out and\nA1 < α1, . . ., αi1−1, β1, αk\n,\n. . .\n,\nAh < α1, . . ., αih−1, βh, αk\n,\nand the rest of the proof is as in Frege.\nRemark 5.2. We could have equivalently defined an xFrege proof of α as a Frege deriva-\ntion with conclusion α and premisses {A1 ↔β1, . . ., Ah ↔βh} such that A1, ̄A1, ..., Ah, ̄Ah\nare mutually distinct and A1 < β1, α and . . . and Ah < β1, . . ., βh, α. Notice that xFrege\nis indeed a proof system in the sense that it proves tautologies. In fact, given the xFrege\nproof just mentioned, we obtain a Frege proof by applying to it, in order, the substitutions\nσh = Ah/βh, ..., σ1 = A1/β1, and by prepending to it proofs of the tautologies β1 ↔β1,\n(β2 ↔β2)σ1, ..., (βh ↔βh)σh−1 · · · σ1. In general, a proof so obtained is exponentially bigger\nthan the xFrege one it derives from.\nSKSg can analogously be extended, but there is no need to create a special rule; we\nonly need to broaden the criterion by which we recognize a proof.\nDefinition 5.3. An extended SKSg proof of α is an SKSg derivation with conclusion α\nand premiss [ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah], where A1, ̄A1, ..., Ah, ̄Ah are\nmutually distinct and A1 < β1, α and . . . and Ah < β1, . . ., βh, α. We denote by xSKSg the\nproof system whose proofs are extended SKSg proofs.\nTheorem 5.4. For every xFrege proof of length l and size n there exists an xSKSg proof\nof the same formula and whose length and size are, respectively, O(l) and O(n2).\nProof. Consider an xFrege proof as in Definition 5.1. By Remark 5.2 and Theorem 4.6,\nthere exists the following xSKSg proof, whose length and size are yielded by 4.6:\n[ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah]\n∥\n∥SKSg\nαk\n.\n□\nAlthough not strictly necessary to establish the equivalence of the four extended for-\nmalisms (see diagram in the Introduction), the following theorem is very easy to prove.\nTheorem 5.5. For every xSKSg proof of size n there exists an xFrege proof of the same\nformula and whose length and size are, respectively, O(n4) and O(n5).\nProof. Consider an xSKSg proof as in Definition 5.3. The statement is an immediate\nconsequence of Theorem 4.11, after observing that there is an O(h)-length and O(hn)-size\nxFrege proof\nA1 ↔β1, . . ., Ah ↔βh, . . . , (A1 ↔β1) ∧· · · ∧(Ah ↔βh)\n.\n□\nCorollary 5.6. Systems xFrege and xSKSg are p-equivalent.\nWe now move to the substitution rule.\nDefinition 5.7. A substitution Frege (proof) system is a Frege system augmented with\nthe substitution rule\nA\nsub Aσ. We denote by sFrege the proof system where a proof is a\nderivation with no premisses, conclusion αk, and shape\nα1, . . ., αi1−1,\nαi1 ≡\nz}|{\nα j1σ1 , αi1+1, . . ., αih−1,\nαih ≡\nz}|{\nα jhσh , αih+1, . . . , αk\n,\nwhere all the conclusions of substitution instances αi1, ..., αih are singled out, α j1 ∈\n{α1, . . ., αi1−1}, ..., α jh ∈{α1, . . . , αih−1}, and the rest of the proof is as in Frege.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n21\nWe rely on the following result.\nTheorem 5.8. (Cook-Reckhow and Kraj ́ıˇcek-Pudl ́ak, [CR79, KP89])\nSystems xFrege\nand sFrege are p-equivalent.\nWe can extend SKSg with the same substitution rule as for Frege. The rule is used like\nother proper rules of system SKSg, so its instances are interleaved with =-rule instances.\nDefinition 5.9. An sSKSg proof is a proof of SKSg where, in addition to the inference\nsteps generated by rules of SKSg, we admit inference steps obtained as instances of the\nsubstitution rule\nA\nsub Aσ.\nThis rule does not fit any of the usual deep-inference rule classes (see Section 2), and (as\nin Frege systems) is not sound, in the sense that the premiss does not imply the conclusion.\nHowever, of course, if the premiss is provable the conclusion also is.\nRemark 5.10. Notice that instances of the substitution rule cannot be used inside a context;\nfor example, the expression on the left is not a valid sSKSg proof, while the one on the\nright is:\nt\ni↓A ∨ ̄A\nsub? (B ∧C) ∨ ̄A\n,\nt\ni↓A ∨ ̄A\nsub (B ∧C) ∨[ ̄B ∨ ̄C]\n.\nIn the so-called ‘Formalism B’ of deep inference, which is currently under development\n[Gug04], and for which all the proof-complexity results in this article apply unchanged,\nsubstitution becomes part of the composition mechanism of proofs, rather than an odd\nextension to the set of rules.\nFor the time being, we can establish the promised p-equivalence of all extended systems\nby completing the diagram in the Introduction with the last two missing steps.\nTheorem 5.11. For every xSKSg proof of size n there exists an sSKSg proof of the same\nformula and whose length and size are, respectively, O(n) and O(n2).\nProof. Consider the xSKSg proof\n[ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah]\nΦ ∥\n∥SKSg\nα\n,\nwhere\nA1 < β1, α\n,\n. . .\n,\nAh < β1, . . . , βh, α\n,\nand let us call its premiss γ. We can build the sSKSg proof\nt\ni↓ ̄γ ∨γ\n ̄γ∨Φ ∥\n∥SKSg\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(Ah ∧ ̄βh) ∨(βh ∧ ̄Ah)] ∨α\nsub [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(βh ∧ ̄βh) ∨(βh ∧ ̄βh)] ∨α\nc↓\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(βh ∧ ̄βh)] ∨α\ni↑\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨f] ∨α\n=\n...\n= [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1)] ∨α\nsub [(β1 ∧ ̄β1) ∨(β1 ∧ ̄β1)] ∨α\nc↓\n(β1 ∧ ̄β1) ∨α\ni↑\nf ∨α\n=\nα\n.\n\n22\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\n□\nCorollary 5.12. sSKSg p-simulates xSKSg.\nTheorem 5.13. For every sSKSg proof of size n there exists a proof of the same formula\nin sFrege, whose length and size are, respectively, O(n4) and O(n5).\nProof. Every sSKSg proof has shape\nt\nΦ0 ∥\n∥SKSg\nα1\nsub α1σ1\nΦ1 ∥\n∥SKSg\n...\nΦh−1 ∥\n∥SKSg\nαh\nsub αhσh\nΦh ∥\n∥SKSg\nαh+1\n.\nBy Theorem 4.11, for each of Φ0, ..., Φh there exist Frege derivations Υ0, ..., Υh with\nthe same premiss and conclusion, respectively. We can then build the proof\nΥ0\nz }| {\n. . . , α1 ,\nΥ1\nz }| {\nα1σ1, . . ., . . . ,\nΥh−1\nz }| {\n. . . , αh ,\nΥh\nz }| {\nαhσh, . . . , αh+1\nin sFrege; the cited theorem also yields its length and size.\n□\nCorollary 5.14. sFrege p-simulates sSKSg.\nNothing prevents us from using Tseitin extension and the substitution rule with system\nSKS, or any other atomic or nonatomic CoS system. The integration of these mechanisms\ninto CoS is similar to their integration into Frege systems, as the simplicity of the argu-\nments showing p-equivalence testifies.\n6. Open Problems\nWe conclude the article with a list of open problems, some of which are currently in-\nvestigated by us and other researchers.\n6.1. Relation with Resolution and Other Formalisms. In this article, we explored the\nrelation between CoS and Frege systems, and in the cited literature the relation between\nCoS and Gentzen systems has been explored in depth. There are, of course, other for-\nmalisms, like resolution, whose relation with CoS might lead to some interesting research\ndirections. For example, the note [Gug03] shows how simply, compared to Gentzen sys-\ntems, KS expresses resolution (analytically, of course).\n6.2. Does Cocontraction Provide for an Exponential Speedup? As we argued in Re-\nmark 3.6, we do not know whether KSg p-simulates KSg ∪{c↑}, or, equivalently, whether\nKS p-simulates KS ∪{ac↑}.\nOur intuition, as well as some clues, like the mutual behaviour of the ‘atomic flows’\nof contraction and cocontraction (see [GG08]) would lead us to believe that cocontraction\nindeed provides for an exponential speedup. However, we know that in similar situations,\nlike for dag-like versus tree-like Frege systems, intuition was fallacious.\nIf cocontraction yields an exponential speedup, we obtain an even stronger analytic\nsystem than KSg, which is, in turn, stronger than analytic Gentzen. This would draw\ninterest to a hierarchy of analytic proof systems of different strength.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n23\nUnless we prove the p-equivalence of KSg and KSg ∪{c↑}, we tend to consider cocon-\ntraction a simple rule-based mechanism for compressing proofs, like cut, extension, and\nsubstitution.\n6.3. Pigeonhole in Analytic CoS. Does the pigeonhole principle, in particular in its rela-\ntional variety, admit polynomially growing proofs in KS? If not, does it in KS ∪{ac↑}?\nInvestigating this problem could be relevant to the more general, following one (Prob-\nlem 6.4), about the ability of analytic CoS to simulate CoS, and so Frege. In fact, the\npigeonhole principle generates some of the hardest classes of tautologies known.\nWe note that in [Jap08], Japaridze shows polynomially growing proofs for the pigeon-\nhole class of tautologies in a deep-inference system over certain circuit-like sequents,\ncalled ‘cirquents’. In this case, the speedup is obtained by the sharing of logical expres-\nsions in circuits.\nIn [Jeˇr09], Jeˇr ́abek shows that there are polynomial-time constructible proofs in KS ∪\n{ac↑} of the functional and onto variants of the pigeonhole principle.\n6.4. Relative Strength of Analytic CoS and CoS. Some recent major progress has been\nmade in [Jeˇr09]. There, Jeˇr ́abek uses a construction on threshold formulae in the mono-\ntone sequent calculus, by Atserias, Galesi and Pudl ́ak [AGP02], to show that analytic CoS\nquasipolynomially simulates CoS. In [BGGP09], we provide a direct and simplified con-\nstruction based on atomic flows [GG08] and threshold formulae.\nBecause of these recent advances, we expect that analytic CoS p-simulates CoS. A more\nin-depth discussion of this subject is in [BGGP09]. If analytic CoS p-simulates CoS, then\nthere are polynomially growing proofs of the pigeonhole principle in analytic CoS, though\nnot necessarily in KS.\nWe think that investigating this problem will help us to better understand analyticity, in\norder to obtain for it a general and more useful definition than the one we have now. We\nfeel that the current notion is not satisfactory because it depends on the formalism and must\nbe defined by resorting to the syntactic structure of inference rules (or, worse, by indicating\nwhich rules are analytic and which are not).\nFor example, a more general, nonsyntactic definition of analyticity could be the follow-\ning: a rule is analytic if, given an instance of its conclusion, the set of possible instances\nof the premiss is finite (this is what we call a finitely generating rule in Section 2). In this\nsense, an atomic ‘finitary’ cut rule\nξ{a ∧ ̄a}\nfai↑\nξ{t}\n, such that a appears in ξ{ }, would be ana-\nlytic. However, [BG04] shows that we can easily transform proofs in SKS into smaller- or\nequal-size proofs that only use fai↑wherever ai↑was used. So, we could deem fai↑an ana-\nlytic rule, and the system obtained from SKS by substituting ai↑with fai↑an analytic one,\nand we could immediately conclude that analytic CoS p-simulates CoS. This ‘solution’,\nhowever, is way too cheap.\nWe prefer to think that fai↑is not an analytic rule, in some sense to be made precise. A\npossible point of attack is offered by the fact that fai↑is not a local rule: it requires checking\nthat a appears in its context, whose size is unbounded (see Remark 2.17). So, we think it\ncould be productive to look for a notion of analyticity that is based on boundedness instead\nof finiteness, and tackle the separation problem between analytic CoS and CoS under that\nnotion. The note [BG07] further explores this direction, but much more work is necessary.\n6.5. Strength of Analytic CoS Systems Plus Substitution. We showed that CoS and\nFrege systems are p-equivalent, and both remain p-equivalent when extended either with\nTseitin extension or substitution. However, CoS is more flexible than Frege, because it\nallows to ‘switch off’ two mechanisms that potentially provide for an exponential com-\npression of proofs: cut and cocontraction (see Problem 6.2).\nIt might be interesting to study the relative strength of systems obtained by removing\nfrom SKS ∪{sub} either ai↑or ac↑or both. (Rule aw↑can also be removed, but we do\n\n24\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nnot see a crucial role for it.) Notice that systems KS ∪{sub} and KS ∪{ac↑, sub} could be\nconsidered, in some sense, analytic, and we do not know their relative strength.\n6.6. Speedup of Deep Inference Over Any Bounded-Depth System. We saw in The-\norem 3.12 that analytic CoS exhibits an exponential speedup over analytic Gentzen, for\nStatman tautologies. We argued that, in this case, the speedup is obtained by a rather triv-\nial use of deep inference, because the depth at which inference has to be performed, in\norder to get the speedup, is constant. So, a natural question is whether there exists a class\nof tautologies that requires full-fledged deep inference in order to obtain efficient proofs.\nWe think we found such a class, which is defined as follows.\nConsider, for every propositional formula α, the following set of second-order formulae,\nfor n > 0:\ng(1, α) ≡∀β.[((α ∧β) ∧β) ∨( ̄β ∧ ̄β)]\n,\ng(n + 1, α) ≡∀β.[(g(n, α ∧β) ∧g(n, β)) ∨(g(n, ̄β) ∧g(n, ̄β))]\n.\nBy using these formulae as a template, we can generate a set of first-order formulae, where\nthe (complex) management of indices ensures their uniqueness:\nDefinition 6.1. Consider, for m, n ≥0\nh(1, m, α) ≡((α ∧βm+1) ∧βm+1) ∨( ̄βm+1 ∧ ̄βm+1)\n,\nh(n + 2, m, α) ≡(h(n + 1, m, α ∧β5n+1+m) ∧h(n + 1, 5n + m, β5n+1+m)) ∨\n(h(n + 1, 2 · 5n + m, ̄β5n+1+m) ∧h(n + 1, 3 · 5n + m, ̄β5n+1+m))\n.\nConsider now\nf(n) ≡h(n, 0, t)\n,\nfor n > 0\n.\nWe define the set DT = { f(n) | n > 0 }.\nThe program [Gug07a] can help in understanding the nature of these formulae.\nRemark 6.2. It is not difficult to verify that DT contains tautologies possessing analytic\nCoS proofs that grow polynomially in the size of the tautologies.\nThe analytic CoS proofs of the DT tautologies, when read bottom-up, work by applying\ninteractions starting from the deepest subformulae. When this cannot be the case, we\nconjecture that the size of the proofs grows exponentially.\nDefinition 6.3. The and/or depth of a formula is the maximum number of alternations of\nconjunctions and disjunctions in the formula tree; the and/or depth of a context ξ{ } is the\nnumber of alternations of conjunctions and disjunctions between the hole and the root of\nthe context tree. We define a bounded-depth CoS proof system as a CoS proof system\nwhose inference rules only generate inference steps at a bounded depth, namely inference\nsteps\nξ{γ}\nν ξ{δ} are such that, if\nγ\nν δ is a rule instance then the and/or depth of ξ{ } is bounded\nby a given constant, and the same restriction holds for the contexts in the context closure\ncondition of relation =.\nRemark 6.4. Note that the nonatomic rules interaction (identity), cointeraction (cut),\ncontraction and cocontraction require establishing duality or identity of formulae of un-\nbounded and/or depth. So, their adoption might be considered an implicit use of deep in-\nference. However, the atomic counterparts of these rules do not suffer this problem because\nthe ‘deep checking’ is delegated to the inference mechanism. For this reason, proving the\nfollowing conjecture is better done in the analytic part of system SKS.\nConjecture 6.5. In any analytic bounded-depth CoS proof system, the tautologies in DT\nonly have proofs that grow exponentially in their size.\n\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\n25\n7. Conclusion\nIn this article, we showed that the calculus of structures (CoS) has the same character-\nistics of the Frege formalism in terms of proof complexity, including when extended with\nTseitin extension and substitution.\nWe know that, contrary to Frege, CoS has a rich proof theory, and its proof systems\nenjoy several properties, arguably relevant to proof complexity, that cannot be observed in\nother formalisms, like locality for all inference rules. We also know that other logics, like\nmodal logics, enjoy simple and modular presentations in deep inference, which should help\nin proof complexity investigations. This article establishes the basic connection between\nproof theory in deep inference and proof complexity.\nAs a consequence of its flexibility in inference rule design, CoS admits a notion of\nanalyticity that is more flexible than its counterpart for Gentzen systems. We can then\nexplore the strength of analytic systems in finer detail than possible in Gentzen systems. In\nthis article, we moved forward the boundary between polynomial and exponential analytic\nproofs by proving Statman tautologies with polynomial, analytic deep-inference proofs.\nWe included a list of open problems and currently active research directions.\nAcknowledgements. 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Theoretical Computer Science, 309:213–285,\n2003.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/els.pdf.\n[Tiu06a]\nAlwen Tiu. A local system for intuitionistic logic. In M. Hermann and A. Voronkov, editors, LPAR\n2006, volume 4246 of Lecture Notes in Artificial Intelligence, pages 242–256. Springer-Verlag, 2006.\nhttp://users.rsise.anu.edu.au/ ̃tiu/localint.pdf.\n[Tiu06b]\nAlwen Tiu. A system of interaction and structure II: The need for deep inference. Logical Methods\nin Computer Science, 2(2:4):1–24, 2006.\nhttp://arxiv.org/pdf/cs.LO/0512036.\n[TS96]\nA.S. Troelstra and H. Schwichtenberg. Basic Proof Theory, volume 43 of Cambridge Tracts in The-\noretical Computer Science. Cambridge University Press, 1996.\nUniversity of Bath, Bath BA2 7AY, UK,\nhttp://cs.bath.ac.uk/pb/ and\nhttp://alessio.guglielmi.name/res","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.1201v3 [cs.CC] 19 Apr 2009\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nAbstract. We obtain two results about the proof complexity of deep inference: 1) deep-\ninference proof systems are as powerful as Frege ones, even when both are extended with\nthe Tseitin extension rule or with the substitution rule; 2) there are analytic deep-inference\nproof systems that exhibit an exponential speedup over analytic Gentzen proof systems\nthat they polynomially simulate.\n1. Introduction\nDeep inference is a relatively new methodology in proof theory, consisting in deal-\ning with proof systems whose inference rules are applicable at any depth inside formulae\n[Gug07b]. We obtain two results about the proof complexity of deep inference:\n• deep-inference proof systems are as powerful as Frege ones, even when both are\nextended with the Tseitin extension rule or with the substitution rule;\n• there are analytic deep-inference proof systems that exhibit an exponential speed-\nup over analytic Gentzen proof systems that they polynomially simulate.\nThese results are established for the calculus of structures, or CoS, the simplest formal-\nism in deep inference [Gug07b], and in particular for its proof system SKS, introduced\nby Br ̈unnler in [Br ̈u04] and then extensively studied [Br ̈u03a, Br ̈u03b, Br ̈u06a, Br ̈u06d,\nBG04, BT01].\nOur contributions fit in the following picture.\nCoS +\nextension\nCoS +\nsubstitution\nFrege +\nextension\nFrege +\nsubstitution\n⋆\n4\n3\nKraj ́ıˇcek-Pudl ́ak ’89\n⋆\n5\nCook-Reckhow ’79\nFrege\nCoS\nGentzen\nopen\n2\nCook-\nReckhow ’74\nanalytic\nCoS\nanalytic\nGentzen\nBr ̈unnler\n’04\n1 ×\nStatman ’78\n×\nopen\nThe notation F\nG indicates that formalism F polynomially simulates formalism G ;\nthe notation F\nG\n×\nindicates that it is known that this does not happen.\nThe left side of the picture represents, in part, the following. Analytic Gentzen systems,\ni.e., Gentzen proof systems without the cut rule, can only prove certain formulae, which\nwe call ‘Statman tautologies’, with proofs that grow exponentially in the size of the for-\nmulae. On the contrary, Gentzen systems with the cut rule can prove Statman tautologies\nby polynomially growing proofs. So, Gentzen systems p-simulate analytic Gentzen ones,\nDate: October 24, 2018.\nThis research was partially supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in\nDeep Inference.\nc⃝ACM, 2009. This is the authors’ version of the work. It is posted here by permission of ACM for your\npersonal use. Not for redistribution. The definitive version was published in ACM Transactions on Computational\nLogic 10 (2:14) 2009, pp. 1–34, http://doi.acm.org/10.1145/1462179.1462186.\n1"},{"paragraph_id":"p2","order":2,"text":"2\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nbut not vice versa [Sta78]. Cook and Reckhow proved that Frege and Gentzen systems\nare polynomially equivalent, i.e., each Frege system polynomially simulates any Gentzen\nsystem and vice versa [CR74].\nIn the box at the right of the figure, ‘extension’ refers to the Tseitin extension rule,\nand ‘substitution’ to the substitution rule. The works of Cook and Reckhow [CR79] and\nKraj ́ıˇcek and Pudl ́ak [KP89] established that Frege + extension and Frege + substitution\nare polynomially equivalent. It is immediate to see that these formalisms polynomially\nsimulate Frege and Gentzen, which, in turn, polynomially simulate analytic Gentzen. It is\na major open problem to establish whether Frege polynomially simulates Frege + exten-\nsion/substitution.\nIn this work, we establish the following results (numbered as in the previous figure):\n(1) Analytic Gentzen does not polynomially simulate analytic CoS (essentially in the\nform of system SKS without cut); in fact, Statman tautologies admit polynomially\ngrowing proofs in analytic CoS (Theorem 3.12).\n(2) CoS and Frege are polynomially equivalent (Theorems 4.7 and 4.12).\n(3) There is a natural notion of (Tseitin) extension for CoS, and CoS + extension is\npolynomially equivalent to Frege + extension (Theorem 5.6).\n(4) There is a natural notion of substitution for CoS, and CoS + substitution polyno-\nmially simulates CoS + extension (Theorem 5.12).\n(5) Frege + substitution polynomially simulates CoS + substitution; this way, we\nknow that all extended formalisms are polynomially equivalent (Theorem 5.14).\nThe polynomial simulations indicated by ⋆arcs in the picture follow from the others.\nEstablishing whether analytic CoS polynomially simulates CoS is an open problem.\nAfter the necessary preliminaries, in Section 2, we see how CoS expresses Gentzen\nsystems, including their properties, like analyticity, and then in Section 3 how it provides\nfor exponentially more compact proofs. The relation between CoS and Frege systems is\nexplored in Section 4 and the extensions are studied in Section 5. We conclude the article\nwith a list of open problems, in Section 6.\n2. Preliminaries\nIn this section, we quickly introduce the necessary deep-inference notions. A more\nextensive treatment of much of this material is in Br ̈unnler’s [Br ̈u04].\nWe only need the following, basic proof complexity notions (see [CR79]).\nDefinition 2.1. A (propositional) proof system is a binary relation S between formulae\nα and proofs Π such that S is computable in polynomial time, and the formula α is a\ntautology if and only if there is a proof Π such that S(α, Π); in this case we say that Π is\na proof of α in S. We say that proof system S p-simulates proof system S′ if there is a\npolynomial-time computable algorithm that transforms every proof in S′ into a proof in S\nof the same tautology. Two proof systems are p-equivalent if each p-simulates the other.\nRemark 2.2. In the following, we state theorems on the existence of proofs in one proof\nsystem when proofs exist in another proof system, such that their size is polynomially\nrelated. Implicitly, we always mean that the new proofs are obtained by transforming the\nold ones by way of a polynomial-time computable algorithm.\nDeep inference is a relatively recent development in proof theory. Its main idea is\nto provide a finer analysis of inference than possible with traditional methods, and one\nof the main objectives is to obtain a geometric semantics for proofs, inspired by linear\nlogic’s proof nets [Gir87]. Another objective is to provide a uniform and useful syntactic\ntreatment of several logics, especially modal ones, for which no satisfactory proof theory\nexisted before.\nIn deep inference, several formalisms can be defined with excellent structural proper-\nties, like locality for all the inference rules. The calculus of structures [Gug07b] is one"},{"paragraph_id":"p3","order":3,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n3\nof them and is now well developed for classical [Br ̈u03a, Br ̈u06a, Br ̈u06d, BT01], intu-\nitionistic [Tiu06a], linear [Str02, Str03b], modal [Br ̈u06c, GT07, Sto07] and commuta-\ntive/noncommutative logics [Gug07b, Tiu06b, Str03a, Bru02, DG04, GS01, GS02, GS09,\nKah06b, Kah07]; for all these logics, quantification can be defined at any order. We em-\nphasise that deep inference is developing the first reasonable proof theory for modal logics;\nthe large number of different modal logic systems can be studied in simple and modular\ndeep-inference systems, which are similar to their propositional logic counterparts and en-\njoy the same locality properties. The calculus of structures promoted the discovery of a\nnew class of proof nets for classical and linear logic [LS05a, LS05b, LS06, SL04] (see\nalso [Gui06]). Moreover, there exist implementations in Maude of deep-inference proof\nsystems [Kah08].\nIn this article, we focus on the calculus of structures because it is well developed and is\nprobably the simplest formalism definable in deep inference. The complexity results that\nwe present here are not dependent on the choice of formalism; rather, they only depend on\nthe deep-inference methodology and the finer granularity of inference rules that it yields.\nAdopting deep inference basically means that it is possible to replace subformulae inside\nformulae by other, implied subformulae, and that there is no limit to the nesting depth of\nsubformulae. Formalisms like Gentzen’s sequent calculus differ because they only rewrite\nformulae, or sequents, around their root connectives, and (we argue) they suffer excessive\nrigidity in the syntax and they do not sufficiently support geometric semantics.\nBecause of its geometric nature, it is important, in deep inference, to control whether\npropositional variables can be instantiated by formulae. In particular, normalisation (cut\nelimination) in deep inference crucially depends on the availability of ‘atomic’ inference\nrules, which are rules related to some topological invariants (see, for example, [GG08]\nfor normalisation in propositional logic). In practice, we need two kinds of propositional\nvariables: the atoms, only subject to renaming, and the formula variables, subject to (unre-\nstricted) substitution. This distinction does not bear dramatic effects on proof complexity,\nbut it does allow for some finer measures than otherwise possible.\nDefinition 2.3. Formulae of the calculus of structures, or CoS, are denoted by α, β, γ, δ\nand are freely built from: units, like f (false) and t (true); atoms a, b, c, d and ̄a, ̄b, ̄c, ̄d;\n(formula) variables A, B, C, D and ̄A, ̄B, ̄C, ̄D; logical relations, like disjunction [α ∨β]\nand conjunction (α ∧β). A formula is ground if it contains no variables. We usually omit\nexternal brackets of formulae, and sometimes we omit dispensable brackets under associa-\ntivity. We use ≡to denote literal equality of formulae. The size |α| of a formula α is the\nnumber of unit, atom, and variable occurrences appearing in it. On the set of atoms, there\nis an involution ̄·, called negation (i.e., ̄· is a bijection from the set of atoms to itself such\nthat ̄ ̄a ≡a); we require that ̄a . a for every a; when both a and ̄a appear in a formula,\nwe mean that atom a is mapped to ̄a by ̄·. An analogous involution is defined on the set of\nformula variables. The (De Morgan) dual of a formula is obtained by exchanging disjunc-\ntion and conjunction and applying negation to all atoms and variables; we denote duals by\nusing ̄·; for example, the De Morgan dual of α ≡t ∨(a ∧[ ̄B ∨c]) is ̄α ≡f ∧[ ̄a ∨(B ∧ ̄c)]. A\ncontext is a formula where one hole { } appears in the place of a subformula; for example,\nA ∨(b ∧{ }) is a context; the generic context is denoted by ξ{ }. The hole can be filled with\nformulae; for example, if ξ{ } ≡b ∧[{ } ∨c], then ξ{a} ≡b ∧[a ∨c], ξ{b} ≡b ∧[b ∨c] and\nξ{a ∧B} ≡b ∧[(a ∧B) ∨c]. The size of ξ{ } is defined as |ξ{ }| = |ξ{a}| −1.\nRemark 2.4. We do not say that a is positive and ̄a is negative. It only matters that,\nwhen a and ̄a appear in the same formula, if one is negative the other is positive. In\nabsence of disambiguating information, there are two ways in which ξ{b} might correspond\nto b ∧[b ∨c]: one such that ξ{a} ≡a ∧[b ∨c] and another such that ξ{a} ≡b ∧[a ∨c].\nThe language of formulae is redundant because we can choose whether to use atoms\nor formula variables whenever a propositional variable is needed. The distinction between"},{"paragraph_id":"p4","order":4,"text":"4\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\natoms and formula variables only plays a role in the choice of applicable inference rules,\nand this aspect is controlled by renamings and substitutions.\nDefinition 2.5. A renaming is a map from the set of atoms to itself, and is denoted by\n{a1/b1, a2/b2, . . .}; we use ρ for renamings. A renaming of α by ρ = {a1/b1, a2/b2, . . .}\nis indicated by αρ and is obtained by simultaneously substituting every occurrence of ai\nin α by bi and every occurrence of ̄ai by ̄bi; for example, if α ≡a ∧[b ∨(a ∧[ ̄a ∨C])]\nthen α{a/ ̄b, ̄b/c} ≡ ̄b ∧[ ̄c ∨( ̄b ∧[b ∨C])]. A substitution is a map from the set of for-\nmula variables to formulae, denoted by {A1/β1, A2/β2, . . .}; we use σ for substitutions.\nAn instance of α by σ = {A1/β1, A2/β2, . . . } is indicated by ασ and is obtained by si-\nmultaneously substituting every occurrence of variable Ai in α by formula βi and every\noccurrence of ̄Ai by the De Morgan dual of βi; for example, if α ≡(A ∧[A ∨ ̄A]) ∨b then\nα{A/(c ∧ ̄B)} ≡((c ∧ ̄B) ∧[(c ∧ ̄B) ∨[ ̄c ∨B]]) ∨b.\nDefinition 2.6. A CoS (inference) rule ν is an expression\nα\nν β , where formulae α and β\nare called premiss and conclusion, respectively. A (rule) instance\nγ\nν δ of\nα\nν β is such that\nγ ≡αρσ and δ ≡βρσ, for some renaming ρ and substitution σ. For some context ξ{ }, a\nCoS (inference) step, generated by rule\nα\nν β via its instance\nγ\nν δ , is the expression\nξ{γ}\nν ξ{δ}.\nExample 2.7. Given\nt\ni↓A ∨ ̄A and\nt\nai↓a ∨ ̄a, then\nt\ni↓(A ∧b) ∨[ ̄A ∨ ̄b] is an instance of i↓and\nt\nν b ∨ ̄b is an instance of both i↓and ai↓. The rule\nA ∨A\nc↓\nA\ngenerates the inference step\n[a ∨b] ∧[(c ∧D) ∨(c ∧D)]\nc↓\n[a ∨b] ∧(c ∧D)\n.\nWe usually classify deep-inference rules in three classes. For this, we rely on the notion\nof linearity, which, in this context, essentially means the same as in term-rewriting: a\nrewriting rule is linear if variables appear once in both sides of the rule. In other words, a\nlinear rule does not create or destroy anything. These are the three classes of rules:\n(1) Atomic rules. They usually correspond to structural rules in Gentzen systems; in\nnormalisation and in semantics of proofs, they play a crucial role because they\nexpress the causality relations between atoms, so shaping the geometry of proofs.\nTheir instances are obtained by renaming.\n(2) Noninvertible linear rules. They usually correspond to logical rules in Gentzen\nsystems. Since they are noninvertible, they express proper inference choices, but\nsince they are linear, they do not alter the geometry of causality between atoms.\nTheir instances are obtained by substitution.\n(3) Invertible linear rules. These rules are equivalences between formulae that do not\ncorrespond to proper inference choices and have no impact on the geometry of\nproofs. For this reason, they are usually gathered into one big equivalence relation\nbetween formulae, corresponding to just one rule, defined via substitution.\nThe success of deep inference is due to its ability to separate rules into classes 1 and\n2, which is only possible by adopting deep inference. The references to the ‘geometry of\nproofs’ can be understood by reading [GG08, LS05b]. Class 3 allows us to greatly simplify\nproofs and to hide, so to speak, a great deal of logical complexity (in the sense of size of\nproofs). We start by defining our ‘class 3’ rule, the others being dependent on specific\nproof systems."},{"paragraph_id":"p5","order":5,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n5\nCommutativity\nα ∨β = β ∨α\nα ∧β = β ∧α\nAssociativity\n[α ∨β] ∨γ = α ∨[β ∨γ]\n(α ∧β) ∧γ = α ∧(β ∧γ)\nUnits\nα ∨f = α\nα ∧t = α\nt ∨t = t\nf ∧f = f\nContext closure\nif α = β then ξ{α} = ξ{β}\nFigure 1. Equality = on formulae.\nDefinition 2.8. The equality relation = on formulae is defined by closing the equations in\nFigure 1 by reflexivity, symmetry, transitivity and by applying context closure. We define\nthe inference rule = as\nα\n= β , where α = β.\nThe following remark helps in assessing how much complexity is hidden in =.\nRemark 2.9. It is possible to decide α = β in polynomial time by reducing α and β to\nsome canonical form and comparing the canonical forms. A canonical form under = of\nany given formula can be obtained, for example, by removing as many units as possible\nand ordering units, atoms, and variables according to an arbitrary order; the canonical\nform is normal for associativity and units equations, when these are orientated from left\nto right. Let us assume a total order on the set of units, atoms, and variables; we now\nsee in detail how to find an equivalent canonical formula in the case of a formula only\ncontaining one logical relation. On the formula, use commutativity until the minimal unit,\natom, or variable appears in the leftmost position, then use associativity, orientated from\nleft to right, until normality is reached. For example, on [b ∨d] ∨[c ∨a], we perform the\nsteps\n[b ∨d] ∨[c ∨a] ❀[c ∨a] ∨[b ∨d] ❀[a ∨c] ∨[b ∨d] ❀a ∨[c ∨[b ∨d]]\n.\nThis phase requires O(n) steps, where n is the size of the formula. We proceed the same\nway on the subformula immediately following the first element, and so on recursively; for\nexample,\na ∨[c ∨[b ∨d]] ❀a ∨[[b ∨d] ∨c] ❀a ∨[b ∨[d ∨c]] ❀a ∨[b ∨[c ∨d]]\n.\nThe number of steps of the algorithm for a formula only containing one logical relation is\nthen O(n2). On a generic formula, the same algorithm can be used, with the same number-\nof-steps complexity O(n2) on the size n of the given formula, by adopting the lexicographic\norder induced by the given total order. This is an example, also involving an initial O(n)\nphase of simplification of units:\n(t ∧t) ∧[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f]\n❀t ∧[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f]\n❀[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f] ∧t\n❀[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f\n❀a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])\n❀⋆a ∨([a ∨[b ∨[c ∨d]]] ∧[[B ∨c] ∨[t ∨a]])\n❀⋆a ∨([a ∨[b ∨[c ∨d]]] ∧[t ∨[a ∨[c ∨B]]])\n❀a ∨([t ∨[a ∨[c ∨B]]] ∧[a ∨[b ∨[c ∨d]]])"},{"paragraph_id":"p6","order":6,"text":"6\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\n❀([t ∨[a ∨[c ∨B]]] ∧[a ∨[b ∨[c ∨d]]]) ∨a\n.\nThis way we obtain a (unique, of course) canonical formula in O(n2) steps, given any\nformula of size n, so we can decide the equivalence of two formulae α and β in O(n2)\nsteps, where n = |α| + |β|. Notice that at each step the size of the formula stays the same or\ndiminishes.\nDefinition 2.10. A CoS (proof) system is a finite set of inference rules. A CoS derivation\nΦ of length k in proof system S, whose premiss is α0 and conclusion is αk, is a chain of\ninference steps\nΦ =\nα0\nν1 α1\nν2 ...\nνk−1 αk−1\nνk αk\n,\nsuch that ν1, . . ., νk is a sequence of inference rules that alternate between the = rule and\nany rule of system S, where k ≥0. The same derivation can be indicated by\nα0\nΦ ∥\n∥S\nαk\n, when\nthe details are known or irrelevant; a proof is a derivation whose premiss is t. A derivation\nis ground if it contains no variables. Sometimes, we omit to indicate the inference steps\ngenerated by =. The size |Φ| of derivation Φ is the number of unit, atom, and variable\noccurrences appearing in it. We denote by ξ{Φ} the result of including every formula of\nΦ into the context ξ{ }. We denote by Φρ and Φσ the expression obtained from Φ by\napplying renaming ρ and substitution σ to every formula in Φ. A CoS proof system that,\nfor every valid implication α →β, contains a derivation with premiss α and conclusion β, is\nsaid to be implicationally complete.\nRemark 2.11. If Φ is a derivation, then ξ{Φ}, Φρ, and Φσ are derivations, for every context\nξ{ }, renaming ρ, and substitution σ.\nWe use the notion of groundness to relate the complexity of deep-inference proof sys-\ntems with atomic rules to proof systems without atomic rules, including those outside of\ndeep inference. Due to the aforementioned redundancy in the language, groundness is not\nreally a restriction.\nRemark 2.12. Every nonground derivation can be transformed into an equivalent,\nground one, by replacing variables with atoms in such a way that newly introduced atoms\nare different from the already present one.\nWe can now define some deep-inference proof systems. System SKS is the most impor-\ntant for the proof theory of classical logic, because of its atomic structural rules. System\nSKSg relates SKS to proof systems in other formalisms, like Frege.\nDefinition 2.13. CoS proof systems KSg = {i↓, w↓, c↓, s}, SKSg = KSg ∪{i↑, w↑,\nc↑}, KS = {ai↓, aw↓, ac↓, s, m} and SKS = KS ∪{ai↑, aw↑, ac↑} are defined in Figures 2\nand 3, for a language containing f, t, disjunction, and conjunction. Proof systems where\nnone of the rules i↑, ai↑, w↑, and aw↑appear are said to be analytic."},{"paragraph_id":"p7","order":7,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n7\nStructural rules\nLogical rule\nSKSg"},{"paragraph_id":"p8","order":8,"text":"A ∧ ̄A\ni↑\nf\nA\nw↑t\nA\nc↑A ∧A\ncointeraction\ncoweakening\ncocontraction\nor cut\nt\ni↓A ∨ ̄A\nf\nw↓A\nA ∨A\nc↓\nA\nA ∧[B ∨C]\ns (A ∧B) ∨C"},{"paragraph_id":"p9","order":9,"text":"KSg\ninteraction\nweakening\ncontraction\nswitch\nor identity\nFigure 2. Systems SKSg and KSg.\nAtomic structural rules\nLogical rules\nSKS"},{"paragraph_id":"p10","order":10,"text":"a ∧ ̄a\nai↑\nf\na\naw↑t\na\nac↑a ∧a\ncointeraction\ncoweakening\ncocontraction\nor cut\nt\nai↓a ∨ ̄a\nf\naw↓a\na ∨a\nac↓\na\nA ∧[B ∨C]\ns (A ∧B) ∨C\n(A ∧B) ∨(C ∧D)\nm [A ∨C] ∧[B ∨D]"},{"paragraph_id":"p11","order":11,"text":"KS\ninteraction\nweakening\ncontraction\nswitch\nmedial\nor identity\nFigure 3. Systems SKS and KS.\nExample 2.14. This is a valid derivation in all CoS proof systems defined previously (and\nit plays a role in the proof of Lemma 3.11):\nγ ∨[(([ ̄α ∨α] ∧c) ∧(α ∧d)) ∨δ]\n= γ ∨[(((α ∧d) ∧c) ∧[α ∨ ̄α]) ∨δ]\ns γ ∨[[(((α ∧d) ∧c) ∧α) ∨ ̄α] ∨δ]\n= [ ̄α ∨γ] ∨[((α ∧c) ∧(α ∧d)) ∨δ]\n.\nNote that SKSg, KSg, SKS, and KS are closed under renaming and substitution (see\nRemark 2.11). This is so because of the distinction between atoms and formula variables.\nObtaining the closure of these and other systems under renaming and substitution is one of\nthe main technical reasons for distinguishing between atoms and variables.\nThe following theorem is proved in [Br ̈u04], and follows immediately from Section 3.1,\nwhere we prove that CoS systems p-simulate Gentzen systems.\nTheorem 2.15. (Br ̈unnler)\nSystems SKSg, KSg, SKS, and KS are complete; systems\nSKSg and SKS are implicationally complete.\nThe theorem holds also when restricting the language to ground derivations, since sys-\ntems SKS and KS apply to them.\nIn the presence of cut, the coweakening and cocontraction rules do not play a major role\nin terms of proof complexity:\nTheorem 2.16. Systems SKSg and KSg∪i↑are p-equivalent, and systems SKS and KS∪\nai↑are p-equivalent."},{"paragraph_id":"p12","order":12,"text":"8\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nProof. Observe that the rules w↑and c↑can be derived in KSg ∪i↑:\nA\n= A ∧[f ∨t]\ns (A ∧f) ∨t\nw↓(A ∧ ̄A) ∨t\ni↑\nf ∨t\n=\nt\nand\nA\n= A ∧t\ni↓A ∧[[ ̄A ∨ ̄A] ∨(A ∧A)]\ns (A ∧[ ̄A ∨ ̄A]) ∨(A ∧A)\nc↓\n(A ∧ ̄A) ∨(A ∧A)\ni↑\nf ∨(A ∧A)\n=\nA ∧A\n.\nSimilar constructions hold in KS ∪i↑for aw↑and ac↑.\n□\nIt turns out that all the systems mentioned in the previous theorem are p-equivalent, as\na consequence of Corollary 2.23.\nAnalytic systems are formally defined in Definition 2.13 for CoS, and 3.2 for Gentzen.\nThose definitions are specific to different systems in different formalisms, which is not\nnecessarily satisfactory. Defining a general, syntax-independent concept of analyticity is a\nsubject of ongoing research (see Problem 6.4). We briefly discuss now the notion of ana-\nlyticity and its connections with the proof complexity of deep inference, as an introduction\nto our result on Statman tautologies.\nA Gentzen system is said to be analytic when it does not contain the cut rule. Analytic\nGentzen systems enjoy the ‘subformula property’, i.e., proofs in these systems only contain\nsubformulae of their conclusions. In fact, we might stipulate that enjoying the subformula\nproperty is a primitive notion of analyticity, which we can use to exclude the cut rule, as\ndesired. In analytic Gentzen proofs, all formulae have lower or equal complexity than\nthat of the conclusion, when complexity is measured, for example, as the and/or depth of a\nformula (i.e., the number of alternations of conjunction and disjunction; see Definition 6.3).\nThere is another property, of interest to us, that analytic Gentzen systems enjoy: given an\ninference rule and its conclusion, there are only finitely many premisses to choose from;\nwe call such rules ‘finitely generating’. The cut rule in Gentzen does not possess the\nsubformula property nor is it finitely generating.\nThe primitive notion of analyticity that we are currently adopting for CoS is different\nfrom the one for Gentzen. We stipulate that a rule is analytic if its premiss is a formula\nobtained from a formula scheme by instantiating it with subformulae of the conclusion\n(so, the premiss is not just a subformula of the conclusion). This means that no atom or\nvariable can appear in the premiss of an analytic rule that does not appear in its conclusion.\nIt is, of course, a weaker condition than asking for the subformula property of Gentzen\nsystems, but doing so is necessary if we want to adopt deep inference and obtain linear\nrules. Like the subformula property does for Gentzen, this weaker notion for CoS excludes\nthe cut rule, but also the coweakening one. However, this is not an important difference\nwith the sequent calculus because coweakening is irrelevant for the proof complexity of\na CoS system (see, for example [GG08]). The reason for dealing with coweakening is\nthat, given the potential importance of cocontraction for proof complexity, we preferred to\nintroduce top-down-symmetric CoS systems (so, closed by duality), even if coweakening\nand cocontraction are not required for completeness.\nSo, the two notions of analyticity, for Gentzen and for CoS, are such that the only\nimportant rules that are not analytic are the respective cut rules. Note that in both cases,\nanalytic systems are made of finitely generating rules. However, there is an important\ndifference: in CoS, the complexity of formulae in an analytic proof can be unboundedly"},{"paragraph_id":"p13","order":13,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n9\ngreater than the complexity of the conclusion. Consider, for example, the derivation\nc ∨(a ∧[b ∨[c ∨(a ∧b)])\ns c ∨[(a ∧b) ∨[c ∨(a ∧b)]]\n= [c ∨(a ∧b)] ∨[c ∨(a ∧b)]\nc↓\nc ∨(a ∧b)\n.\nThe and/or depth of the conclusion is 1, while that of the premiss is 3. We could repeat the\nconstruction on top of itself and further increase the and/or depth of the premiss at will.\nDeep-inference systems can be top-down symmetric in the sense that a derivation can\nbe flipped upside-down and negated and still be a valid derivation (we say that two such\nderivations are dual). Accordingly, some forms of analyticity can be defined in a symmetric\nway. Then, typically asymmetric theorems that depend on the notion of analyticity, like\ncut elimination, can be generalised to symmetric statements that imply cut elimination.\nThis is not the place to be detailed about this aspect; suffice to say that we can obtain for\nCoS systems much stronger normalisation (and cut elimination) results than for Gentzen\nsystems (see [Br ̈u06b, GG08]).\nAs we see in Section 3.1, analyticity in CoS faithfully captures analyticity in Gentzen,\nin the sense that analytic CoS can produce isomorphic proofs to Gentzen ones (almost\namounting to a change of notation). However, analytic CoS admits more proofs than an-\nalytic Gentzen, and, among CoS proofs, we can find some remarkably small ones, which\nanalytic Gentzen cannot express; this is the subject of Section 3.2 on Statman tautologies.\nRemark 2.17. The rules of SKS are local, in the sense that, for any language with a\nfinite number of atoms, checking that a given expression is an instance of any of these\nrules requires time bounded by a constant (adopting a tree representation of formulae,\nfor example). This property is peculiar to deep inference; it cannot be obtained in other\nformalisms. For example, a traditional, nonatomic contraction rule is not local because\nit requires checking the identity of two unbounded formulae. Contrary to other nonlocal\nrules, like identity in a Gentzen system, contraction cannot be replaced by its local, atomic\ncounterpart without losing completeness. A counterexample showing this is in [Br ̈u03b].\nLocality can possibly lead to a new, general, productive notion of analyticity, as argued in\nProblem 6.4.\nWe conclude the section by showing the p-equivalence of systems with atomic rules to\nsystems without atomic rules. We start by proving the result on ground derivations.\nLemma 2.18. For every ground instance\nt\ni↓α ∨ ̄α there is a derivation\nt\nΦ ∥\n∥{ai↓,s}\nα ∨ ̄α\nand for\nevery ground instance\nα ∧ ̄α\ni↑\nf\nthere is a derivation\nα ∧ ̄α\nΦ ∥\n∥{ai↑,s}\nf\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. Let us see the case for i↑, the other being its dual. We make an induction on the\nstructure of α. The cases when α is a unit or an atom are trivial: in the former case Φ\nconsists of an instance of = and in the latter the instance of i↑is also an instance of ai↑. We\nonly have to consider the case when α ≡β ∨γ: we apply the induction hypothesis on the\nderivation\n[β ∨γ] ∧( ̄β ∧ ̄γ)\n= ( ̄β ∧[β ∨γ]) ∧ ̄γ\ns [( ̄β ∧β) ∨γ] ∧ ̄γ\ni↑\n[f ∨γ] ∧ ̄γ\n=\nγ ∧ ̄γ\ni↑\nf\n,"},{"paragraph_id":"p14","order":14,"text":"10\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nand we obtain a derivation whose length is O(n), and so its size is O(n2), where n = |α|.\n□\nLemma 2.19. For every ground instance\nf\nw↓α there is a derivation\nf\nΦ ∥\n∥{aw↓,s}\nα\nand for\nevery ground instance\nα\nw↑t there is a derivation\nα\nΦ ∥\n∥{aw↑,s}\nt\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. The proof is similar to the one for Lemma 2.18. In case α ≡t an instance of w↓\nyields\nf\n= f ∧[f ∨t]\ns (f ∧f) ∨t\n=\nt\n;\nwe can do similarly if α ≡f in an instance of w↑. These are the derivations for the inductive\ncases about w↓(those about w↑are dual):\nf\nw↓γ\n= f ∨γ\nw↓β ∨γ\nand\nf\n= f ∧f\nw↓f ∧γ\nw↓β ∧γ\n.\nFrom these we obtain a derivation whose length is O(n), and so its size is O(n2), where\nn = |α|.\n□\nRemark 2.20. In the statement of Lemma 2.19, instead of {aw↓, s} and {aw↑, s} we could\nhave used {aw↓, m} and {aw↑, m}, respectively.\nLemma 2.21. For every ground instance\nα ∨α\nc↓\nα\nthere is a derivation\nα ∨α\nΦ ∥\n∥{ac↓,m}\nα\nand for\nevery ground instance\nα\nc↑α ∧α there is a derivation\nα\nΦ ∥\n∥{ac↑,m}\nα ∧α\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. The proof is similar to the one for Lemma 2.18. These are the derivations for the\ninductive cases about c↓(those about c↑are dual):\n[β ∨γ] ∨[β ∨γ]\n= [β ∨β] ∨[γ ∨γ]\nc↓\nβ ∨[γ ∨γ]\nc↓\nβ ∨γ\nand\n(β ∧γ) ∨(β ∧γ)\nm [β ∨β] ∧[γ ∨γ]\nc↓\nβ ∧[γ ∨γ]\nc↓\nβ ∧γ\n.\nFrom these we obtain a derivation whose length is O(n), and so its size is O(n2), where\nn = |α|.\n□\nTheorem 2.22. For every ground SKSg derivation Φ there is a ground SKS derivation\nΦ′ with the same premiss and conclusion of Φ; if n is the size of Φ then the size of Φ′ is\nO(n2); moreover, if Φ is in KSg then Φ′ is in KS.\nProof. The theorem follows immediately from Lemmas 2.18, 2.19, and 2.21.\n□\nBy Remark 2.12, every derivation can be ‘grounded’, so:\nCorollary 2.23. KS and KSg are p-equivalent and SKS and SKSg are p-equivalent."},{"paragraph_id":"p15","order":15,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n11\nφ, A\n ̄A, ψ\ncut\nφ, ψ\ncut\nid A, ̄A\nt t\nφ\nw φ, A\nφ, A, A\nc\nφ, A\nφ, A, B\n∨\nφ, A ∨B\nφ, A\nB, ψ\n∧\nφ, A ∧B, ψ\nidentity\ntrue\nweakening\ncontraction\ndisjunction\nconjunction\nFigure 4. System Gentzen.\nRemark 2.24. Sometimes, we use nonatomic structural rule instances in SKS and KS\nderivations: those instances actually stand for the SKS and KS derivations that would\nbe obtained according to the proofs of Lemmas 2.18, 2.19, and 2.21. In this sense, we\nsay that i↓, i↑, w↓, w↑, c↓, and c↑are ‘macro’ rules for SKS and KS. The reason we\nmight want to work with macro rules in SKS and KS instead of working in SKSg and\nKSg and then appealing to Theorem 2.22 is to obtain finer upper bounds. This is because\nthe size of formulae over which nonatomic structural rules operate can be much smaller\nthan the square root of the size of a derivation, which is the pessimistic assumption of\nTheorem 2.22.\nRemark 2.25. All implicationally complete CoS proof systems are p-equivalent. This can\nbe proved analogously to, or resorting to, a similar ‘robustness’ result for Frege systems\n(Theorem 4.2), as argued in Remark 4.13. This means that studying proof complexity for\nSKSg and SKS has universal value for all CoS systems for propositional logic.\n3. Calculus of Structures, Gentzen Proof Systems and Statman Tautologies\nThere are two parts in this section. In the first part, we show how CoS naturally p-\nsimulates Gentzen systems, and in particular how it realizes Gentzen’s notion of analyticity.\nIn the second part, we show that analytic CoS admits polynomial proofs when analytic\nGentzen only has exponential ones, in the case of Statman tautologies.\n3.1. Calculus of Structures and Gentzen Proof Systems. In this section, we adopt a\nspecific one-sided (Gentzen-Sch ̈utte) sequent system that we call Gentzen (and that is\ncalled GS1p in [TS96]). We could have adopted any other style of presentation without\naffecting our results. In fact, for Gentzen systems an analogous ‘robustness’ theorem to\nthat for Frege systems (Theorem 4.2) can be established. This means that studying the\nproof complexity of Gentzen has universal value for the class of Gentzen systems.\nDefinition 3.1. Over the language of SKS formulae, the sequent-calculus proof system\nGentzen is defined by the inference rules in Figure 4, where φ and ψ stand for multisets\nof formulae and the symbol ‘,’ represents multiset union. We interpret multisets of for-\nmulae as their disjunction (where associativity is irrelevant). Derivations, denoted by ∆,\nare trees obtained by composing instances of inference rules; the leaves of a derivation are\nits premisses and the root is its conclusion; a derivation ∆with premisses φ1, ..., φh and\nconclusion ψ is denoted by\nφ1 . . . φh\n∆\nψ\n.\nA derivation with no premisses is a proof. The size |∆| of derivation ∆is the number of\nunit, atom, and variable occurrences appearing in it. In the following, every SKS formula\nis translated into a Gentzen formula in the obvious way, and vice versa; in particular, we\ntranslate a Gentzen multiset φ = α1, . . ., αh into α1 ∨· · · ∨αh."},{"paragraph_id":"p16","order":16,"text":"12\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nIn the language, we keep the distinction between atoms and variables because, thanks\nto atoms, we obtain a better upper bound for the size of Statman tautologies proofs, in\nthe Section 3.2. As we said in the case of CoS, the redundancy in the language has no\nconsequences outside of the possibility of using certain CoS rules instead of others.\nDefinition 3.2. The proof system analytic Gentzen is proof system Gentzen without the\ncut rule; analytic derivations and proofs are those derivations and proofs in Gentzen where\nno instances of the cut rule appear.\nWe know, of course, that both Gentzen and analytic Gentzen are complete, and that\nGentzen proofs can be transformed into analytic Gentzen proofs by a cut-elimination\nprocedure, which, in general, blows-up a given proof exponentially.\nEvery Gentzen derivation has natural counterparts in CoS: the idea is to (arbitrarily)\nsequentialise its tree structure. This is possible because the natural logical relation between\ntree branches is conjunction, which CoS can represent, of course. In doing so, we pay\nin terms of complexity because the tree structure is less redundant than CoS contexts:\nthe size of derivations grows quadratically. Other deep-inference formalisms (currently\nunder development, see [BL05, Gug04, Gug05]) are more efficient than CoS and Gentzen\nformalisms in dealing with this so-called ‘bureaucracy’.\nRemark 3.3. In the following, we assume that an empty conjunction can be represented\nby a nonempty conjunction of t units.\nTheorem 3.4. For every Gentzen derivation ∆with premisses φ1, ..., φh and conclusion\nψ there is a derivation\nφ1 ∧· · · ∧φh\nΦ ∥\n∥SKSg\nψ\n; if n is the size of ∆, the size of Φ is O(n2); moreover,\nif ∆is analytic then Φ is in KSg.\nProof. We proceed by induction on the tree structure of ∆. The base cases id A, ̄A and t t\nare, respectively, translated into\nt\ni↓A, ̄A and t. The derivations\nφ1 . . . φh\n∆1\nφ\nw φ, A\n,\nφ1 . . . φh\n∆1\nφ, A, A\nc\nφ, A\n,\nand\nφ1 . . . φh\n∆1\nφ, A, B\n∨\nφ, A ∨B\nare, respectively, translated into\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ\n= φ ∨f\nw↓φ ∨A\n,\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ ∨[A ∨A]\nc↓\nφ ∨A\n,\nand\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ ∨[A ∨B]\n,\nwhere Φ1 is obtained by induction from ∆1, and some possibly necessary instances of the\n= rule have been omitted (they depend on the exact translation of Gentzen multisets into\nSKSg formulae). The derivations\nφ1 . . . φh\n∆1\nφ, A\nφh+1 . . . φk\n∆2\nB, ψ\n∧\nφ, A ∧B, ψ\nand\nφ1 . . . φh\n∆1\nφ, A\nφh+1 . . . φk\n∆2\n ̄A, ψ\ncut\nφ, ψ"},{"paragraph_id":"p17","order":17,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n13\nare, respectively, translated into\nφ1 ∧· · · ∧φk\nΦ1∧Φ2 ∥\n∥\n[φ ∨A] ∧[B ∨ψ]\n= [B ∨ψ] ∧[A ∨φ]\ns ([B ∨ψ] ∧A) ∨φ\n= φ ∨(A ∧[B ∨ψ])\ns φ ∨[(A ∧B) ∨ψ]\nand\nφ1 ∧· · · ∧φk\nΦ1∧Φ2 ∥\n∥\n[φ ∨A] ∧[ ̄A ∨ψ]\n= [ ̄A ∨ψ] ∧[A ∨φ]\ns ([ ̄A ∨ψ] ∧A) ∨φ\n= φ ∨(A ∧[ ̄A ∨ψ])\ns φ ∨[(A ∧ ̄A) ∨ψ]\ni↑\nφ ∨[f ∨ψ]\n=\nφ ∨ψ\n,\nwhere Φ1 and Φ2 are obtained by induction from ∆1 and ∆2, some possibly necessary\ninstances of the = rule have been omitted, and Φ1 ∧Φ2 stands for the derivation\nφ1 ∧· · · ∧φk\nΦ1∧(φh+1∧···∧φk) ∥\n∥\n[φ ∨A] ∧(φh+1 ∧· · · ∧φk)\n[φ∨A]∧Φ2 ∥\n∥\n[φ ∨A] ∧[B ∨ψ]\n,\nwhere B is possibly instantiated by ̄A; the length of this derivation and the size of the largest\nformula appearing in it are both O(n). The resulting O(n2) measure of these last two cases\ndominates the others.\n□\nCorollary 3.5. SKSg p-simulates Gentzen and KSg p-simulates analytic Gentzen.\nAlthough it does not explicitly address complexity, [Br ̈u04] is more exhaustive than the\naforesaid on the two-way translation between SKSg and Gentzen. Translating from SKSg\nto Gentzen crucially employs the cut rule: for every inference step in SKSg, a cut instance\nis used in Gentzen. So, while it is very natural and easy to show that Gentzen p-simulates\nSKSg (see [Br ̈u04]), we are left with the question: does analytic Gentzen p-simulate KSg?\n3.2. Analytic Calculus of Structures on Statman Tautologies. We prove here that an-\nalytic Gentzen does not p-simulate KSg. In fact, the CoS (polynomial) inefficiency in\ndealing with context bureaucracy is compensated by its freedom in applying inference\nrules, which leads to exponential speedups on certain classes of tautologies. Here, we\nstudy Statman tautologies, which have been used to provide the classic lower bound for\nanalytic Gentzen systems: no cut-free proofs of Statman tautologies are possible in an-\nalytic Gentzen without the proofs growing exponentially over the size of the tautologies\nthey prove [Sta78]. We show that, on the contrary, KSg and KS prove Statman tautologies\nwith polynomially growing analytic proofs.\nRemark 3.6. The subset of SKSg only containing analytic rules is equal to KSg plus\nthe cocontraction rule. We do not know whether cocontraction provides for exponential\nspeedups, so separating, proof-complexity-wise, the class of KSg from that of ‘analytic\nCoS’; about this, see Problem 6.2. In our opinion, the very notion of analyticity would\nbenefit from some further analysis; about this, see Problem 6.4.\nDefinition 3.7. For n ≥1, consider the following formulae:\nαi ≡ ̄ci ∨ ̄di\nfor i ≥1,\nβn\nk ≡Vk\ni=n αi ≡αn ∧βn−1\nk\nfor n ≥k > 1,\nγn\nk ≡βn\nk+1 ∧ck\nfor n > k ≥1,\nδn\nk ≡βn\nk+1 ∧dk\nfor n > k ≥1."},{"paragraph_id":"p18","order":18,"text":"14\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nStatman tautologies are, for n ≥1, the formulae:\nSn ≡ ̄αn ∨[(γn\nn−1 ∧δn\nn−1) ∨[· · · ∨[(γn\n1 ∧δn\n1) ∨α1] · · ·]]\n.\nExample 3.8. These are the first three Statman tautologies:\nS1 ≡(c1 ∧d1) ∨[ ̄c1 ∨ ̄d1]\n,\nS2 ≡(c2 ∧d2) ∨[(([ ̄c2 ∨ ̄d2] ∧c1) ∧([ ̄c2 ∨ ̄d2] ∧d1)) ∨[ ̄c1 ∨ ̄d1]]\n,\nS3 ≡(c3 ∧d3) ∨[(([ ̄c3 ∨ ̄d3] ∧c2) ∧([ ̄c3 ∨ ̄d3] ∧d2)) ∨\n[((([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) ∧c1) ∧(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) ∧d1)) ∨\n[ ̄c1 ∨ ̄d1]]]\n.\nIt is perhaps easier to understand their meaning by using implication, as in\nS′\n3 ≡[ ̄c3 ∨ ̄d3] →([([ ̄c3 ∨ ̄d3] → ̄c2) ∨([ ̄c3 ∨ ̄d3] → ̄d2)] →\n([(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) → ̄c1) ∨(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) → ̄d1)] →\n[ ̄c1 ∨ ̄d1]))\n.\nRemark 3.9. |Sn| = 2 + 2 Pn−1\nk=1 |γn\nk| + 2 = 2 Pn\nk=2(|βn\nk| + 1) + 4 = 2n2 + 2.\nIt is not difficult to see why analytic Gentzen proofs of Statman tautologies grow expo-\nnentially (this is a classic argument that can be found in many textbooks; see, for example,\n[CK02]). Basically, all what analytic Gentzen can do while building a proof of Sn is to\ngenerate a proof tree with O(2n) branches. The next lemma shows the crucial advantage\nof deep inference over Gentzen systems: Statman tautologies can be proved ‘from the in-\nside out’, which is precisely what Gentzen systems can only do by resorting to convoluted\nproofs involving cuts (so, nonanalytic proofs).\nRemark 3.10. In the following, for brevity, we label inference steps with expressions like\nn · ν, to denote n inference steps involving rule ν.\nLemma 3.11. For Statman tautologies Sn and Sn+1 there exists a derivation\nSn\n∥\n∥KS\nSn+1\nwhose\nlength is O(n) and size is O(n3).\nProof. We refer to Definition 3.7. The requested derivation is\nΦ =\n(cn ∧dn) ∨\n[((βn\nn ∧cn−1) ∧(βn\nn ∧dn−1)) ∨[· · · ∨\n[((βn\n2 ∧c1) ∧(βn\n2 ∧d1)) ∨α1] · · ·]]\n2n · i↓(([αn+1 ∨ ̄αn+1] ∧cn) ∧([αn+1 ∨ ̄αn+1] ∧dn)) ∨\n[((([αn+1 ∨ ̄αn+1] ∧βn\nn) ∧cn−1) ∧(([αn+1 ∨ ̄αn+1] ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[((([αn+1 ∨ ̄αn+1] ∧βn\n2) ∧c1) ∧(([αn+1 ∨ ̄αn+1] ∧βn\n2) ∧d1)) ∨α1] · · ·]]\n2n · s\n2n\nz }| {\n[ ̄αn+1 ∨[· · · ∨[ ̄αn+1 ∨ ̄αn+1] · · ·]] ∨\n[((αn+1 ∧cn) ∧(αn+1 ∧dn)) ∨\n[(((αn+1 ∧βn\nn) ∧cn−1) ∧((αn+1 ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[(((αn+1 ∧βn\n2) ∧c1) ∧((αn+1 ∧βn\n2) ∧d1)) ∨α1] · · ·]]]\n(2n −1) · c↓ ̄αn+1 ∨\n[((αn+1 ∧cn) ∧(αn+1 ∧dn)) ∨\n[(((αn+1 ∧βn\nn) ∧cn−1) ∧((αn+1 ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[(((αn+1 ∧βn\n2) ∧c1) ∧((αn+1 ∧βn\n2) ∧d1)) ∨α1] · · ·]]]\n,\nwhere we use macro inference rules as explained in Remark 2.22. (Example 2.14 explains\nthe central step in the preceding derivation.) The formulae appearing in the middle of the\nprevious derivation are the largest. Since |αn+1| = 2 and |βn\nk| = 2(n −k + 1), their size is"},{"paragraph_id":"p19","order":19,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n15\n2n·2+6+2(3n−3+Pn\nk=2 |βn\nk|)+2 = 2n2+8n+2. Each i↓macro inference step involves one s\nand two ai↓steps in KS, for a total of six steps, including = ones; each c↓macro inference\nstep involves one m and two ac↓steps in KS, for a total of six steps. So, the length of Φ is\n2n · 6 + 2n · 2 + (2n −1) · 6 = 28n −6, and so |Φ| ≤(28n −6)(2n2 + 8n + 2) ∈O(n3).\n□\nNote that in the previous proof, by working with macro inference rules, we get a better\nupper bound for KS than if we worked in KSg and then applied Theorem 2.22.\nTheorem 3.12. There are KS proofs of Statman tautologies whose size is quadratic in the\nsize of the tautologies they prove.\nProof. Tautology S1 is trivially provable by an instance of the i↓macro rule. By repeatedly\napplying the previous lemma, we obtain proofs of all Statman tautologies Sn, whose size\nis O(n4). Since |Sn| ∈O(n2) (see Remark 3.9), the statement follows.\n□\nThis is enough to conclude that analytic Gentzen does not polynomially simulate KSg\nand KS. Some could argue that Statman tautologies are artificial in their forcing exponen-\ntial Gentzen proofs into ‘wildly’ branching. However, notice that both notions of proof\nand analyticity in Gentzen systems ‘get into tautologies’ from the outside inwards. In\nother words, the restricted notion of analyticity in Gentzen systems is strongly correlated\nto the restricted notion of proof that leads to exponential-size proofs. In CoS, both notions\nare more liberal, to the advantage of proof complexity. We pay a price for this in terms\nof proof-search complexity: there is research aimed at improving the situation, with very\npromising results; see [Kah06a].\nWe note that polynomial proofs on Statman tautologies are obtained by a very small\ndose of deep inference. In fact, the trick is done by the switch and interaction instances in\nthe proof in Lemma 3.11: they all operate just below the ‘surface’ of a formula. This leads\nus to state a currently open problem, in Section 6.6.\n4. Calculus of Structures and Frege Systems\nIn this section, we prove that CoS and Frege systems are p-equivalent.\nIn the following definitions about Frege systems, we do not assume that the language of\nformulae coincides with the CoS one, but, as always, this is not a very important issue.\nDefinition 4.1. Given a language of propositional logic formulae built over a complete\nbase of connectives, a Frege (proof) system is a finite collection of sound inference rules,\neach of which is a tuple of n > 0 formulae such that from n −1 premisses one conclusion\nis derived; inference rules with 0 premisses are called axioms. Given a Frege system, a\nFrege derivation of length l with premisses α1, ..., αh and conclusion βl is a sequence of\nformulae β1, . . ., βl, such that each βi either belongs to {α1, . . ., αh} or is the conclusion of\nan instance of an inference rule whose premisses belong to β1, . . ., βi−1, where 1 ≤i ≤l;\na Frege proof of β is a Frege derivation with no premisses and conclusion β; we use Υ for\nderivations. We require of each Frege system to be implicationally complete, i.e., whenever\n(α1 ∧· · · ∧αh) →β is valid there is a derivation with premisses α1, ..., αh and conclusion\nβ in the proof system. The size of a Frege derivation Υ is the number of unit, atom, and\nvariable occurrences that it contains, and is indicated by |Υ|.\nThe following ‘robustness’ theorem can easily be proved.\nTheorem 4.2. (Robustness, Cook-Reckhow, [CR79])\nAll Frege systems in the same lan-\nguage are p-equivalent.\nThe theorem has been generalised by Reckhow to Frege systems in any language (under\n‘natural translations’) [Rec76], but we do not need this level of generality in our article.\nThe robustness theorem allows us to work with just one Frege system, and we arbitrarily\nchoose the following, taken from [Bus87] and modified by adding axioms F14, F15, F16,\nand F17 in order to deal with units."},{"paragraph_id":"p20","order":20,"text":"16\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nAxioms:\nF1 ≡A →(B →(A ∧B))\nF2 ≡(A ∧B) →A\nF3 ≡(A ∧B) →B\nF4 ≡A →[A ∨B]\nF5 ≡B →[A ∨B]\nF6 ≡¬¬A →A\nF7 ≡A →¬¬A\nF8 ≡A →(B →A)\nF9 ≡¬A →(A →B)\nF10 ≡(A →(B →C)) →((A →B) →(A →C))\nF11 ≡(A →C) →((B →C) →([A ∨B] →C))\nF12 ≡(A →(B →C)) →(B →(A →C))\nF13 ≡(A →B) →(¬B →¬A)\nF14 ≡f →(A ∧¬A)\nF15 ≡(A ∧¬A) →f\nF16 ≡t →[A ∨¬A]\nF17 ≡[A ∨¬A] →t\nInference rule:\nA\nA →B\nmp\nB\nFigure 5. System Frege.\nDefinition 4.3. Frege system Frege, over the language of formulae freely generated by\nunits, non-negated formula variables, and the connectives ∨, ∧, →, and ¬, has inference\nrules as shown in Figure 5, where the formulae F1, ..., F17 are axioms and the inference\nrule mp is called modus ponens.\nRemark 4.4. In the following, every SKS formula is implicitly translated into a Frege\nformula in the obvious way, and vice versa; in particular, we translate Frege’s formulae of\nthe kind α →β into SKS formulae ̄α ∨β.\nRemark 4.5. In Frege systems, distinguishing atoms from formula variables is unnec-\nessary because we only instantiate rules by general substitution (as opposed to renam-\ning). So, from now on, we assume that CoS atoms correspond to Frege formula vari-\nables. Since in system Frege we have a connective for negation, we can also assume that\nwhen dual atoms and formula variables appear in an SKSg formula, their Frege translation\nonly uses ¬; for example, the SKSg formula [A ∨ ̄A] ∧[a ∨ ̄a] is translated into Frege for-\nmula [A ∨¬A] ∧[B ∨¬B] or [A ∨¬A] ∧[¬B ∨B] or [¬A ∨A] ∧[B ∨¬B] or [¬A ∨A] ∧[¬B ∨B].\nConversely, Frege formula [A ∨¬A] ∧[B ∨¬B] is translated into SKSg or SKS formula\n[a ∨ ̄a] ∧[b ∨ ̄b] or [a ∨ ̄a] ∧[ ̄b ∨b] or [ ̄a ∨a] ∧[b ∨ ̄b] or [ ̄a ∨a] ∧[ ̄b ∨b] or one such for-\nmula with formula variables in the place of some of the atoms. As always, we use atoms\nwhen we need to use SKS atomic structural rules, we use formula variables when we need\nto instantiate formulae and derivations, and otherwise we can choose both.\nTranslating Frege into SKSg derivations is straightforward, given that the cut rule of\nSKSg can easily simulate modus ponens.\nTheorem 4.6. For every Frege derivation Υ with premisses α1, ..., αh, where h ≥0, and\nconclusion β, there is a derivation\nα1 ∧· · · ∧αh\nΦ ∥\n∥SKSg\nβ\n; if l and n are, respectively, the length and\nsize of Υ, then the length and size of Φ are, respectively, O(l) and O(n2)."},{"paragraph_id":"p21","order":21,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n17\nProof. The axioms Fi of Frege are tautologies, so each one has a proof Φi in SKSg, for\n1 ≤i ≤17; for example F1 and F10 are, respectively, proved by\nΦ1 =\nt\ni↓[ ̄A ∨ ̄B] ∨(A ∧B)\n= ̄A ∨[ ̄B ∨(A ∧B)]\nand\nΦ10 =\nt\ni↓(A ∧(B ∧ ̄C)) ∨[ ̄A ∨[ ̄B ∨C]]\n= (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧t) ∨C]]\ni↓(A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧[A ∨ ̄A]) ∨C]]\ns (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[[( ̄B ∧A) ∨ ̄A] ∨C]]\n= (A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[[ ̄A ∨ ̄A] ∨C]]\nc↓\n(A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[ ̄A ∨C]]\n.\nWe proceed by induction on the length of Υ = β1, . . ., βk, β and we prove the existence\nof a derivation\nα1 ∧· · · ∧αh\nΦ′ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n. The base case k = 0 is as follows: 1) if β is a premiss,\nthen Φ′ = β; 2) if β ≡Fiσ, for some i and σ, then Φ′ = Φiσ. For the inductive step,\ngiven Υk = β1, . . . , βk and\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk\n, where γk is the conjunction of premisses of Υk, we\nconsider the following cases:\n• if β is a premiss, then Φ′ =\nγk ∧β\nΦ′\nk∧β ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n;\n• if β ≡Fiσ, for some i and substitution σ, then Φ′ =\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk\n= (β1 ∧· · · ∧βk) ∧t\n(β1∧···∧βk)∧Φiσ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n;\n• if β is the conclusion of an instance\nβk′\nβk′ →β\nmp\nβ\n, where βk′′ ≡βk′ →β and\n1 ≤k′, k′′ ≤k, then\nΦ′ =\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk′′ ∧· · · ∧βk\nc↑β1 ∧· · · ∧βk′ ∧· · · ∧(βk′′ ∧βk′′) ∧· · · ∧βk\nc↑β1 ∧· · · ∧(βk′ ∧βk′) ∧· · · ∧(βk′′ ∧βk′′) ∧· · · ∧βk\n=\n(β1 ∧· · · ∧βk) ∧(βk′ ∧[ ̄βk′ ∨β])\ns (β1 ∧· · · ∧βk) ∧[(βk′ ∧ ̄βk′) ∨β]\ni↑\n(β1 ∧· · · ∧βk) ∧[f ∨β]\n=\n(β1 ∧· · · ∧βk) ∧β\n,\nwhere, without loss of generality, we assumed k′ < k′′.\nAt every inductive step the length of the SKSg derivation is only increased by an O(1)\nnumber of inference steps. From\nα1 ∧· · · ∧αh\nΦ′ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\nwe can obtain the desired derivation\nα1 ∧· · · ∧αh\nΦ ∥\n∥SKSg\nβ\nby applying once the w↑rule. So, the length of Φ is O(k). From this, and after\ninspecting the aforesaid derivations, it follows that |Φ| ∈O(k2m), where m is the maximum\nsize of a formula appearing in Υ, and so |Φ| ∈O(n2), where |Υ| = n.\n□\nCorollary 4.7. SKSg and SKS p-simulate Frege."},{"paragraph_id":"p22","order":22,"text":"18\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nProof. The statement for SKSg follows from Theorem 4.6, and that for SKS from this and\nCorollary 2.23.\n□\nTranslating derivations from SKSg to Frege requires more effort than the converse,\npartly because of the need to simulate deep inference, and partly because of the large\n‘amount of inference’ of =-rule instances. The next two lemmas take care of these two\nissues.\nLemma 4.8. For every SKS context ξ{ } and formulae α and β, there is a Frege derivation\nwith premiss α→β and conclusion ξ{α}→ξ{β} whose length is O(m) and size is O(n2), where\nm = |ξ{ }| and n = |ξ{α} →ξ{β}|.\nProof. Consider four Frege proofs Υ′, Υ′′, Υ′′′, and Υ′′′′, respectively of the four tautolo-\ngies\n(A →B) →([A ∨C] →[B ∨C])\n,\n(A →B) →([C ∨A] →[C ∨B])\n,\n(A →B) →((A ∧C) →(B ∧C))\n,\n(A →B) →((C ∧A) →(C ∧B))\n.\nWe proceed by induction on the structure of ξ{ }. If ξ{ } ≡ξ1{{ } ∨γ1}, we build Frege\nderivation\nΥ1 = α →β, Υ′{A/α, B/β,C/γ1}, α ∨γ1 →β ∨γ1\n;\nwe build Υ1 similarly if ξ{ } ≡ξ1{{ } ∧γ1} or ξ{ } ≡ξ1{γ1 ∨{ }} or ξ{ } ≡ξ1{γ1 ∧{ }}.\nGiven ξ1{ } ≡ξ2{{ } ∨γ2} or ξ1{ } ≡ξ2{{ } ∧γ2} or ξ1{ } ≡ξ2{γ2 ∨{ }} or ξ1{ } ≡\nξ2{γ2 ∧{ }} we build Υ2 analogously to Υ1, and the premiss of Υ2 is the conclusion of\nΥ1. We proceed this way until we build Υl, whose conclusion is ξ{α} →ξ{β}, where l ≤m.\nWe obtain the desired derivation Υ by concatenating Υ1, ..., Υl. Since the length and size\nof Υ′, Υ′′, Υ′′′, and Υ′′′′ are independent of ξ{ }, α, and β, the length of Υ is O(m) and its\nsize is O(mn), and so O(n2).\n□\nLemma 4.9. For every SKS formulae α and β such that α = β there is a Frege derivation\nwith premiss α, conclusion β, length O(n3), and size O(n4), where n = |α| + |β|.\nProof. Consider the following tautologies, derived from the equations in Figure 1:\n(1)\n[A ∨B] ↔[B ∨A]\n,\n[A ∨f] ↔A\n,\n(A ∧B) ↔(B ∧A)\n,\n(A ∧t) ↔A\n,\n[[A ∨B] ∨C] ↔[A ∨[B ∨C]]\n,\n[t ∨t] ↔t\n,\n((A ∧B) ∧C) ↔(A ∧(B ∧C))\n,\n(f ∧f) ↔f\n,\nwhere each expression corresponds to the two tautologies obtained by orientating each\ndouble implication. Every such tautology can be proved in Frege with a constant-size\nproof, so every instance γ →γ′ of any of these tautologies has a Frege proof of length O(1)\nand size O(m′), where m′ = |γ| + |γ′|. By Lemma 4.8, for every ξ{ } there is a derivation\nwith premiss γ →γ′ and conclusion ξ{γ} →ξ{γ′} whose length is O(m) and size is O(m2),\nwhere m = |ξ{γ} →ξ{γ′}|. By concatenating the proof and derivation so obtained, we can\nbuild a proof of ξ{γ} →ξ{γ′} whose length is O(m) and size is O(m2). By Remark 2.9, we\ncan build a chain of implications\nα ≡α1 →· · · →αh ≡δ ≡βk →· · · →β1 ≡β\n,\nwhere δ is a canonical form for α and β, h + k is O(n2), and each implication αi →αi+1 and\nβi+1 →βi is a tautology of the form ξ{γ} →ξ{γ′}, such that γ →γ′ is an instance of one of\nthe tautologies 1. By concatenating the proofs of every ξ{γ} →ξ{γ′} by mp, we obtain a\nderivation with premiss α, conclusion β, length O(n3), and size O(n4).\n□\nLemma 4.10. For every inference step\nα\nν β , where ν is a rule of SKSg, there is a Frege\nderivation with premiss α, conclusion β, length O(n), and size O(n2), where n = |α| + |β|."},{"paragraph_id":"p23","order":23,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n19\nProof. Each of the following tautologies, corresponding to the inference rules in Figure 2,\ncan be proved in Frege with a constant-size proof:\n(2)\n(A ∧¬A) →f\n,\nA →t\n,\nA →(A ∧A)\n,\nf →[A ∨¬A]\n,\nf →A\n,\n[A ∨A] →A\n,\n(A ∧[B ∨C]) →[(A ∧B) ∨C]\n.\nLet\nα\nν β =\nξ{γ}\nν ξ{δ}, where\nγ\nν δ is an instance of ν. There is a Frege proof Υ of γ →δ,\nwhose length is O(1) and size is O(n), obtained by instantiating the corresponding proof to\nν among those in 2. By Lemma 4.8, there exists a Frege derivation Υ′ with premiss γ →δ,\nconclusion ξ{γ} →ξ{δ}, length O(n), and size O(n2). By concatenating Υ and Υ′ we obtain\na proof Υ′′ of ξ{γ} →ξ{δ}. From Υ′′, by using mp, we obtain the desired derivation with\npremiss α ≡ξ{γ} and conclusion β ≡ξ{δ}.\n□\nTheorem 4.11. For every derivation\nα\nΦ ∥\n∥SKSg\nβ\nthere is a Frege derivation Υ with premiss\nα and conclusion β; if n is the size of Φ, then the length and size of Υ are, respectively,\nO(n4) and O(n5).\nProof. The statement immediately follows from Lemmas 4.9 and 4.10, after assuming that\nthe length of Φ is O(n).\n□\nCorollary 4.12. Frege p-simulates SKSg and SKS.\nRemark 4.13. As evidenced by the proofs of Theorems 4.6 and 4.11, it does not really\nmatter, for establishing the p-simulations, precisely which inference rules are adopted by\nthe CoS and Frege systems. In fact, the simulations work because the simulating systems\nare implicationally complete and their set of proofs is closed under substitution. This way,\nthe constant-size proofs in one system, simulating the rules of the other system, can be\ninstantiated at a linear cost in order to simulate instances of rules. We can then use a ro-\nbustness theorem (see Theorem 4.2) for Frege in order to establish a robustness theorem\nfor CoS, possibly also for systems on mutually different languages: given two implication-\nally complete CoS systems, we p-simulate each in two appropriate Frege systems and use\nFrege robustness.\n5. Extension and Substitution\nIn this section, we show how CoS systems can be extended with the Tseitin exten-\nsion rule and with the substitution rule, analogously to Frege systems. We also show the\np-equivalence of all these systems, as described in the box of the diagram in the Introduc-\ntion. As always, we operate under robustness theorems (relying on the mentioned one,\nTheorem 4.2) that ensure that the proof complexity properties we establish for the specific\nsystems actually hold for the formalisms they belong to.\nDefinition 5.1. An extended Frege (proof) system is a Frege system augmented with the\n(Tseitin) extension rule, which is a rule with no premisses and whose instances A ↔β are\nsuch that the variable A does not appear before in the derivation, nor appears in β or in\nthe conclusion of the proof. We write A < α to state that variables A and ̄A do not appear\nin formula α. The symbol ↔stands for logical equivalence, and the specific syntax of the\nexpressions A ↔β depends on the language of the Frege system in use. In the following,\nwe consider A ↔β a shortcut for (A →β) ∧(β →A). We denote by xFrege the proof system\nwhere a proof is a derivation with no premisses, conclusion αk, and shape\nα1, . . ., αi1−1,\nαi1 ≡\nz }| {\nA1 ↔β1 , αi1+1, . . ., αih−1,\nαih ≡\nz }| {\nAh ↔βh , αih+1, . . . , αk\n,"},{"paragraph_id":"p24","order":24,"text":"20\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nwhere all the conclusions of extension instances αi1, ..., αih are singled out and\nA1 < α1, . . ., αi1−1, β1, αk\n,\n. . .\n,\nAh < α1, . . ., αih−1, βh, αk\n,\nand the rest of the proof is as in Frege.\nRemark 5.2. We could have equivalently defined an xFrege proof of α as a Frege deriva-\ntion with conclusion α and premisses {A1 ↔β1, . . ., Ah ↔βh} such that A1, ̄A1, ..., Ah, ̄Ah\nare mutually distinct and A1 < β1, α and . . . and Ah < β1, . . ., βh, α. Notice that xFrege\nis indeed a proof system in the sense that it proves tautologies. In fact, given the xFrege\nproof just mentioned, we obtain a Frege proof by applying to it, in order, the substitutions\nσh = Ah/βh, ..., σ1 = A1/β1, and by prepending to it proofs of the tautologies β1 ↔β1,\n(β2 ↔β2)σ1, ..., (βh ↔βh)σh−1 · · · σ1. In general, a proof so obtained is exponentially bigger\nthan the xFrege one it derives from.\nSKSg can analogously be extended, but there is no need to create a special rule; we\nonly need to broaden the criterion by which we recognize a proof.\nDefinition 5.3. An extended SKSg proof of α is an SKSg derivation with conclusion α\nand premiss [ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah], where A1, ̄A1, ..., Ah, ̄Ah are\nmutually distinct and A1 < β1, α and . . . and Ah < β1, . . ., βh, α. We denote by xSKSg the\nproof system whose proofs are extended SKSg proofs.\nTheorem 5.4. For every xFrege proof of length l and size n there exists an xSKSg proof\nof the same formula and whose length and size are, respectively, O(l) and O(n2).\nProof. Consider an xFrege proof as in Definition 5.1. By Remark 5.2 and Theorem 4.6,\nthere exists the following xSKSg proof, whose length and size are yielded by 4.6:\n[ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah]\n∥\n∥SKSg\nαk\n.\n□\nAlthough not strictly necessary to establish the equivalence of the four extended for-\nmalisms (see diagram in the Introduction), the following theorem is very easy to prove.\nTheorem 5.5. For every xSKSg proof of size n there exists an xFrege proof of the same\nformula and whose length and size are, respectively, O(n4) and O(n5).\nProof. Consider an xSKSg proof as in Definition 5.3. The statement is an immediate\nconsequence of Theorem 4.11, after observing that there is an O(h)-length and O(hn)-size\nxFrege proof\nA1 ↔β1, . . ., Ah ↔βh, . . . , (A1 ↔β1) ∧· · · ∧(Ah ↔βh)\n.\n□\nCorollary 5.6. Systems xFrege and xSKSg are p-equivalent.\nWe now move to the substitution rule.\nDefinition 5.7. A substitution Frege (proof) system is a Frege system augmented with\nthe substitution rule\nA\nsub Aσ. We denote by sFrege the proof system where a proof is a\nderivation with no premisses, conclusion αk, and shape\nα1, . . ., αi1−1,\nαi1 ≡\nz}|{\nα j1σ1 , αi1+1, . . ., αih−1,\nαih ≡\nz}|{\nα jhσh , αih+1, . . . , αk\n,\nwhere all the conclusions of substitution instances αi1, ..., αih are singled out, α j1 ∈\n{α1, . . ., αi1−1}, ..., α jh ∈{α1, . . . , αih−1}, and the rest of the proof is as in Frege."},{"paragraph_id":"p25","order":25,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n21\nWe rely on the following result.\nTheorem 5.8. (Cook-Reckhow and Kraj ́ıˇcek-Pudl ́ak, [CR79, KP89])\nSystems xFrege\nand sFrege are p-equivalent.\nWe can extend SKSg with the same substitution rule as for Frege. The rule is used like\nother proper rules of system SKSg, so its instances are interleaved with =-rule instances.\nDefinition 5.9. An sSKSg proof is a proof of SKSg where, in addition to the inference\nsteps generated by rules of SKSg, we admit inference steps obtained as instances of the\nsubstitution rule\nA\nsub Aσ.\nThis rule does not fit any of the usual deep-inference rule classes (see Section 2), and (as\nin Frege systems) is not sound, in the sense that the premiss does not imply the conclusion.\nHowever, of course, if the premiss is provable the conclusion also is.\nRemark 5.10. Notice that instances of the substitution rule cannot be used inside a context;\nfor example, the expression on the left is not a valid sSKSg proof, while the one on the\nright is:\nt\ni↓A ∨ ̄A\nsub? (B ∧C) ∨ ̄A\n,\nt\ni↓A ∨ ̄A\nsub (B ∧C) ∨[ ̄B ∨ ̄C]\n.\nIn the so-called ‘Formalism B’ of deep inference, which is currently under development\n[Gug04], and for which all the proof-complexity results in this article apply unchanged,\nsubstitution becomes part of the composition mechanism of proofs, rather than an odd\nextension to the set of rules.\nFor the time being, we can establish the promised p-equivalence of all extended systems\nby completing the diagram in the Introduction with the last two missing steps.\nTheorem 5.11. For every xSKSg proof of size n there exists an sSKSg proof of the same\nformula and whose length and size are, respectively, O(n) and O(n2).\nProof. Consider the xSKSg proof\n[ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah]\nΦ ∥\n∥SKSg\nα\n,\nwhere\nA1 < β1, α\n,\n. . .\n,\nAh < β1, . . . , βh, α\n,\nand let us call its premiss γ. We can build the sSKSg proof\nt\ni↓ ̄γ ∨γ\n ̄γ∨Φ ∥\n∥SKSg\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(Ah ∧ ̄βh) ∨(βh ∧ ̄Ah)] ∨α\nsub [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(βh ∧ ̄βh) ∨(βh ∧ ̄βh)] ∨α\nc↓\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(βh ∧ ̄βh)] ∨α\ni↑\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨f] ∨α\n=\n...\n= [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1)] ∨α\nsub [(β1 ∧ ̄β1) ∨(β1 ∧ ̄β1)] ∨α\nc↓\n(β1 ∧ ̄β1) ∨α\ni↑\nf ∨α\n=\nα\n."},{"paragraph_id":"p26","order":26,"text":"22\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\n□\nCorollary 5.12. sSKSg p-simulates xSKSg.\nTheorem 5.13. For every sSKSg proof of size n there exists a proof of the same formula\nin sFrege, whose length and size are, respectively, O(n4) and O(n5).\nProof. Every sSKSg proof has shape\nt\nΦ0 ∥\n∥SKSg\nα1\nsub α1σ1\nΦ1 ∥\n∥SKSg\n...\nΦh−1 ∥\n∥SKSg\nαh\nsub αhσh\nΦh ∥\n∥SKSg\nαh+1\n.\nBy Theorem 4.11, for each of Φ0, ..., Φh there exist Frege derivations Υ0, ..., Υh with\nthe same premiss and conclusion, respectively. We can then build the proof\nΥ0\nz }| {\n. . . , α1 ,\nΥ1\nz }| {\nα1σ1, . . ., . . . ,\nΥh−1\nz }| {\n. . . , αh ,\nΥh\nz }| {\nαhσh, . . . , αh+1\nin sFrege; the cited theorem also yields its length and size.\n□\nCorollary 5.14. sFrege p-simulates sSKSg.\nNothing prevents us from using Tseitin extension and the substitution rule with system\nSKS, or any other atomic or nonatomic CoS system. The integration of these mechanisms\ninto CoS is similar to their integration into Frege systems, as the simplicity of the argu-\nments showing p-equivalence testifies.\n6. Open Problems\nWe conclude the article with a list of open problems, some of which are currently in-\nvestigated by us and other researchers.\n6.1. Relation with Resolution and Other Formalisms. In this article, we explored the\nrelation between CoS and Frege systems, and in the cited literature the relation between\nCoS and Gentzen systems has been explored in depth. There are, of course, other for-\nmalisms, like resolution, whose relation with CoS might lead to some interesting research\ndirections. For example, the note [Gug03] shows how simply, compared to Gentzen sys-\ntems, KS expresses resolution (analytically, of course).\n6.2. Does Cocontraction Provide for an Exponential Speedup? As we argued in Re-\nmark 3.6, we do not know whether KSg p-simulates KSg ∪{c↑}, or, equivalently, whether\nKS p-simulates KS ∪{ac↑}.\nOur intuition, as well as some clues, like the mutual behaviour of the ‘atomic flows’\nof contraction and cocontraction (see [GG08]) would lead us to believe that cocontraction\nindeed provides for an exponential speedup. However, we know that in similar situations,\nlike for dag-like versus tree-like Frege systems, intuition was fallacious.\nIf cocontraction yields an exponential speedup, we obtain an even stronger analytic\nsystem than KSg, which is, in turn, stronger than analytic Gentzen. This would draw\ninterest to a hierarchy of analytic proof systems of different strength."},{"paragraph_id":"p27","order":27,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n23\nUnless we prove the p-equivalence of KSg and KSg ∪{c↑}, we tend to consider cocon-\ntraction a simple rule-based mechanism for compressing proofs, like cut, extension, and\nsubstitution.\n6.3. Pigeonhole in Analytic CoS. Does the pigeonhole principle, in particular in its rela-\ntional variety, admit polynomially growing proofs in KS? If not, does it in KS ∪{ac↑}?\nInvestigating this problem could be relevant to the more general, following one (Prob-\nlem 6.4), about the ability of analytic CoS to simulate CoS, and so Frege. In fact, the\npigeonhole principle generates some of the hardest classes of tautologies known.\nWe note that in [Jap08], Japaridze shows polynomially growing proofs for the pigeon-\nhole class of tautologies in a deep-inference system over certain circuit-like sequents,\ncalled ‘cirquents’. In this case, the speedup is obtained by the sharing of logical expres-\nsions in circuits.\nIn [Jeˇr09], Jeˇr ́abek shows that there are polynomial-time constructible proofs in KS ∪\n{ac↑} of the functional and onto variants of the pigeonhole principle.\n6.4. Relative Strength of Analytic CoS and CoS. Some recent major progress has been\nmade in [Jeˇr09]. There, Jeˇr ́abek uses a construction on threshold formulae in the mono-\ntone sequent calculus, by Atserias, Galesi and Pudl ́ak [AGP02], to show that analytic CoS\nquasipolynomially simulates CoS. In [BGGP09], we provide a direct and simplified con-\nstruction based on atomic flows [GG08] and threshold formulae.\nBecause of these recent advances, we expect that analytic CoS p-simulates CoS. A more\nin-depth discussion of this subject is in [BGGP09]. If analytic CoS p-simulates CoS, then\nthere are polynomially growing proofs of the pigeonhole principle in analytic CoS, though\nnot necessarily in KS.\nWe think that investigating this problem will help us to better understand analyticity, in\norder to obtain for it a general and more useful definition than the one we have now. We\nfeel that the current notion is not satisfactory because it depends on the formalism and must\nbe defined by resorting to the syntactic structure of inference rules (or, worse, by indicating\nwhich rules are analytic and which are not).\nFor example, a more general, nonsyntactic definition of analyticity could be the follow-\ning: a rule is analytic if, given an instance of its conclusion, the set of possible instances\nof the premiss is finite (this is what we call a finitely generating rule in Section 2). In this\nsense, an atomic ‘finitary’ cut rule\nξ{a ∧ ̄a}\nfai↑\nξ{t}\n, such that a appears in ξ{ }, would be ana-\nlytic. However, [BG04] shows that we can easily transform proofs in SKS into smaller- or\nequal-size proofs that only use fai↑wherever ai↑was used. So, we could deem fai↑an ana-\nlytic rule, and the system obtained from SKS by substituting ai↑with fai↑an analytic one,\nand we could immediately conclude that analytic CoS p-simulates CoS. This ‘solution’,\nhowever, is way too cheap.\nWe prefer to think that fai↑is not an analytic rule, in some sense to be made precise. A\npossible point of attack is offered by the fact that fai↑is not a local rule: it requires checking\nthat a appears in its context, whose size is unbounded (see Remark 2.17). So, we think it\ncould be productive to look for a notion of analyticity that is based on boundedness instead\nof finiteness, and tackle the separation problem between analytic CoS and CoS under that\nnotion. The note [BG07] further explores this direction, but much more work is necessary.\n6.5. Strength of Analytic CoS Systems Plus Substitution. We showed that CoS and\nFrege systems are p-equivalent, and both remain p-equivalent when extended either with\nTseitin extension or substitution. However, CoS is more flexible than Frege, because it\nallows to ‘switch off’ two mechanisms that potentially provide for an exponential com-\npression of proofs: cut and cocontraction (see Problem 6.2).\nIt might be interesting to study the relative strength of systems obtained by removing\nfrom SKS ∪{sub} either ai↑or ac↑or both. (Rule aw↑can also be removed, but we do"},{"paragraph_id":"p28","order":28,"text":"24\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nnot see a crucial role for it.) Notice that systems KS ∪{sub} and KS ∪{ac↑, sub} could be\nconsidered, in some sense, analytic, and we do not know their relative strength.\n6.6. Speedup of Deep Inference Over Any Bounded-Depth System. We saw in The-\norem 3.12 that analytic CoS exhibits an exponential speedup over analytic Gentzen, for\nStatman tautologies. We argued that, in this case, the speedup is obtained by a rather triv-\nial use of deep inference, because the depth at which inference has to be performed, in\norder to get the speedup, is constant. So, a natural question is whether there exists a class\nof tautologies that requires full-fledged deep inference in order to obtain efficient proofs.\nWe think we found such a class, which is defined as follows.\nConsider, for every propositional formula α, the following set of second-order formulae,\nfor n > 0:\ng(1, α) ≡∀β.[((α ∧β) ∧β) ∨( ̄β ∧ ̄β)]\n,\ng(n + 1, α) ≡∀β.[(g(n, α ∧β) ∧g(n, β)) ∨(g(n, ̄β) ∧g(n, ̄β))]\n.\nBy using these formulae as a template, we can generate a set of first-order formulae, where\nthe (complex) management of indices ensures their uniqueness:\nDefinition 6.1. Consider, for m, n ≥0\nh(1, m, α) ≡((α ∧βm+1) ∧βm+1) ∨( ̄βm+1 ∧ ̄βm+1)\n,\nh(n + 2, m, α) ≡(h(n + 1, m, α ∧β5n+1+m) ∧h(n + 1, 5n + m, β5n+1+m)) ∨\n(h(n + 1, 2 · 5n + m, ̄β5n+1+m) ∧h(n + 1, 3 · 5n + m, ̄β5n+1+m))\n.\nConsider now\nf(n) ≡h(n, 0, t)\n,\nfor n > 0\n.\nWe define the set DT = { f(n) | n > 0 }.\nThe program [Gug07a] can help in understanding the nature of these formulae.\nRemark 6.2. It is not difficult to verify that DT contains tautologies possessing analytic\nCoS proofs that grow polynomially in the size of the tautologies.\nThe analytic CoS proofs of the DT tautologies, when read bottom-up, work by applying\ninteractions starting from the deepest subformulae. When this cannot be the case, we\nconjecture that the size of the proofs grows exponentially.\nDefinition 6.3. The and/or depth of a formula is the maximum number of alternations of\nconjunctions and disjunctions in the formula tree; the and/or depth of a context ξ{ } is the\nnumber of alternations of conjunctions and disjunctions between the hole and the root of\nthe context tree. We define a bounded-depth CoS proof system as a CoS proof system\nwhose inference rules only generate inference steps at a bounded depth, namely inference\nsteps\nξ{γ}\nν ξ{δ} are such that, if\nγ\nν δ is a rule instance then the and/or depth of ξ{ } is bounded\nby a given constant, and the same restriction holds for the contexts in the context closure\ncondition of relation =.\nRemark 6.4. Note that the nonatomic rules interaction (identity), cointeraction (cut),\ncontraction and cocontraction require establishing duality or identity of formulae of un-\nbounded and/or depth. So, their adoption might be considered an implicit use of deep in-\nference. However, the atomic counterparts of these rules do not suffer this problem because\nthe ‘deep checking’ is delegated to the inference mechanism. For this reason, proving the\nfollowing conjecture is better done in the analytic part of system SKS.\nConjecture 6.5. In any analytic bounded-depth CoS proof system, the tautologies in DT\nonly have proofs that grow exponentially in their size."},{"paragraph_id":"p29","order":29,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n25\n7. Conclusion\nIn this article, we showed that the calculus of structures (CoS) has the same character-\nistics of the Frege formalism in terms of proof complexity, including when extended with\nTseitin extension and substitution.\nWe know that, contrary to Frege, CoS has a rich proof theory, and its proof systems\nenjoy several properties, arguably relevant to proof complexity, that cannot be observed in\nother formalisms, like locality for all inference rules. We also know that other logics, like\nmodal logics, enjoy simple and modular presentations in deep inference, which should help\nin proof complexity investigations. This article establishes the basic connection between\nproof theory in deep inference and proof complexity.\nAs a consequence of its flexibility in inference rule design, CoS admits a notion of\nanalyticity that is more flexible than its counterpart for Gentzen systems. We can then\nexplore the strength of analytic systems in finer detail than possible in Gentzen systems. In\nthis article, we moved forward the boundary between polynomial and exponential analytic\nproofs by proving Statman tautologies with polynomial, analytic deep-inference proofs.\nWe included a list of open problems and currently active research directions.\nAcknowledgements. 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IEEE,\n2005.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/FreeBool-long.pdf.\n[LS05b]\nFranc ̧ois Lamarche and Lutz Straßburger. Naming proofs in classical propositional logic. In Paweł\nUrzyczyn, editor, Typed Lambda Calculi and Applications, volume 3461 of Lecture Notes in Com-\nputer Science, pages 246–261. Springer-Verlag, 2005.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/namingproofsCL.pdf.\n[LS06]\nFranc ̧ois Lamarche and Lutz Straßburger. From proof nets to the free *-autonomous category. Logical\nMethods in Computer Science, 2(4):3:1–44, 2006.\nhttp://arxiv.org/pdf/cs.LO/0605054.\n[Rec76]\nRobert A. Reckhow. On the Lengths of Proofs in the Propositional Calculus. PhD thesis, University\nof Toronto, 1976.\n[SL04]\nLutz Straßburger and Franc ̧ois Lamarche. On proof nets for multiplicative linear logic with units.\nIn J. Marcinkowski and A. Tarlecki, editors, CSL 2004, volume 3210 of Lecture Notes in Computer\nScience, pages 145–159. Springer-Verlag, 2004.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/multPN.pdf.\n[Sta78]\nRichard Statman. Bounds for proof-search and speed-up in the predicate calculus. Annals of Mathe-\nmatical Logic, 15:225–287, 1978.\n[Sto07]\nPhiniki Stouppa. A deep inference system for the modal logic S5. Studia Logica, 85(2):199–214,\n2007.\nhttp://www.iam.unibe.ch/til/publications/pubitems/pdfs/sto07.pdf.\n[Str02]\nLutz Straßburger. A local system for linear logic. In M. Baaz and A. Voronkov, editors, LPAR 2002,\nvolume 2514 of Lecture Notes in Artificial Intelligence, pages 388–402. Springer-Verlag, 2002.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/lls-lpar.pdf.\n[Str03a]\nLutz Straßburger. Linear Logic and Noncommutativity in the Calculus of Structures. PhD thesis,\nTechnische Universit ̈at Dresden, 2003.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/dissvonlutz.pdf.\n[Str03b]\nLutz Straßburger. MELL in the calculus of structures. Theoretical Computer Science, 309:213–285,\n2003.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/els.pdf.\n[Tiu06a]\nAlwen Tiu. A local system for intuitionistic logic. In M. Hermann and A. Voronkov, editors, LPAR\n2006, volume 4246 of Lecture Notes in Artificial Intelligence, pages 242–256. Springer-Verlag, 2006.\nhttp://users.rsise.anu.edu.au/ ̃tiu/localint.pdf.\n[Tiu06b]\nAlwen Tiu. A system of interaction and structure II: The need for deep inference. Logical Methods\nin Computer Science, 2(2:4):1–24, 2006.\nhttp://arxiv.org/pdf/cs.LO/0512036.\n[TS96]\nA.S. Troelstra and H. Schwichtenberg. Basic Proof Theory, volume 43 of Cambridge Tracts in The-\noretical Computer Science. Cambridge University Press, 1996.\nUniversity of Bath, Bath BA2 7AY, UK,\nhttp://cs.bath.ac.uk/pb/ and\nhttp://alessio.guglielmi.name/res"}],"pages":[{"page":1,"text":"arXiv:0709.1201v3 [cs.CC] 19 Apr 2009\nON THE PROOF COMPLEXITY OF DEEP INFERENCE\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nAbstract. We obtain two results about the proof complexity of deep inference: 1) deep-\ninference proof systems are as powerful as Frege ones, even when both are extended with\nthe Tseitin extension rule or with the substitution rule; 2) there are analytic deep-inference\nproof systems that exhibit an exponential speedup over analytic Gentzen proof systems\nthat they polynomially simulate.\n1. Introduction\nDeep inference is a relatively new methodology in proof theory, consisting in deal-\ning with proof systems whose inference rules are applicable at any depth inside formulae\n[Gug07b]. We obtain two results about the proof complexity of deep inference:\n• deep-inference proof systems are as powerful as Frege ones, even when both are\nextended with the Tseitin extension rule or with the substitution rule;\n• there are analytic deep-inference proof systems that exhibit an exponential speed-\nup over analytic Gentzen proof systems that they polynomially simulate.\nThese results are established for the calculus of structures, or CoS, the simplest formal-\nism in deep inference [Gug07b], and in particular for its proof system SKS, introduced\nby Br ̈unnler in [Br ̈u04] and then extensively studied [Br ̈u03a, Br ̈u03b, Br ̈u06a, Br ̈u06d,\nBG04, BT01].\nOur contributions fit in the following picture.\nCoS +\nextension\nCoS +\nsubstitution\nFrege +\nextension\nFrege +\nsubstitution\n⋆\n4\n3\nKraj ́ıˇcek-Pudl ́ak ’89\n⋆\n5\nCook-Reckhow ’79\nFrege\nCoS\nGentzen\nopen\n2\nCook-\nReckhow ’74\nanalytic\nCoS\nanalytic\nGentzen\nBr ̈unnler\n’04\n1 ×\nStatman ’78\n×\nopen\nThe notation F\nG indicates that formalism F polynomially simulates formalism G ;\nthe notation F\nG\n×\nindicates that it is known that this does not happen.\nThe left side of the picture represents, in part, the following. Analytic Gentzen systems,\ni.e., Gentzen proof systems without the cut rule, can only prove certain formulae, which\nwe call ‘Statman tautologies’, with proofs that grow exponentially in the size of the for-\nmulae. On the contrary, Gentzen systems with the cut rule can prove Statman tautologies\nby polynomially growing proofs. So, Gentzen systems p-simulate analytic Gentzen ones,\nDate: October 24, 2018.\nThis research was partially supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in\nDeep Inference.\nc⃝ACM, 2009. This is the authors’ version of the work. It is posted here by permission of ACM for your\npersonal use. Not for redistribution. The definitive version was published in ACM Transactions on Computational\nLogic 10 (2:14) 2009, pp. 1–34, http://doi.acm.org/10.1145/1462179.1462186.\n1"},{"page":2,"text":"2\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nbut not vice versa [Sta78]. Cook and Reckhow proved that Frege and Gentzen systems\nare polynomially equivalent, i.e., each Frege system polynomially simulates any Gentzen\nsystem and vice versa [CR74].\nIn the box at the right of the figure, ‘extension’ refers to the Tseitin extension rule,\nand ‘substitution’ to the substitution rule. The works of Cook and Reckhow [CR79] and\nKraj ́ıˇcek and Pudl ́ak [KP89] established that Frege + extension and Frege + substitution\nare polynomially equivalent. It is immediate to see that these formalisms polynomially\nsimulate Frege and Gentzen, which, in turn, polynomially simulate analytic Gentzen. It is\na major open problem to establish whether Frege polynomially simulates Frege + exten-\nsion/substitution.\nIn this work, we establish the following results (numbered as in the previous figure):\n(1) Analytic Gentzen does not polynomially simulate analytic CoS (essentially in the\nform of system SKS without cut); in fact, Statman tautologies admit polynomially\ngrowing proofs in analytic CoS (Theorem 3.12).\n(2) CoS and Frege are polynomially equivalent (Theorems 4.7 and 4.12).\n(3) There is a natural notion of (Tseitin) extension for CoS, and CoS + extension is\npolynomially equivalent to Frege + extension (Theorem 5.6).\n(4) There is a natural notion of substitution for CoS, and CoS + substitution polyno-\nmially simulates CoS + extension (Theorem 5.12).\n(5) Frege + substitution polynomially simulates CoS + substitution; this way, we\nknow that all extended formalisms are polynomially equivalent (Theorem 5.14).\nThe polynomial simulations indicated by ⋆arcs in the picture follow from the others.\nEstablishing whether analytic CoS polynomially simulates CoS is an open problem.\nAfter the necessary preliminaries, in Section 2, we see how CoS expresses Gentzen\nsystems, including their properties, like analyticity, and then in Section 3 how it provides\nfor exponentially more compact proofs. The relation between CoS and Frege systems is\nexplored in Section 4 and the extensions are studied in Section 5. We conclude the article\nwith a list of open problems, in Section 6.\n2. Preliminaries\nIn this section, we quickly introduce the necessary deep-inference notions. A more\nextensive treatment of much of this material is in Br ̈unnler’s [Br ̈u04].\nWe only need the following, basic proof complexity notions (see [CR79]).\nDefinition 2.1. A (propositional) proof system is a binary relation S between formulae\nα and proofs Π such that S is computable in polynomial time, and the formula α is a\ntautology if and only if there is a proof Π such that S(α, Π); in this case we say that Π is\na proof of α in S. We say that proof system S p-simulates proof system S′ if there is a\npolynomial-time computable algorithm that transforms every proof in S′ into a proof in S\nof the same tautology. Two proof systems are p-equivalent if each p-simulates the other.\nRemark 2.2. In the following, we state theorems on the existence of proofs in one proof\nsystem when proofs exist in another proof system, such that their size is polynomially\nrelated. Implicitly, we always mean that the new proofs are obtained by transforming the\nold ones by way of a polynomial-time computable algorithm.\nDeep inference is a relatively recent development in proof theory. Its main idea is\nto provide a finer analysis of inference than possible with traditional methods, and one\nof the main objectives is to obtain a geometric semantics for proofs, inspired by linear\nlogic’s proof nets [Gir87]. Another objective is to provide a uniform and useful syntactic\ntreatment of several logics, especially modal ones, for which no satisfactory proof theory\nexisted before.\nIn deep inference, several formalisms can be defined with excellent structural proper-\nties, like locality for all the inference rules. The calculus of structures [Gug07b] is one"},{"page":3,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n3\nof them and is now well developed for classical [Br ̈u03a, Br ̈u06a, Br ̈u06d, BT01], intu-\nitionistic [Tiu06a], linear [Str02, Str03b], modal [Br ̈u06c, GT07, Sto07] and commuta-\ntive/noncommutative logics [Gug07b, Tiu06b, Str03a, Bru02, DG04, GS01, GS02, GS09,\nKah06b, Kah07]; for all these logics, quantification can be defined at any order. We em-\nphasise that deep inference is developing the first reasonable proof theory for modal logics;\nthe large number of different modal logic systems can be studied in simple and modular\ndeep-inference systems, which are similar to their propositional logic counterparts and en-\njoy the same locality properties. The calculus of structures promoted the discovery of a\nnew class of proof nets for classical and linear logic [LS05a, LS05b, LS06, SL04] (see\nalso [Gui06]). Moreover, there exist implementations in Maude of deep-inference proof\nsystems [Kah08].\nIn this article, we focus on the calculus of structures because it is well developed and is\nprobably the simplest formalism definable in deep inference. The complexity results that\nwe present here are not dependent on the choice of formalism; rather, they only depend on\nthe deep-inference methodology and the finer granularity of inference rules that it yields.\nAdopting deep inference basically means that it is possible to replace subformulae inside\nformulae by other, implied subformulae, and that there is no limit to the nesting depth of\nsubformulae. Formalisms like Gentzen’s sequent calculus differ because they only rewrite\nformulae, or sequents, around their root connectives, and (we argue) they suffer excessive\nrigidity in the syntax and they do not sufficiently support geometric semantics.\nBecause of its geometric nature, it is important, in deep inference, to control whether\npropositional variables can be instantiated by formulae. In particular, normalisation (cut\nelimination) in deep inference crucially depends on the availability of ‘atomic’ inference\nrules, which are rules related to some topological invariants (see, for example, [GG08]\nfor normalisation in propositional logic). In practice, we need two kinds of propositional\nvariables: the atoms, only subject to renaming, and the formula variables, subject to (unre-\nstricted) substitution. This distinction does not bear dramatic effects on proof complexity,\nbut it does allow for some finer measures than otherwise possible.\nDefinition 2.3. Formulae of the calculus of structures, or CoS, are denoted by α, β, γ, δ\nand are freely built from: units, like f (false) and t (true); atoms a, b, c, d and ̄a, ̄b, ̄c, ̄d;\n(formula) variables A, B, C, D and ̄A, ̄B, ̄C, ̄D; logical relations, like disjunction [α ∨β]\nand conjunction (α ∧β). A formula is ground if it contains no variables. We usually omit\nexternal brackets of formulae, and sometimes we omit dispensable brackets under associa-\ntivity. We use ≡to denote literal equality of formulae. The size |α| of a formula α is the\nnumber of unit, atom, and variable occurrences appearing in it. On the set of atoms, there\nis an involution ̄·, called negation (i.e., ̄· is a bijection from the set of atoms to itself such\nthat ̄ ̄a ≡a); we require that ̄a . a for every a; when both a and ̄a appear in a formula,\nwe mean that atom a is mapped to ̄a by ̄·. An analogous involution is defined on the set of\nformula variables. The (De Morgan) dual of a formula is obtained by exchanging disjunc-\ntion and conjunction and applying negation to all atoms and variables; we denote duals by\nusing ̄·; for example, the De Morgan dual of α ≡t ∨(a ∧[ ̄B ∨c]) is ̄α ≡f ∧[ ̄a ∨(B ∧ ̄c)]. A\ncontext is a formula where one hole { } appears in the place of a subformula; for example,\nA ∨(b ∧{ }) is a context; the generic context is denoted by ξ{ }. The hole can be filled with\nformulae; for example, if ξ{ } ≡b ∧[{ } ∨c], then ξ{a} ≡b ∧[a ∨c], ξ{b} ≡b ∧[b ∨c] and\nξ{a ∧B} ≡b ∧[(a ∧B) ∨c]. The size of ξ{ } is defined as |ξ{ }| = |ξ{a}| −1.\nRemark 2.4. We do not say that a is positive and ̄a is negative. It only matters that,\nwhen a and ̄a appear in the same formula, if one is negative the other is positive. In\nabsence of disambiguating information, there are two ways in which ξ{b} might correspond\nto b ∧[b ∨c]: one such that ξ{a} ≡a ∧[b ∨c] and another such that ξ{a} ≡b ∧[a ∨c].\nThe language of formulae is redundant because we can choose whether to use atoms\nor formula variables whenever a propositional variable is needed. The distinction between"},{"page":4,"text":"4\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\natoms and formula variables only plays a role in the choice of applicable inference rules,\nand this aspect is controlled by renamings and substitutions.\nDefinition 2.5. A renaming is a map from the set of atoms to itself, and is denoted by\n{a1/b1, a2/b2, . . .}; we use ρ for renamings. A renaming of α by ρ = {a1/b1, a2/b2, . . .}\nis indicated by αρ and is obtained by simultaneously substituting every occurrence of ai\nin α by bi and every occurrence of ̄ai by ̄bi; for example, if α ≡a ∧[b ∨(a ∧[ ̄a ∨C])]\nthen α{a/ ̄b, ̄b/c} ≡ ̄b ∧[ ̄c ∨( ̄b ∧[b ∨C])]. A substitution is a map from the set of for-\nmula variables to formulae, denoted by {A1/β1, A2/β2, . . .}; we use σ for substitutions.\nAn instance of α by σ = {A1/β1, A2/β2, . . . } is indicated by ασ and is obtained by si-\nmultaneously substituting every occurrence of variable Ai in α by formula βi and every\noccurrence of ̄Ai by the De Morgan dual of βi; for example, if α ≡(A ∧[A ∨ ̄A]) ∨b then\nα{A/(c ∧ ̄B)} ≡((c ∧ ̄B) ∧[(c ∧ ̄B) ∨[ ̄c ∨B]]) ∨b.\nDefinition 2.6. A CoS (inference) rule ν is an expression\nα\nν β , where formulae α and β\nare called premiss and conclusion, respectively. A (rule) instance\nγ\nν δ of\nα\nν β is such that\nγ ≡αρσ and δ ≡βρσ, for some renaming ρ and substitution σ. For some context ξ{ }, a\nCoS (inference) step, generated by rule\nα\nν β via its instance\nγ\nν δ , is the expression\nξ{γ}\nν ξ{δ}.\nExample 2.7. Given\nt\ni↓A ∨ ̄A and\nt\nai↓a ∨ ̄a, then\nt\ni↓(A ∧b) ∨[ ̄A ∨ ̄b] is an instance of i↓and\nt\nν b ∨ ̄b is an instance of both i↓and ai↓. The rule\nA ∨A\nc↓\nA\ngenerates the inference step\n[a ∨b] ∧[(c ∧D) ∨(c ∧D)]\nc↓\n[a ∨b] ∧(c ∧D)\n.\nWe usually classify deep-inference rules in three classes. For this, we rely on the notion\nof linearity, which, in this context, essentially means the same as in term-rewriting: a\nrewriting rule is linear if variables appear once in both sides of the rule. In other words, a\nlinear rule does not create or destroy anything. These are the three classes of rules:\n(1) Atomic rules. They usually correspond to structural rules in Gentzen systems; in\nnormalisation and in semantics of proofs, they play a crucial role because they\nexpress the causality relations between atoms, so shaping the geometry of proofs.\nTheir instances are obtained by renaming.\n(2) Noninvertible linear rules. They usually correspond to logical rules in Gentzen\nsystems. Since they are noninvertible, they express proper inference choices, but\nsince they are linear, they do not alter the geometry of causality between atoms.\nTheir instances are obtained by substitution.\n(3) Invertible linear rules. These rules are equivalences between formulae that do not\ncorrespond to proper inference choices and have no impact on the geometry of\nproofs. For this reason, they are usually gathered into one big equivalence relation\nbetween formulae, corresponding to just one rule, defined via substitution.\nThe success of deep inference is due to its ability to separate rules into classes 1 and\n2, which is only possible by adopting deep inference. The references to the ‘geometry of\nproofs’ can be understood by reading [GG08, LS05b]. Class 3 allows us to greatly simplify\nproofs and to hide, so to speak, a great deal of logical complexity (in the sense of size of\nproofs). We start by defining our ‘class 3’ rule, the others being dependent on specific\nproof systems."},{"page":5,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n5\nCommutativity\nα ∨β = β ∨α\nα ∧β = β ∧α\nAssociativity\n[α ∨β] ∨γ = α ∨[β ∨γ]\n(α ∧β) ∧γ = α ∧(β ∧γ)\nUnits\nα ∨f = α\nα ∧t = α\nt ∨t = t\nf ∧f = f\nContext closure\nif α = β then ξ{α} = ξ{β}\nFigure 1. Equality = on formulae.\nDefinition 2.8. The equality relation = on formulae is defined by closing the equations in\nFigure 1 by reflexivity, symmetry, transitivity and by applying context closure. We define\nthe inference rule = as\nα\n= β , where α = β.\nThe following remark helps in assessing how much complexity is hidden in =.\nRemark 2.9. It is possible to decide α = β in polynomial time by reducing α and β to\nsome canonical form and comparing the canonical forms. A canonical form under = of\nany given formula can be obtained, for example, by removing as many units as possible\nand ordering units, atoms, and variables according to an arbitrary order; the canonical\nform is normal for associativity and units equations, when these are orientated from left\nto right. Let us assume a total order on the set of units, atoms, and variables; we now\nsee in detail how to find an equivalent canonical formula in the case of a formula only\ncontaining one logical relation. On the formula, use commutativity until the minimal unit,\natom, or variable appears in the leftmost position, then use associativity, orientated from\nleft to right, until normality is reached. For example, on [b ∨d] ∨[c ∨a], we perform the\nsteps\n[b ∨d] ∨[c ∨a] ❀[c ∨a] ∨[b ∨d] ❀[a ∨c] ∨[b ∨d] ❀a ∨[c ∨[b ∨d]]\n.\nThis phase requires O(n) steps, where n is the size of the formula. We proceed the same\nway on the subformula immediately following the first element, and so on recursively; for\nexample,\na ∨[c ∨[b ∨d]] ❀a ∨[[b ∨d] ∨c] ❀a ∨[b ∨[d ∨c]] ❀a ∨[b ∨[c ∨d]]\n.\nThe number of steps of the algorithm for a formula only containing one logical relation is\nthen O(n2). On a generic formula, the same algorithm can be used, with the same number-\nof-steps complexity O(n2) on the size n of the given formula, by adopting the lexicographic\norder induced by the given total order. This is an example, also involving an initial O(n)\nphase of simplification of units:\n(t ∧t) ∧[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f]\n❀t ∧[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f]\n❀[[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f] ∧t\n❀[a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])] ∨f\n❀a ∨([[b ∨d] ∨[c ∨a]] ∧[[B ∨c] ∨[t ∨a]])\n❀⋆a ∨([a ∨[b ∨[c ∨d]]] ∧[[B ∨c] ∨[t ∨a]])\n❀⋆a ∨([a ∨[b ∨[c ∨d]]] ∧[t ∨[a ∨[c ∨B]]])\n❀a ∨([t ∨[a ∨[c ∨B]]] ∧[a ∨[b ∨[c ∨d]]])"},{"page":6,"text":"6\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\n❀([t ∨[a ∨[c ∨B]]] ∧[a ∨[b ∨[c ∨d]]]) ∨a\n.\nThis way we obtain a (unique, of course) canonical formula in O(n2) steps, given any\nformula of size n, so we can decide the equivalence of two formulae α and β in O(n2)\nsteps, where n = |α| + |β|. Notice that at each step the size of the formula stays the same or\ndiminishes.\nDefinition 2.10. A CoS (proof) system is a finite set of inference rules. A CoS derivation\nΦ of length k in proof system S, whose premiss is α0 and conclusion is αk, is a chain of\ninference steps\nΦ =\nα0\nν1 α1\nν2 ...\nνk−1 αk−1\nνk αk\n,\nsuch that ν1, . . ., νk is a sequence of inference rules that alternate between the = rule and\nany rule of system S, where k ≥0. The same derivation can be indicated by\nα0\nΦ ∥\n∥S\nαk\n, when\nthe details are known or irrelevant; a proof is a derivation whose premiss is t. A derivation\nis ground if it contains no variables. Sometimes, we omit to indicate the inference steps\ngenerated by =. The size |Φ| of derivation Φ is the number of unit, atom, and variable\noccurrences appearing in it. We denote by ξ{Φ} the result of including every formula of\nΦ into the context ξ{ }. We denote by Φρ and Φσ the expression obtained from Φ by\napplying renaming ρ and substitution σ to every formula in Φ. A CoS proof system that,\nfor every valid implication α →β, contains a derivation with premiss α and conclusion β, is\nsaid to be implicationally complete.\nRemark 2.11. If Φ is a derivation, then ξ{Φ}, Φρ, and Φσ are derivations, for every context\nξ{ }, renaming ρ, and substitution σ.\nWe use the notion of groundness to relate the complexity of deep-inference proof sys-\ntems with atomic rules to proof systems without atomic rules, including those outside of\ndeep inference. Due to the aforementioned redundancy in the language, groundness is not\nreally a restriction.\nRemark 2.12. Every nonground derivation can be transformed into an equivalent,\nground one, by replacing variables with atoms in such a way that newly introduced atoms\nare different from the already present one.\nWe can now define some deep-inference proof systems. System SKS is the most impor-\ntant for the proof theory of classical logic, because of its atomic structural rules. System\nSKSg relates SKS to proof systems in other formalisms, like Frege.\nDefinition 2.13. CoS proof systems KSg = {i↓, w↓, c↓, s}, SKSg = KSg ∪{i↑, w↑,\nc↑}, KS = {ai↓, aw↓, ac↓, s, m} and SKS = KS ∪{ai↑, aw↑, ac↑} are defined in Figures 2\nand 3, for a language containing f, t, disjunction, and conjunction. Proof systems where\nnone of the rules i↑, ai↑, w↑, and aw↑appear are said to be analytic."},{"page":7,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n7\nStructural rules\nLogical rule\nSKSg\n \nA ∧ ̄A\ni↑\nf\nA\nw↑t\nA\nc↑A ∧A\ncointeraction\ncoweakening\ncocontraction\nor cut\nt\ni↓A ∨ ̄A\nf\nw↓A\nA ∨A\nc↓\nA\nA ∧[B ∨C]\ns (A ∧B) ∨C\n \nKSg\ninteraction\nweakening\ncontraction\nswitch\nor identity\nFigure 2. Systems SKSg and KSg.\nAtomic structural rules\nLogical rules\nSKS\n \na ∧ ̄a\nai↑\nf\na\naw↑t\na\nac↑a ∧a\ncointeraction\ncoweakening\ncocontraction\nor cut\nt\nai↓a ∨ ̄a\nf\naw↓a\na ∨a\nac↓\na\nA ∧[B ∨C]\ns (A ∧B) ∨C\n(A ∧B) ∨(C ∧D)\nm [A ∨C] ∧[B ∨D]\n \nKS\ninteraction\nweakening\ncontraction\nswitch\nmedial\nor identity\nFigure 3. Systems SKS and KS.\nExample 2.14. This is a valid derivation in all CoS proof systems defined previously (and\nit plays a role in the proof of Lemma 3.11):\nγ ∨[(([ ̄α ∨α] ∧c) ∧(α ∧d)) ∨δ]\n= γ ∨[(((α ∧d) ∧c) ∧[α ∨ ̄α]) ∨δ]\ns γ ∨[[(((α ∧d) ∧c) ∧α) ∨ ̄α] ∨δ]\n= [ ̄α ∨γ] ∨[((α ∧c) ∧(α ∧d)) ∨δ]\n.\nNote that SKSg, KSg, SKS, and KS are closed under renaming and substitution (see\nRemark 2.11). This is so because of the distinction between atoms and formula variables.\nObtaining the closure of these and other systems under renaming and substitution is one of\nthe main technical reasons for distinguishing between atoms and variables.\nThe following theorem is proved in [Br ̈u04], and follows immediately from Section 3.1,\nwhere we prove that CoS systems p-simulate Gentzen systems.\nTheorem 2.15. (Br ̈unnler)\nSystems SKSg, KSg, SKS, and KS are complete; systems\nSKSg and SKS are implicationally complete.\nThe theorem holds also when restricting the language to ground derivations, since sys-\ntems SKS and KS apply to them.\nIn the presence of cut, the coweakening and cocontraction rules do not play a major role\nin terms of proof complexity:\nTheorem 2.16. Systems SKSg and KSg∪i↑are p-equivalent, and systems SKS and KS∪\nai↑are p-equivalent."},{"page":8,"text":"8\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nProof. Observe that the rules w↑and c↑can be derived in KSg ∪i↑:\nA\n= A ∧[f ∨t]\ns (A ∧f) ∨t\nw↓(A ∧ ̄A) ∨t\ni↑\nf ∨t\n=\nt\nand\nA\n= A ∧t\ni↓A ∧[[ ̄A ∨ ̄A] ∨(A ∧A)]\ns (A ∧[ ̄A ∨ ̄A]) ∨(A ∧A)\nc↓\n(A ∧ ̄A) ∨(A ∧A)\ni↑\nf ∨(A ∧A)\n=\nA ∧A\n.\nSimilar constructions hold in KS ∪i↑for aw↑and ac↑.\n□\nIt turns out that all the systems mentioned in the previous theorem are p-equivalent, as\na consequence of Corollary 2.23.\nAnalytic systems are formally defined in Definition 2.13 for CoS, and 3.2 for Gentzen.\nThose definitions are specific to different systems in different formalisms, which is not\nnecessarily satisfactory. Defining a general, syntax-independent concept of analyticity is a\nsubject of ongoing research (see Problem 6.4). We briefly discuss now the notion of ana-\nlyticity and its connections with the proof complexity of deep inference, as an introduction\nto our result on Statman tautologies.\nA Gentzen system is said to be analytic when it does not contain the cut rule. Analytic\nGentzen systems enjoy the ‘subformula property’, i.e., proofs in these systems only contain\nsubformulae of their conclusions. In fact, we might stipulate that enjoying the subformula\nproperty is a primitive notion of analyticity, which we can use to exclude the cut rule, as\ndesired. In analytic Gentzen proofs, all formulae have lower or equal complexity than\nthat of the conclusion, when complexity is measured, for example, as the and/or depth of a\nformula (i.e., the number of alternations of conjunction and disjunction; see Definition 6.3).\nThere is another property, of interest to us, that analytic Gentzen systems enjoy: given an\ninference rule and its conclusion, there are only finitely many premisses to choose from;\nwe call such rules ‘finitely generating’. The cut rule in Gentzen does not possess the\nsubformula property nor is it finitely generating.\nThe primitive notion of analyticity that we are currently adopting for CoS is different\nfrom the one for Gentzen. We stipulate that a rule is analytic if its premiss is a formula\nobtained from a formula scheme by instantiating it with subformulae of the conclusion\n(so, the premiss is not just a subformula of the conclusion). This means that no atom or\nvariable can appear in the premiss of an analytic rule that does not appear in its conclusion.\nIt is, of course, a weaker condition than asking for the subformula property of Gentzen\nsystems, but doing so is necessary if we want to adopt deep inference and obtain linear\nrules. Like the subformula property does for Gentzen, this weaker notion for CoS excludes\nthe cut rule, but also the coweakening one. However, this is not an important difference\nwith the sequent calculus because coweakening is irrelevant for the proof complexity of\na CoS system (see, for example [GG08]). The reason for dealing with coweakening is\nthat, given the potential importance of cocontraction for proof complexity, we preferred to\nintroduce top-down-symmetric CoS systems (so, closed by duality), even if coweakening\nand cocontraction are not required for completeness.\nSo, the two notions of analyticity, for Gentzen and for CoS, are such that the only\nimportant rules that are not analytic are the respective cut rules. Note that in both cases,\nanalytic systems are made of finitely generating rules. However, there is an important\ndifference: in CoS, the complexity of formulae in an analytic proof can be unboundedly"},{"page":9,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n9\ngreater than the complexity of the conclusion. Consider, for example, the derivation\nc ∨(a ∧[b ∨[c ∨(a ∧b)])\ns c ∨[(a ∧b) ∨[c ∨(a ∧b)]]\n= [c ∨(a ∧b)] ∨[c ∨(a ∧b)]\nc↓\nc ∨(a ∧b)\n.\nThe and/or depth of the conclusion is 1, while that of the premiss is 3. We could repeat the\nconstruction on top of itself and further increase the and/or depth of the premiss at will.\nDeep-inference systems can be top-down symmetric in the sense that a derivation can\nbe flipped upside-down and negated and still be a valid derivation (we say that two such\nderivations are dual). Accordingly, some forms of analyticity can be defined in a symmetric\nway. Then, typically asymmetric theorems that depend on the notion of analyticity, like\ncut elimination, can be generalised to symmetric statements that imply cut elimination.\nThis is not the place to be detailed about this aspect; suffice to say that we can obtain for\nCoS systems much stronger normalisation (and cut elimination) results than for Gentzen\nsystems (see [Br ̈u06b, GG08]).\nAs we see in Section 3.1, analyticity in CoS faithfully captures analyticity in Gentzen,\nin the sense that analytic CoS can produce isomorphic proofs to Gentzen ones (almost\namounting to a change of notation). However, analytic CoS admits more proofs than an-\nalytic Gentzen, and, among CoS proofs, we can find some remarkably small ones, which\nanalytic Gentzen cannot express; this is the subject of Section 3.2 on Statman tautologies.\nRemark 2.17. The rules of SKS are local, in the sense that, for any language with a\nfinite number of atoms, checking that a given expression is an instance of any of these\nrules requires time bounded by a constant (adopting a tree representation of formulae,\nfor example). This property is peculiar to deep inference; it cannot be obtained in other\nformalisms. For example, a traditional, nonatomic contraction rule is not local because\nit requires checking the identity of two unbounded formulae. Contrary to other nonlocal\nrules, like identity in a Gentzen system, contraction cannot be replaced by its local, atomic\ncounterpart without losing completeness. A counterexample showing this is in [Br ̈u03b].\nLocality can possibly lead to a new, general, productive notion of analyticity, as argued in\nProblem 6.4.\nWe conclude the section by showing the p-equivalence of systems with atomic rules to\nsystems without atomic rules. We start by proving the result on ground derivations.\nLemma 2.18. For every ground instance\nt\ni↓α ∨ ̄α there is a derivation\nt\nΦ ∥\n∥{ai↓,s}\nα ∨ ̄α\nand for\nevery ground instance\nα ∧ ̄α\ni↑\nf\nthere is a derivation\nα ∧ ̄α\nΦ ∥\n∥{ai↑,s}\nf\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. Let us see the case for i↑, the other being its dual. We make an induction on the\nstructure of α. The cases when α is a unit or an atom are trivial: in the former case Φ\nconsists of an instance of = and in the latter the instance of i↑is also an instance of ai↑. We\nonly have to consider the case when α ≡β ∨γ: we apply the induction hypothesis on the\nderivation\n[β ∨γ] ∧( ̄β ∧ ̄γ)\n= ( ̄β ∧[β ∨γ]) ∧ ̄γ\ns [( ̄β ∧β) ∨γ] ∧ ̄γ\ni↑\n[f ∨γ] ∧ ̄γ\n=\nγ ∧ ̄γ\ni↑\nf\n,"},{"page":10,"text":"10\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nand we obtain a derivation whose length is O(n), and so its size is O(n2), where n = |α|.\n□\nLemma 2.19. For every ground instance\nf\nw↓α there is a derivation\nf\nΦ ∥\n∥{aw↓,s}\nα\nand for\nevery ground instance\nα\nw↑t there is a derivation\nα\nΦ ∥\n∥{aw↑,s}\nt\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. The proof is similar to the one for Lemma 2.18. In case α ≡t an instance of w↓\nyields\nf\n= f ∧[f ∨t]\ns (f ∧f) ∨t\n=\nt\n;\nwe can do similarly if α ≡f in an instance of w↑. These are the derivations for the inductive\ncases about w↓(those about w↑are dual):\nf\nw↓γ\n= f ∨γ\nw↓β ∨γ\nand\nf\n= f ∧f\nw↓f ∧γ\nw↓β ∧γ\n.\nFrom these we obtain a derivation whose length is O(n), and so its size is O(n2), where\nn = |α|.\n□\nRemark 2.20. In the statement of Lemma 2.19, instead of {aw↓, s} and {aw↑, s} we could\nhave used {aw↓, m} and {aw↑, m}, respectively.\nLemma 2.21. For every ground instance\nα ∨α\nc↓\nα\nthere is a derivation\nα ∨α\nΦ ∥\n∥{ac↓,m}\nα\nand for\nevery ground instance\nα\nc↑α ∧α there is a derivation\nα\nΦ ∥\n∥{ac↑,m}\nα ∧α\n; in both cases |Φ| ∈O(n2),\nwhere n = |α|.\nProof. The proof is similar to the one for Lemma 2.18. These are the derivations for the\ninductive cases about c↓(those about c↑are dual):\n[β ∨γ] ∨[β ∨γ]\n= [β ∨β] ∨[γ ∨γ]\nc↓\nβ ∨[γ ∨γ]\nc↓\nβ ∨γ\nand\n(β ∧γ) ∨(β ∧γ)\nm [β ∨β] ∧[γ ∨γ]\nc↓\nβ ∧[γ ∨γ]\nc↓\nβ ∧γ\n.\nFrom these we obtain a derivation whose length is O(n), and so its size is O(n2), where\nn = |α|.\n□\nTheorem 2.22. For every ground SKSg derivation Φ there is a ground SKS derivation\nΦ′ with the same premiss and conclusion of Φ; if n is the size of Φ then the size of Φ′ is\nO(n2); moreover, if Φ is in KSg then Φ′ is in KS.\nProof. The theorem follows immediately from Lemmas 2.18, 2.19, and 2.21.\n□\nBy Remark 2.12, every derivation can be ‘grounded’, so:\nCorollary 2.23. KS and KSg are p-equivalent and SKS and SKSg are p-equivalent."},{"page":11,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n11\nφ, A\n ̄A, ψ\ncut\nφ, ψ\ncut\nid A, ̄A\nt t\nφ\nw φ, A\nφ, A, A\nc\nφ, A\nφ, A, B\n∨\nφ, A ∨B\nφ, A\nB, ψ\n∧\nφ, A ∧B, ψ\nidentity\ntrue\nweakening\ncontraction\ndisjunction\nconjunction\nFigure 4. System Gentzen.\nRemark 2.24. Sometimes, we use nonatomic structural rule instances in SKS and KS\nderivations: those instances actually stand for the SKS and KS derivations that would\nbe obtained according to the proofs of Lemmas 2.18, 2.19, and 2.21. In this sense, we\nsay that i↓, i↑, w↓, w↑, c↓, and c↑are ‘macro’ rules for SKS and KS. The reason we\nmight want to work with macro rules in SKS and KS instead of working in SKSg and\nKSg and then appealing to Theorem 2.22 is to obtain finer upper bounds. This is because\nthe size of formulae over which nonatomic structural rules operate can be much smaller\nthan the square root of the size of a derivation, which is the pessimistic assumption of\nTheorem 2.22.\nRemark 2.25. All implicationally complete CoS proof systems are p-equivalent. This can\nbe proved analogously to, or resorting to, a similar ‘robustness’ result for Frege systems\n(Theorem 4.2), as argued in Remark 4.13. This means that studying proof complexity for\nSKSg and SKS has universal value for all CoS systems for propositional logic.\n3. Calculus of Structures, Gentzen Proof Systems and Statman Tautologies\nThere are two parts in this section. In the first part, we show how CoS naturally p-\nsimulates Gentzen systems, and in particular how it realizes Gentzen’s notion of analyticity.\nIn the second part, we show that analytic CoS admits polynomial proofs when analytic\nGentzen only has exponential ones, in the case of Statman tautologies.\n3.1. Calculus of Structures and Gentzen Proof Systems. In this section, we adopt a\nspecific one-sided (Gentzen-Sch ̈utte) sequent system that we call Gentzen (and that is\ncalled GS1p in [TS96]). We could have adopted any other style of presentation without\naffecting our results. In fact, for Gentzen systems an analogous ‘robustness’ theorem to\nthat for Frege systems (Theorem 4.2) can be established. This means that studying the\nproof complexity of Gentzen has universal value for the class of Gentzen systems.\nDefinition 3.1. Over the language of SKS formulae, the sequent-calculus proof system\nGentzen is defined by the inference rules in Figure 4, where φ and ψ stand for multisets\nof formulae and the symbol ‘,’ represents multiset union. We interpret multisets of for-\nmulae as their disjunction (where associativity is irrelevant). Derivations, denoted by ∆,\nare trees obtained by composing instances of inference rules; the leaves of a derivation are\nits premisses and the root is its conclusion; a derivation ∆with premisses φ1, ..., φh and\nconclusion ψ is denoted by\nφ1 . . . φh\n∆\nψ\n.\nA derivation with no premisses is a proof. The size |∆| of derivation ∆is the number of\nunit, atom, and variable occurrences appearing in it. In the following, every SKS formula\nis translated into a Gentzen formula in the obvious way, and vice versa; in particular, we\ntranslate a Gentzen multiset φ = α1, . . ., αh into α1 ∨· · · ∨αh."},{"page":12,"text":"12\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nIn the language, we keep the distinction between atoms and variables because, thanks\nto atoms, we obtain a better upper bound for the size of Statman tautologies proofs, in\nthe Section 3.2. As we said in the case of CoS, the redundancy in the language has no\nconsequences outside of the possibility of using certain CoS rules instead of others.\nDefinition 3.2. The proof system analytic Gentzen is proof system Gentzen without the\ncut rule; analytic derivations and proofs are those derivations and proofs in Gentzen where\nno instances of the cut rule appear.\nWe know, of course, that both Gentzen and analytic Gentzen are complete, and that\nGentzen proofs can be transformed into analytic Gentzen proofs by a cut-elimination\nprocedure, which, in general, blows-up a given proof exponentially.\nEvery Gentzen derivation has natural counterparts in CoS: the idea is to (arbitrarily)\nsequentialise its tree structure. This is possible because the natural logical relation between\ntree branches is conjunction, which CoS can represent, of course. In doing so, we pay\nin terms of complexity because the tree structure is less redundant than CoS contexts:\nthe size of derivations grows quadratically. Other deep-inference formalisms (currently\nunder development, see [BL05, Gug04, Gug05]) are more efficient than CoS and Gentzen\nformalisms in dealing with this so-called ‘bureaucracy’.\nRemark 3.3. In the following, we assume that an empty conjunction can be represented\nby a nonempty conjunction of t units.\nTheorem 3.4. For every Gentzen derivation ∆with premisses φ1, ..., φh and conclusion\nψ there is a derivation\nφ1 ∧· · · ∧φh\nΦ ∥\n∥SKSg\nψ\n; if n is the size of ∆, the size of Φ is O(n2); moreover,\nif ∆is analytic then Φ is in KSg.\nProof. We proceed by induction on the tree structure of ∆. The base cases id A, ̄A and t t\nare, respectively, translated into\nt\ni↓A, ̄A and t. The derivations\nφ1 . . . φh\n∆1\nφ\nw φ, A\n,\nφ1 . . . φh\n∆1\nφ, A, A\nc\nφ, A\n,\nand\nφ1 . . . φh\n∆1\nφ, A, B\n∨\nφ, A ∨B\nare, respectively, translated into\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ\n= φ ∨f\nw↓φ ∨A\n,\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ ∨[A ∨A]\nc↓\nφ ∨A\n,\nand\nφ1 ∧· · · ∧φh\nΦ1 ∥\n∥SKSg\nφ ∨[A ∨B]\n,\nwhere Φ1 is obtained by induction from ∆1, and some possibly necessary instances of the\n= rule have been omitted (they depend on the exact translation of Gentzen multisets into\nSKSg formulae). The derivations\nφ1 . . . φh\n∆1\nφ, A\nφh+1 . . . φk\n∆2\nB, ψ\n∧\nφ, A ∧B, ψ\nand\nφ1 . . . φh\n∆1\nφ, A\nφh+1 . . . φk\n∆2\n ̄A, ψ\ncut\nφ, ψ"},{"page":13,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n13\nare, respectively, translated into\nφ1 ∧· · · ∧φk\nΦ1∧Φ2 ∥\n∥\n[φ ∨A] ∧[B ∨ψ]\n= [B ∨ψ] ∧[A ∨φ]\ns ([B ∨ψ] ∧A) ∨φ\n= φ ∨(A ∧[B ∨ψ])\ns φ ∨[(A ∧B) ∨ψ]\nand\nφ1 ∧· · · ∧φk\nΦ1∧Φ2 ∥\n∥\n[φ ∨A] ∧[ ̄A ∨ψ]\n= [ ̄A ∨ψ] ∧[A ∨φ]\ns ([ ̄A ∨ψ] ∧A) ∨φ\n= φ ∨(A ∧[ ̄A ∨ψ])\ns φ ∨[(A ∧ ̄A) ∨ψ]\ni↑\nφ ∨[f ∨ψ]\n=\nφ ∨ψ\n,\nwhere Φ1 and Φ2 are obtained by induction from ∆1 and ∆2, some possibly necessary\ninstances of the = rule have been omitted, and Φ1 ∧Φ2 stands for the derivation\nφ1 ∧· · · ∧φk\nΦ1∧(φh+1∧···∧φk) ∥\n∥\n[φ ∨A] ∧(φh+1 ∧· · · ∧φk)\n[φ∨A]∧Φ2 ∥\n∥\n[φ ∨A] ∧[B ∨ψ]\n,\nwhere B is possibly instantiated by ̄A; the length of this derivation and the size of the largest\nformula appearing in it are both O(n). The resulting O(n2) measure of these last two cases\ndominates the others.\n□\nCorollary 3.5. SKSg p-simulates Gentzen and KSg p-simulates analytic Gentzen.\nAlthough it does not explicitly address complexity, [Br ̈u04] is more exhaustive than the\naforesaid on the two-way translation between SKSg and Gentzen. Translating from SKSg\nto Gentzen crucially employs the cut rule: for every inference step in SKSg, a cut instance\nis used in Gentzen. So, while it is very natural and easy to show that Gentzen p-simulates\nSKSg (see [Br ̈u04]), we are left with the question: does analytic Gentzen p-simulate KSg?\n3.2. Analytic Calculus of Structures on Statman Tautologies. We prove here that an-\nalytic Gentzen does not p-simulate KSg. In fact, the CoS (polynomial) inefficiency in\ndealing with context bureaucracy is compensated by its freedom in applying inference\nrules, which leads to exponential speedups on certain classes of tautologies. Here, we\nstudy Statman tautologies, which have been used to provide the classic lower bound for\nanalytic Gentzen systems: no cut-free proofs of Statman tautologies are possible in an-\nalytic Gentzen without the proofs growing exponentially over the size of the tautologies\nthey prove [Sta78]. We show that, on the contrary, KSg and KS prove Statman tautologies\nwith polynomially growing analytic proofs.\nRemark 3.6. The subset of SKSg only containing analytic rules is equal to KSg plus\nthe cocontraction rule. We do not know whether cocontraction provides for exponential\nspeedups, so separating, proof-complexity-wise, the class of KSg from that of ‘analytic\nCoS’; about this, see Problem 6.2. In our opinion, the very notion of analyticity would\nbenefit from some further analysis; about this, see Problem 6.4.\nDefinition 3.7. For n ≥1, consider the following formulae:\nαi ≡ ̄ci ∨ ̄di\nfor i ≥1,\nβn\nk ≡Vk\ni=n αi ≡αn ∧βn−1\nk\nfor n ≥k > 1,\nγn\nk ≡βn\nk+1 ∧ck\nfor n > k ≥1,\nδn\nk ≡βn\nk+1 ∧dk\nfor n > k ≥1."},{"page":14,"text":"14\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nStatman tautologies are, for n ≥1, the formulae:\nSn ≡ ̄αn ∨[(γn\nn−1 ∧δn\nn−1) ∨[· · · ∨[(γn\n1 ∧δn\n1) ∨α1] · · ·]]\n.\nExample 3.8. These are the first three Statman tautologies:\nS1 ≡(c1 ∧d1) ∨[ ̄c1 ∨ ̄d1]\n,\nS2 ≡(c2 ∧d2) ∨[(([ ̄c2 ∨ ̄d2] ∧c1) ∧([ ̄c2 ∨ ̄d2] ∧d1)) ∨[ ̄c1 ∨ ̄d1]]\n,\nS3 ≡(c3 ∧d3) ∨[(([ ̄c3 ∨ ̄d3] ∧c2) ∧([ ̄c3 ∨ ̄d3] ∧d2)) ∨\n[((([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) ∧c1) ∧(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) ∧d1)) ∨\n[ ̄c1 ∨ ̄d1]]]\n.\nIt is perhaps easier to understand their meaning by using implication, as in\nS′\n3 ≡[ ̄c3 ∨ ̄d3] →([([ ̄c3 ∨ ̄d3] → ̄c2) ∨([ ̄c3 ∨ ̄d3] → ̄d2)] →\n([(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) → ̄c1) ∨(([ ̄c3 ∨ ̄d3] ∧[ ̄c2 ∨ ̄d2]) → ̄d1)] →\n[ ̄c1 ∨ ̄d1]))\n.\nRemark 3.9. |Sn| = 2 + 2 Pn−1\nk=1 |γn\nk| + 2 = 2 Pn\nk=2(|βn\nk| + 1) + 4 = 2n2 + 2.\nIt is not difficult to see why analytic Gentzen proofs of Statman tautologies grow expo-\nnentially (this is a classic argument that can be found in many textbooks; see, for example,\n[CK02]). Basically, all what analytic Gentzen can do while building a proof of Sn is to\ngenerate a proof tree with O(2n) branches. The next lemma shows the crucial advantage\nof deep inference over Gentzen systems: Statman tautologies can be proved ‘from the in-\nside out’, which is precisely what Gentzen systems can only do by resorting to convoluted\nproofs involving cuts (so, nonanalytic proofs).\nRemark 3.10. In the following, for brevity, we label inference steps with expressions like\nn · ν, to denote n inference steps involving rule ν.\nLemma 3.11. For Statman tautologies Sn and Sn+1 there exists a derivation\nSn\n∥\n∥KS\nSn+1\nwhose\nlength is O(n) and size is O(n3).\nProof. We refer to Definition 3.7. The requested derivation is\nΦ =\n(cn ∧dn) ∨\n[((βn\nn ∧cn−1) ∧(βn\nn ∧dn−1)) ∨[· · · ∨\n[((βn\n2 ∧c1) ∧(βn\n2 ∧d1)) ∨α1] · · ·]]\n2n · i↓(([αn+1 ∨ ̄αn+1] ∧cn) ∧([αn+1 ∨ ̄αn+1] ∧dn)) ∨\n[((([αn+1 ∨ ̄αn+1] ∧βn\nn) ∧cn−1) ∧(([αn+1 ∨ ̄αn+1] ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[((([αn+1 ∨ ̄αn+1] ∧βn\n2) ∧c1) ∧(([αn+1 ∨ ̄αn+1] ∧βn\n2) ∧d1)) ∨α1] · · ·]]\n2n · s\n2n\nz }| {\n[ ̄αn+1 ∨[· · · ∨[ ̄αn+1 ∨ ̄αn+1] · · ·]] ∨\n[((αn+1 ∧cn) ∧(αn+1 ∧dn)) ∨\n[(((αn+1 ∧βn\nn) ∧cn−1) ∧((αn+1 ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[(((αn+1 ∧βn\n2) ∧c1) ∧((αn+1 ∧βn\n2) ∧d1)) ∨α1] · · ·]]]\n(2n −1) · c↓ ̄αn+1 ∨\n[((αn+1 ∧cn) ∧(αn+1 ∧dn)) ∨\n[(((αn+1 ∧βn\nn) ∧cn−1) ∧((αn+1 ∧βn\nn) ∧dn−1)) ∨[· · · ∨\n[(((αn+1 ∧βn\n2) ∧c1) ∧((αn+1 ∧βn\n2) ∧d1)) ∨α1] · · ·]]]\n,\nwhere we use macro inference rules as explained in Remark 2.22. (Example 2.14 explains\nthe central step in the preceding derivation.) The formulae appearing in the middle of the\nprevious derivation are the largest. Since |αn+1| = 2 and |βn\nk| = 2(n −k + 1), their size is"},{"page":15,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n15\n2n·2+6+2(3n−3+Pn\nk=2 |βn\nk|)+2 = 2n2+8n+2. Each i↓macro inference step involves one s\nand two ai↓steps in KS, for a total of six steps, including = ones; each c↓macro inference\nstep involves one m and two ac↓steps in KS, for a total of six steps. So, the length of Φ is\n2n · 6 + 2n · 2 + (2n −1) · 6 = 28n −6, and so |Φ| ≤(28n −6)(2n2 + 8n + 2) ∈O(n3).\n□\nNote that in the previous proof, by working with macro inference rules, we get a better\nupper bound for KS than if we worked in KSg and then applied Theorem 2.22.\nTheorem 3.12. There are KS proofs of Statman tautologies whose size is quadratic in the\nsize of the tautologies they prove.\nProof. Tautology S1 is trivially provable by an instance of the i↓macro rule. By repeatedly\napplying the previous lemma, we obtain proofs of all Statman tautologies Sn, whose size\nis O(n4). Since |Sn| ∈O(n2) (see Remark 3.9), the statement follows.\n□\nThis is enough to conclude that analytic Gentzen does not polynomially simulate KSg\nand KS. Some could argue that Statman tautologies are artificial in their forcing exponen-\ntial Gentzen proofs into ‘wildly’ branching. However, notice that both notions of proof\nand analyticity in Gentzen systems ‘get into tautologies’ from the outside inwards. In\nother words, the restricted notion of analyticity in Gentzen systems is strongly correlated\nto the restricted notion of proof that leads to exponential-size proofs. In CoS, both notions\nare more liberal, to the advantage of proof complexity. We pay a price for this in terms\nof proof-search complexity: there is research aimed at improving the situation, with very\npromising results; see [Kah06a].\nWe note that polynomial proofs on Statman tautologies are obtained by a very small\ndose of deep inference. In fact, the trick is done by the switch and interaction instances in\nthe proof in Lemma 3.11: they all operate just below the ‘surface’ of a formula. This leads\nus to state a currently open problem, in Section 6.6.\n4. Calculus of Structures and Frege Systems\nIn this section, we prove that CoS and Frege systems are p-equivalent.\nIn the following definitions about Frege systems, we do not assume that the language of\nformulae coincides with the CoS one, but, as always, this is not a very important issue.\nDefinition 4.1. Given a language of propositional logic formulae built over a complete\nbase of connectives, a Frege (proof) system is a finite collection of sound inference rules,\neach of which is a tuple of n > 0 formulae such that from n −1 premisses one conclusion\nis derived; inference rules with 0 premisses are called axioms. Given a Frege system, a\nFrege derivation of length l with premisses α1, ..., αh and conclusion βl is a sequence of\nformulae β1, . . ., βl, such that each βi either belongs to {α1, . . ., αh} or is the conclusion of\nan instance of an inference rule whose premisses belong to β1, . . ., βi−1, where 1 ≤i ≤l;\na Frege proof of β is a Frege derivation with no premisses and conclusion β; we use Υ for\nderivations. We require of each Frege system to be implicationally complete, i.e., whenever\n(α1 ∧· · · ∧αh) →β is valid there is a derivation with premisses α1, ..., αh and conclusion\nβ in the proof system. The size of a Frege derivation Υ is the number of unit, atom, and\nvariable occurrences that it contains, and is indicated by |Υ|.\nThe following ‘robustness’ theorem can easily be proved.\nTheorem 4.2. (Robustness, Cook-Reckhow, [CR79])\nAll Frege systems in the same lan-\nguage are p-equivalent.\nThe theorem has been generalised by Reckhow to Frege systems in any language (under\n‘natural translations’) [Rec76], but we do not need this level of generality in our article.\nThe robustness theorem allows us to work with just one Frege system, and we arbitrarily\nchoose the following, taken from [Bus87] and modified by adding axioms F14, F15, F16,\nand F17 in order to deal with units."},{"page":16,"text":"16\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nAxioms:\nF1 ≡A →(B →(A ∧B))\nF2 ≡(A ∧B) →A\nF3 ≡(A ∧B) →B\nF4 ≡A →[A ∨B]\nF5 ≡B →[A ∨B]\nF6 ≡¬¬A →A\nF7 ≡A →¬¬A\nF8 ≡A →(B →A)\nF9 ≡¬A →(A →B)\nF10 ≡(A →(B →C)) →((A →B) →(A →C))\nF11 ≡(A →C) →((B →C) →([A ∨B] →C))\nF12 ≡(A →(B →C)) →(B →(A →C))\nF13 ≡(A →B) →(¬B →¬A)\nF14 ≡f →(A ∧¬A)\nF15 ≡(A ∧¬A) →f\nF16 ≡t →[A ∨¬A]\nF17 ≡[A ∨¬A] →t\nInference rule:\nA\nA →B\nmp\nB\nFigure 5. System Frege.\nDefinition 4.3. Frege system Frege, over the language of formulae freely generated by\nunits, non-negated formula variables, and the connectives ∨, ∧, →, and ¬, has inference\nrules as shown in Figure 5, where the formulae F1, ..., F17 are axioms and the inference\nrule mp is called modus ponens.\nRemark 4.4. In the following, every SKS formula is implicitly translated into a Frege\nformula in the obvious way, and vice versa; in particular, we translate Frege’s formulae of\nthe kind α →β into SKS formulae ̄α ∨β.\nRemark 4.5. In Frege systems, distinguishing atoms from formula variables is unnec-\nessary because we only instantiate rules by general substitution (as opposed to renam-\ning). So, from now on, we assume that CoS atoms correspond to Frege formula vari-\nables. Since in system Frege we have a connective for negation, we can also assume that\nwhen dual atoms and formula variables appear in an SKSg formula, their Frege translation\nonly uses ¬; for example, the SKSg formula [A ∨ ̄A] ∧[a ∨ ̄a] is translated into Frege for-\nmula [A ∨¬A] ∧[B ∨¬B] or [A ∨¬A] ∧[¬B ∨B] or [¬A ∨A] ∧[B ∨¬B] or [¬A ∨A] ∧[¬B ∨B].\nConversely, Frege formula [A ∨¬A] ∧[B ∨¬B] is translated into SKSg or SKS formula\n[a ∨ ̄a] ∧[b ∨ ̄b] or [a ∨ ̄a] ∧[ ̄b ∨b] or [ ̄a ∨a] ∧[b ∨ ̄b] or [ ̄a ∨a] ∧[ ̄b ∨b] or one such for-\nmula with formula variables in the place of some of the atoms. As always, we use atoms\nwhen we need to use SKS atomic structural rules, we use formula variables when we need\nto instantiate formulae and derivations, and otherwise we can choose both.\nTranslating Frege into SKSg derivations is straightforward, given that the cut rule of\nSKSg can easily simulate modus ponens.\nTheorem 4.6. For every Frege derivation Υ with premisses α1, ..., αh, where h ≥0, and\nconclusion β, there is a derivation\nα1 ∧· · · ∧αh\nΦ ∥\n∥SKSg\nβ\n; if l and n are, respectively, the length and\nsize of Υ, then the length and size of Φ are, respectively, O(l) and O(n2)."},{"page":17,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n17\nProof. The axioms Fi of Frege are tautologies, so each one has a proof Φi in SKSg, for\n1 ≤i ≤17; for example F1 and F10 are, respectively, proved by\nΦ1 =\nt\ni↓[ ̄A ∨ ̄B] ∨(A ∧B)\n= ̄A ∨[ ̄B ∨(A ∧B)]\nand\nΦ10 =\nt\ni↓(A ∧(B ∧ ̄C)) ∨[ ̄A ∨[ ̄B ∨C]]\n= (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧t) ∨C]]\ni↓(A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧[A ∨ ̄A]) ∨C]]\ns (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[[( ̄B ∧A) ∨ ̄A] ∨C]]\n= (A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[[ ̄A ∨ ̄A] ∨C]]\nc↓\n(A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[ ̄A ∨C]]\n.\nWe proceed by induction on the length of Υ = β1, . . ., βk, β and we prove the existence\nof a derivation\nα1 ∧· · · ∧αh\nΦ′ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n. The base case k = 0 is as follows: 1) if β is a premiss,\nthen Φ′ = β; 2) if β ≡Fiσ, for some i and σ, then Φ′ = Φiσ. For the inductive step,\ngiven Υk = β1, . . . , βk and\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk\n, where γk is the conjunction of premisses of Υk, we\nconsider the following cases:\n• if β is a premiss, then Φ′ =\nγk ∧β\nΦ′\nk∧β ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n;\n• if β ≡Fiσ, for some i and substitution σ, then Φ′ =\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk\n= (β1 ∧· · · ∧βk) ∧t\n(β1∧···∧βk)∧Φiσ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\n;\n• if β is the conclusion of an instance\nβk′\nβk′ →β\nmp\nβ\n, where βk′′ ≡βk′ →β and\n1 ≤k′, k′′ ≤k, then\nΦ′ =\nγk\nΦ′\nk\n∥\n∥SKSg\nβ1 ∧· · · ∧βk′′ ∧· · · ∧βk\nc↑β1 ∧· · · ∧βk′ ∧· · · ∧(βk′′ ∧βk′′) ∧· · · ∧βk\nc↑β1 ∧· · · ∧(βk′ ∧βk′) ∧· · · ∧(βk′′ ∧βk′′) ∧· · · ∧βk\n=\n(β1 ∧· · · ∧βk) ∧(βk′ ∧[ ̄βk′ ∨β])\ns (β1 ∧· · · ∧βk) ∧[(βk′ ∧ ̄βk′) ∨β]\ni↑\n(β1 ∧· · · ∧βk) ∧[f ∨β]\n=\n(β1 ∧· · · ∧βk) ∧β\n,\nwhere, without loss of generality, we assumed k′ < k′′.\nAt every inductive step the length of the SKSg derivation is only increased by an O(1)\nnumber of inference steps. From\nα1 ∧· · · ∧αh\nΦ′ ∥\n∥SKSg\n(β1 ∧· · · ∧βk) ∧β\nwe can obtain the desired derivation\nα1 ∧· · · ∧αh\nΦ ∥\n∥SKSg\nβ\nby applying once the w↑rule. So, the length of Φ is O(k). From this, and after\ninspecting the aforesaid derivations, it follows that |Φ| ∈O(k2m), where m is the maximum\nsize of a formula appearing in Υ, and so |Φ| ∈O(n2), where |Υ| = n.\n□\nCorollary 4.7. SKSg and SKS p-simulate Frege."},{"page":18,"text":"18\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nProof. The statement for SKSg follows from Theorem 4.6, and that for SKS from this and\nCorollary 2.23.\n□\nTranslating derivations from SKSg to Frege requires more effort than the converse,\npartly because of the need to simulate deep inference, and partly because of the large\n‘amount of inference’ of =-rule instances. The next two lemmas take care of these two\nissues.\nLemma 4.8. For every SKS context ξ{ } and formulae α and β, there is a Frege derivation\nwith premiss α→β and conclusion ξ{α}→ξ{β} whose length is O(m) and size is O(n2), where\nm = |ξ{ }| and n = |ξ{α} →ξ{β}|.\nProof. Consider four Frege proofs Υ′, Υ′′, Υ′′′, and Υ′′′′, respectively of the four tautolo-\ngies\n(A →B) →([A ∨C] →[B ∨C])\n,\n(A →B) →([C ∨A] →[C ∨B])\n,\n(A →B) →((A ∧C) →(B ∧C))\n,\n(A →B) →((C ∧A) →(C ∧B))\n.\nWe proceed by induction on the structure of ξ{ }. If ξ{ } ≡ξ1{{ } ∨γ1}, we build Frege\nderivation\nΥ1 = α →β, Υ′{A/α, B/β,C/γ1}, α ∨γ1 →β ∨γ1\n;\nwe build Υ1 similarly if ξ{ } ≡ξ1{{ } ∧γ1} or ξ{ } ≡ξ1{γ1 ∨{ }} or ξ{ } ≡ξ1{γ1 ∧{ }}.\nGiven ξ1{ } ≡ξ2{{ } ∨γ2} or ξ1{ } ≡ξ2{{ } ∧γ2} or ξ1{ } ≡ξ2{γ2 ∨{ }} or ξ1{ } ≡\nξ2{γ2 ∧{ }} we build Υ2 analogously to Υ1, and the premiss of Υ2 is the conclusion of\nΥ1. We proceed this way until we build Υl, whose conclusion is ξ{α} →ξ{β}, where l ≤m.\nWe obtain the desired derivation Υ by concatenating Υ1, ..., Υl. Since the length and size\nof Υ′, Υ′′, Υ′′′, and Υ′′′′ are independent of ξ{ }, α, and β, the length of Υ is O(m) and its\nsize is O(mn), and so O(n2).\n□\nLemma 4.9. For every SKS formulae α and β such that α = β there is a Frege derivation\nwith premiss α, conclusion β, length O(n3), and size O(n4), where n = |α| + |β|.\nProof. Consider the following tautologies, derived from the equations in Figure 1:\n(1)\n[A ∨B] ↔[B ∨A]\n,\n[A ∨f] ↔A\n,\n(A ∧B) ↔(B ∧A)\n,\n(A ∧t) ↔A\n,\n[[A ∨B] ∨C] ↔[A ∨[B ∨C]]\n,\n[t ∨t] ↔t\n,\n((A ∧B) ∧C) ↔(A ∧(B ∧C))\n,\n(f ∧f) ↔f\n,\nwhere each expression corresponds to the two tautologies obtained by orientating each\ndouble implication. Every such tautology can be proved in Frege with a constant-size\nproof, so every instance γ →γ′ of any of these tautologies has a Frege proof of length O(1)\nand size O(m′), where m′ = |γ| + |γ′|. By Lemma 4.8, for every ξ{ } there is a derivation\nwith premiss γ →γ′ and conclusion ξ{γ} →ξ{γ′} whose length is O(m) and size is O(m2),\nwhere m = |ξ{γ} →ξ{γ′}|. By concatenating the proof and derivation so obtained, we can\nbuild a proof of ξ{γ} →ξ{γ′} whose length is O(m) and size is O(m2). By Remark 2.9, we\ncan build a chain of implications\nα ≡α1 →· · · →αh ≡δ ≡βk →· · · →β1 ≡β\n,\nwhere δ is a canonical form for α and β, h + k is O(n2), and each implication αi →αi+1 and\nβi+1 →βi is a tautology of the form ξ{γ} →ξ{γ′}, such that γ →γ′ is an instance of one of\nthe tautologies 1. By concatenating the proofs of every ξ{γ} →ξ{γ′} by mp, we obtain a\nderivation with premiss α, conclusion β, length O(n3), and size O(n4).\n□\nLemma 4.10. For every inference step\nα\nν β , where ν is a rule of SKSg, there is a Frege\nderivation with premiss α, conclusion β, length O(n), and size O(n2), where n = |α| + |β|."},{"page":19,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n19\nProof. Each of the following tautologies, corresponding to the inference rules in Figure 2,\ncan be proved in Frege with a constant-size proof:\n(2)\n(A ∧¬A) →f\n,\nA →t\n,\nA →(A ∧A)\n,\nf →[A ∨¬A]\n,\nf →A\n,\n[A ∨A] →A\n,\n(A ∧[B ∨C]) →[(A ∧B) ∨C]\n.\nLet\nα\nν β =\nξ{γ}\nν ξ{δ}, where\nγ\nν δ is an instance of ν. There is a Frege proof Υ of γ →δ,\nwhose length is O(1) and size is O(n), obtained by instantiating the corresponding proof to\nν among those in 2. By Lemma 4.8, there exists a Frege derivation Υ′ with premiss γ →δ,\nconclusion ξ{γ} →ξ{δ}, length O(n), and size O(n2). By concatenating Υ and Υ′ we obtain\na proof Υ′′ of ξ{γ} →ξ{δ}. From Υ′′, by using mp, we obtain the desired derivation with\npremiss α ≡ξ{γ} and conclusion β ≡ξ{δ}.\n□\nTheorem 4.11. For every derivation\nα\nΦ ∥\n∥SKSg\nβ\nthere is a Frege derivation Υ with premiss\nα and conclusion β; if n is the size of Φ, then the length and size of Υ are, respectively,\nO(n4) and O(n5).\nProof. The statement immediately follows from Lemmas 4.9 and 4.10, after assuming that\nthe length of Φ is O(n).\n□\nCorollary 4.12. Frege p-simulates SKSg and SKS.\nRemark 4.13. As evidenced by the proofs of Theorems 4.6 and 4.11, it does not really\nmatter, for establishing the p-simulations, precisely which inference rules are adopted by\nthe CoS and Frege systems. In fact, the simulations work because the simulating systems\nare implicationally complete and their set of proofs is closed under substitution. This way,\nthe constant-size proofs in one system, simulating the rules of the other system, can be\ninstantiated at a linear cost in order to simulate instances of rules. We can then use a ro-\nbustness theorem (see Theorem 4.2) for Frege in order to establish a robustness theorem\nfor CoS, possibly also for systems on mutually different languages: given two implication-\nally complete CoS systems, we p-simulate each in two appropriate Frege systems and use\nFrege robustness.\n5. Extension and Substitution\nIn this section, we show how CoS systems can be extended with the Tseitin exten-\nsion rule and with the substitution rule, analogously to Frege systems. We also show the\np-equivalence of all these systems, as described in the box of the diagram in the Introduc-\ntion. As always, we operate under robustness theorems (relying on the mentioned one,\nTheorem 4.2) that ensure that the proof complexity properties we establish for the specific\nsystems actually hold for the formalisms they belong to.\nDefinition 5.1. An extended Frege (proof) system is a Frege system augmented with the\n(Tseitin) extension rule, which is a rule with no premisses and whose instances A ↔β are\nsuch that the variable A does not appear before in the derivation, nor appears in β or in\nthe conclusion of the proof. We write A < α to state that variables A and ̄A do not appear\nin formula α. The symbol ↔stands for logical equivalence, and the specific syntax of the\nexpressions A ↔β depends on the language of the Frege system in use. In the following,\nwe consider A ↔β a shortcut for (A →β) ∧(β →A). We denote by xFrege the proof system\nwhere a proof is a derivation with no premisses, conclusion αk, and shape\nα1, . . ., αi1−1,\nαi1 ≡\nz }| {\nA1 ↔β1 , αi1+1, . . ., αih−1,\nαih ≡\nz }| {\nAh ↔βh , αih+1, . . . , αk\n,"},{"page":20,"text":"20\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nwhere all the conclusions of extension instances αi1, ..., αih are singled out and\nA1 < α1, . . ., αi1−1, β1, αk\n,\n. . .\n,\nAh < α1, . . ., αih−1, βh, αk\n,\nand the rest of the proof is as in Frege.\nRemark 5.2. We could have equivalently defined an xFrege proof of α as a Frege deriva-\ntion with conclusion α and premisses {A1 ↔β1, . . ., Ah ↔βh} such that A1, ̄A1, ..., Ah, ̄Ah\nare mutually distinct and A1 < β1, α and . . . and Ah < β1, . . ., βh, α. Notice that xFrege\nis indeed a proof system in the sense that it proves tautologies. In fact, given the xFrege\nproof just mentioned, we obtain a Frege proof by applying to it, in order, the substitutions\nσh = Ah/βh, ..., σ1 = A1/β1, and by prepending to it proofs of the tautologies β1 ↔β1,\n(β2 ↔β2)σ1, ..., (βh ↔βh)σh−1 · · · σ1. In general, a proof so obtained is exponentially bigger\nthan the xFrege one it derives from.\nSKSg can analogously be extended, but there is no need to create a special rule; we\nonly need to broaden the criterion by which we recognize a proof.\nDefinition 5.3. An extended SKSg proof of α is an SKSg derivation with conclusion α\nand premiss [ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah], where A1, ̄A1, ..., Ah, ̄Ah are\nmutually distinct and A1 < β1, α and . . . and Ah < β1, . . ., βh, α. We denote by xSKSg the\nproof system whose proofs are extended SKSg proofs.\nTheorem 5.4. For every xFrege proof of length l and size n there exists an xSKSg proof\nof the same formula and whose length and size are, respectively, O(l) and O(n2).\nProof. Consider an xFrege proof as in Definition 5.1. By Remark 5.2 and Theorem 4.6,\nthere exists the following xSKSg proof, whose length and size are yielded by 4.6:\n[ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah]\n∥\n∥SKSg\nαk\n.\n□\nAlthough not strictly necessary to establish the equivalence of the four extended for-\nmalisms (see diagram in the Introduction), the following theorem is very easy to prove.\nTheorem 5.5. For every xSKSg proof of size n there exists an xFrege proof of the same\nformula and whose length and size are, respectively, O(n4) and O(n5).\nProof. Consider an xSKSg proof as in Definition 5.3. The statement is an immediate\nconsequence of Theorem 4.11, after observing that there is an O(h)-length and O(hn)-size\nxFrege proof\nA1 ↔β1, . . ., Ah ↔βh, . . . , (A1 ↔β1) ∧· · · ∧(Ah ↔βh)\n.\n□\nCorollary 5.6. Systems xFrege and xSKSg are p-equivalent.\nWe now move to the substitution rule.\nDefinition 5.7. A substitution Frege (proof) system is a Frege system augmented with\nthe substitution rule\nA\nsub Aσ. We denote by sFrege the proof system where a proof is a\nderivation with no premisses, conclusion αk, and shape\nα1, . . ., αi1−1,\nαi1 ≡\nz}|{\nα j1σ1 , αi1+1, . . ., αih−1,\nαih ≡\nz}|{\nα jhσh , αih+1, . . . , αk\n,\nwhere all the conclusions of substitution instances αi1, ..., αih are singled out, α j1 ∈\n{α1, . . ., αi1−1}, ..., α jh ∈{α1, . . . , αih−1}, and the rest of the proof is as in Frege."},{"page":21,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n21\nWe rely on the following result.\nTheorem 5.8. (Cook-Reckhow and Kraj ́ıˇcek-Pudl ́ak, [CR79, KP89])\nSystems xFrege\nand sFrege are p-equivalent.\nWe can extend SKSg with the same substitution rule as for Frege. The rule is used like\nother proper rules of system SKSg, so its instances are interleaved with =-rule instances.\nDefinition 5.9. An sSKSg proof is a proof of SKSg where, in addition to the inference\nsteps generated by rules of SKSg, we admit inference steps obtained as instances of the\nsubstitution rule\nA\nsub Aσ.\nThis rule does not fit any of the usual deep-inference rule classes (see Section 2), and (as\nin Frege systems) is not sound, in the sense that the premiss does not imply the conclusion.\nHowever, of course, if the premiss is provable the conclusion also is.\nRemark 5.10. Notice that instances of the substitution rule cannot be used inside a context;\nfor example, the expression on the left is not a valid sSKSg proof, while the one on the\nright is:\nt\ni↓A ∨ ̄A\nsub? (B ∧C) ∨ ̄A\n,\nt\ni↓A ∨ ̄A\nsub (B ∧C) ∨[ ̄B ∨ ̄C]\n.\nIn the so-called ‘Formalism B’ of deep inference, which is currently under development\n[Gug04], and for which all the proof-complexity results in this article apply unchanged,\nsubstitution becomes part of the composition mechanism of proofs, rather than an odd\nextension to the set of rules.\nFor the time being, we can establish the promised p-equivalence of all extended systems\nby completing the diagram in the Introduction with the last two missing steps.\nTheorem 5.11. For every xSKSg proof of size n there exists an sSKSg proof of the same\nformula and whose length and size are, respectively, O(n) and O(n2).\nProof. Consider the xSKSg proof\n[ ̄A1 ∨β1] ∧[ ̄β1 ∨A1] ∧· · · ∧[ ̄Ah ∨βh] ∧[ ̄βh ∨Ah]\nΦ ∥\n∥SKSg\nα\n,\nwhere\nA1 < β1, α\n,\n. . .\n,\nAh < β1, . . . , βh, α\n,\nand let us call its premiss γ. We can build the sSKSg proof\nt\ni↓ ̄γ ∨γ\n ̄γ∨Φ ∥\n∥SKSg\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(Ah ∧ ̄βh) ∨(βh ∧ ̄Ah)] ∨α\nsub [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(βh ∧ ̄βh) ∨(βh ∧ ̄βh)] ∨α\nc↓\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨(βh ∧ ̄βh)] ∨α\ni↑\n[(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1) ∨· · · ∨f] ∨α\n=\n...\n= [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1)] ∨α\nsub [(β1 ∧ ̄β1) ∨(β1 ∧ ̄β1)] ∨α\nc↓\n(β1 ∧ ̄β1) ∨α\ni↑\nf ∨α\n=\nα\n."},{"page":22,"text":"22\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\n□\nCorollary 5.12. sSKSg p-simulates xSKSg.\nTheorem 5.13. For every sSKSg proof of size n there exists a proof of the same formula\nin sFrege, whose length and size are, respectively, O(n4) and O(n5).\nProof. Every sSKSg proof has shape\nt\nΦ0 ∥\n∥SKSg\nα1\nsub α1σ1\nΦ1 ∥\n∥SKSg\n...\nΦh−1 ∥\n∥SKSg\nαh\nsub αhσh\nΦh ∥\n∥SKSg\nαh+1\n.\nBy Theorem 4.11, for each of Φ0, ..., Φh there exist Frege derivations Υ0, ..., Υh with\nthe same premiss and conclusion, respectively. We can then build the proof\nΥ0\nz }| {\n. . . , α1 ,\nΥ1\nz }| {\nα1σ1, . . ., . . . ,\nΥh−1\nz }| {\n. . . , αh ,\nΥh\nz }| {\nαhσh, . . . , αh+1\nin sFrege; the cited theorem also yields its length and size.\n□\nCorollary 5.14. sFrege p-simulates sSKSg.\nNothing prevents us from using Tseitin extension and the substitution rule with system\nSKS, or any other atomic or nonatomic CoS system. The integration of these mechanisms\ninto CoS is similar to their integration into Frege systems, as the simplicity of the argu-\nments showing p-equivalence testifies.\n6. Open Problems\nWe conclude the article with a list of open problems, some of which are currently in-\nvestigated by us and other researchers.\n6.1. Relation with Resolution and Other Formalisms. In this article, we explored the\nrelation between CoS and Frege systems, and in the cited literature the relation between\nCoS and Gentzen systems has been explored in depth. There are, of course, other for-\nmalisms, like resolution, whose relation with CoS might lead to some interesting research\ndirections. For example, the note [Gug03] shows how simply, compared to Gentzen sys-\ntems, KS expresses resolution (analytically, of course).\n6.2. Does Cocontraction Provide for an Exponential Speedup? As we argued in Re-\nmark 3.6, we do not know whether KSg p-simulates KSg ∪{c↑}, or, equivalently, whether\nKS p-simulates KS ∪{ac↑}.\nOur intuition, as well as some clues, like the mutual behaviour of the ‘atomic flows’\nof contraction and cocontraction (see [GG08]) would lead us to believe that cocontraction\nindeed provides for an exponential speedup. However, we know that in similar situations,\nlike for dag-like versus tree-like Frege systems, intuition was fallacious.\nIf cocontraction yields an exponential speedup, we obtain an even stronger analytic\nsystem than KSg, which is, in turn, stronger than analytic Gentzen. This would draw\ninterest to a hierarchy of analytic proof systems of different strength."},{"page":23,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n23\nUnless we prove the p-equivalence of KSg and KSg ∪{c↑}, we tend to consider cocon-\ntraction a simple rule-based mechanism for compressing proofs, like cut, extension, and\nsubstitution.\n6.3. Pigeonhole in Analytic CoS. Does the pigeonhole principle, in particular in its rela-\ntional variety, admit polynomially growing proofs in KS? If not, does it in KS ∪{ac↑}?\nInvestigating this problem could be relevant to the more general, following one (Prob-\nlem 6.4), about the ability of analytic CoS to simulate CoS, and so Frege. In fact, the\npigeonhole principle generates some of the hardest classes of tautologies known.\nWe note that in [Jap08], Japaridze shows polynomially growing proofs for the pigeon-\nhole class of tautologies in a deep-inference system over certain circuit-like sequents,\ncalled ‘cirquents’. In this case, the speedup is obtained by the sharing of logical expres-\nsions in circuits.\nIn [Jeˇr09], Jeˇr ́abek shows that there are polynomial-time constructible proofs in KS ∪\n{ac↑} of the functional and onto variants of the pigeonhole principle.\n6.4. Relative Strength of Analytic CoS and CoS. Some recent major progress has been\nmade in [Jeˇr09]. There, Jeˇr ́abek uses a construction on threshold formulae in the mono-\ntone sequent calculus, by Atserias, Galesi and Pudl ́ak [AGP02], to show that analytic CoS\nquasipolynomially simulates CoS. In [BGGP09], we provide a direct and simplified con-\nstruction based on atomic flows [GG08] and threshold formulae.\nBecause of these recent advances, we expect that analytic CoS p-simulates CoS. A more\nin-depth discussion of this subject is in [BGGP09]. If analytic CoS p-simulates CoS, then\nthere are polynomially growing proofs of the pigeonhole principle in analytic CoS, though\nnot necessarily in KS.\nWe think that investigating this problem will help us to better understand analyticity, in\norder to obtain for it a general and more useful definition than the one we have now. We\nfeel that the current notion is not satisfactory because it depends on the formalism and must\nbe defined by resorting to the syntactic structure of inference rules (or, worse, by indicating\nwhich rules are analytic and which are not).\nFor example, a more general, nonsyntactic definition of analyticity could be the follow-\ning: a rule is analytic if, given an instance of its conclusion, the set of possible instances\nof the premiss is finite (this is what we call a finitely generating rule in Section 2). In this\nsense, an atomic ‘finitary’ cut rule\nξ{a ∧ ̄a}\nfai↑\nξ{t}\n, such that a appears in ξ{ }, would be ana-\nlytic. However, [BG04] shows that we can easily transform proofs in SKS into smaller- or\nequal-size proofs that only use fai↑wherever ai↑was used. So, we could deem fai↑an ana-\nlytic rule, and the system obtained from SKS by substituting ai↑with fai↑an analytic one,\nand we could immediately conclude that analytic CoS p-simulates CoS. This ‘solution’,\nhowever, is way too cheap.\nWe prefer to think that fai↑is not an analytic rule, in some sense to be made precise. A\npossible point of attack is offered by the fact that fai↑is not a local rule: it requires checking\nthat a appears in its context, whose size is unbounded (see Remark 2.17). So, we think it\ncould be productive to look for a notion of analyticity that is based on boundedness instead\nof finiteness, and tackle the separation problem between analytic CoS and CoS under that\nnotion. The note [BG07] further explores this direction, but much more work is necessary.\n6.5. Strength of Analytic CoS Systems Plus Substitution. We showed that CoS and\nFrege systems are p-equivalent, and both remain p-equivalent when extended either with\nTseitin extension or substitution. However, CoS is more flexible than Frege, because it\nallows to ‘switch off’ two mechanisms that potentially provide for an exponential com-\npression of proofs: cut and cocontraction (see Problem 6.2).\nIt might be interesting to study the relative strength of systems obtained by removing\nfrom SKS ∪{sub} either ai↑or ac↑or both. (Rule aw↑can also be removed, but we do"},{"page":24,"text":"24\nPAOLA BRUSCOLI AND ALESSIO GUGLIELMI\nnot see a crucial role for it.) Notice that systems KS ∪{sub} and KS ∪{ac↑, sub} could be\nconsidered, in some sense, analytic, and we do not know their relative strength.\n6.6. Speedup of Deep Inference Over Any Bounded-Depth System. We saw in The-\norem 3.12 that analytic CoS exhibits an exponential speedup over analytic Gentzen, for\nStatman tautologies. We argued that, in this case, the speedup is obtained by a rather triv-\nial use of deep inference, because the depth at which inference has to be performed, in\norder to get the speedup, is constant. So, a natural question is whether there exists a class\nof tautologies that requires full-fledged deep inference in order to obtain efficient proofs.\nWe think we found such a class, which is defined as follows.\nConsider, for every propositional formula α, the following set of second-order formulae,\nfor n > 0:\ng(1, α) ≡∀β.[((α ∧β) ∧β) ∨( ̄β ∧ ̄β)]\n,\ng(n + 1, α) ≡∀β.[(g(n, α ∧β) ∧g(n, β)) ∨(g(n, ̄β) ∧g(n, ̄β))]\n.\nBy using these formulae as a template, we can generate a set of first-order formulae, where\nthe (complex) management of indices ensures their uniqueness:\nDefinition 6.1. Consider, for m, n ≥0\nh(1, m, α) ≡((α ∧βm+1) ∧βm+1) ∨( ̄βm+1 ∧ ̄βm+1)\n,\nh(n + 2, m, α) ≡(h(n + 1, m, α ∧β5n+1+m) ∧h(n + 1, 5n + m, β5n+1+m)) ∨\n(h(n + 1, 2 · 5n + m, ̄β5n+1+m) ∧h(n + 1, 3 · 5n + m, ̄β5n+1+m))\n.\nConsider now\nf(n) ≡h(n, 0, t)\n,\nfor n > 0\n.\nWe define the set DT = { f(n) | n > 0 }.\nThe program [Gug07a] can help in understanding the nature of these formulae.\nRemark 6.2. It is not difficult to verify that DT contains tautologies possessing analytic\nCoS proofs that grow polynomially in the size of the tautologies.\nThe analytic CoS proofs of the DT tautologies, when read bottom-up, work by applying\ninteractions starting from the deepest subformulae. When this cannot be the case, we\nconjecture that the size of the proofs grows exponentially.\nDefinition 6.3. The and/or depth of a formula is the maximum number of alternations of\nconjunctions and disjunctions in the formula tree; the and/or depth of a context ξ{ } is the\nnumber of alternations of conjunctions and disjunctions between the hole and the root of\nthe context tree. We define a bounded-depth CoS proof system as a CoS proof system\nwhose inference rules only generate inference steps at a bounded depth, namely inference\nsteps\nξ{γ}\nν ξ{δ} are such that, if\nγ\nν δ is a rule instance then the and/or depth of ξ{ } is bounded\nby a given constant, and the same restriction holds for the contexts in the context closure\ncondition of relation =.\nRemark 6.4. Note that the nonatomic rules interaction (identity), cointeraction (cut),\ncontraction and cocontraction require establishing duality or identity of formulae of un-\nbounded and/or depth. So, their adoption might be considered an implicit use of deep in-\nference. However, the atomic counterparts of these rules do not suffer this problem because\nthe ‘deep checking’ is delegated to the inference mechanism. For this reason, proving the\nfollowing conjecture is better done in the analytic part of system SKS.\nConjecture 6.5. In any analytic bounded-depth CoS proof system, the tautologies in DT\nonly have proofs that grow exponentially in their size."},{"page":25,"text":"ON THE PROOF COMPLEXITY OF DEEP INFERENCE\n25\n7. Conclusion\nIn this article, we showed that the calculus of structures (CoS) has the same character-\nistics of the Frege formalism in terms of proof complexity, including when extended with\nTseitin extension and substitution.\nWe know that, contrary to Frege, CoS has a rich proof theory, and its proof systems\nenjoy several properties, arguably relevant to proof complexity, that cannot be observed in\nother formalisms, like locality for all inference rules. We also know that other logics, like\nmodal logics, enjoy simple and modular presentations in deep inference, which should help\nin proof complexity investigations. This article establishes the basic connection between\nproof theory in deep inference and proof complexity.\nAs a consequence of its flexibility in inference rule design, CoS admits a notion of\nanalyticity that is more flexible than its counterpart for Gentzen systems. We can then\nexplore the strength of analytic systems in finer detail than possible in Gentzen systems. In\nthis article, we moved forward the boundary between polynomial and exponential analytic\nproofs by proving Statman tautologies with polynomial, analytic deep-inference proofs.\nWe included a list of open problems and currently active research directions.\nAcknowledgements. 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IEEE,\n2005.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/FreeBool-long.pdf.\n[LS05b]\nFranc ̧ois Lamarche and Lutz Straßburger. Naming proofs in classical propositional logic. In Paweł\nUrzyczyn, editor, Typed Lambda Calculi and Applications, volume 3461 of Lecture Notes in Com-\nputer Science, pages 246–261. Springer-Verlag, 2005.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/namingproofsCL.pdf.\n[LS06]\nFranc ̧ois Lamarche and Lutz Straßburger. From proof nets to the free *-autonomous category. Logical\nMethods in Computer Science, 2(4):3:1–44, 2006.\nhttp://arxiv.org/pdf/cs.LO/0605054.\n[Rec76]\nRobert A. Reckhow. On the Lengths of Proofs in the Propositional Calculus. PhD thesis, University\nof Toronto, 1976.\n[SL04]\nLutz Straßburger and Franc ̧ois Lamarche. On proof nets for multiplicative linear logic with units.\nIn J. Marcinkowski and A. Tarlecki, editors, CSL 2004, volume 3210 of Lecture Notes in Computer\nScience, pages 145–159. Springer-Verlag, 2004.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/multPN.pdf.\n[Sta78]\nRichard Statman. Bounds for proof-search and speed-up in the predicate calculus. Annals of Mathe-\nmatical Logic, 15:225–287, 1978.\n[Sto07]\nPhiniki Stouppa. A deep inference system for the modal logic S5. Studia Logica, 85(2):199–214,\n2007.\nhttp://www.iam.unibe.ch/til/publications/pubitems/pdfs/sto07.pdf.\n[Str02]\nLutz Straßburger. A local system for linear logic. In M. Baaz and A. Voronkov, editors, LPAR 2002,\nvolume 2514 of Lecture Notes in Artificial Intelligence, pages 388–402. Springer-Verlag, 2002.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/lls-lpar.pdf.\n[Str03a]\nLutz Straßburger. Linear Logic and Noncommutativity in the Calculus of Structures. PhD thesis,\nTechnische Universit ̈at Dresden, 2003.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/dissvonlutz.pdf.\n[Str03b]\nLutz Straßburger. MELL in the calculus of structures. Theoretical Computer Science, 309:213–285,\n2003.\nhttp://www.lix.polytechnique.fr/ ̃lutz/papers/els.pdf.\n[Tiu06a]\nAlwen Tiu. A local system for intuitionistic logic. In M. Hermann and A. Voronkov, editors, LPAR\n2006, volume 4246 of Lecture Notes in Artificial Intelligence, pages 242–256. Springer-Verlag, 2006.\nhttp://users.rsise.anu.edu.au/ ̃tiu/localint.pdf.\n[Tiu06b]\nAlwen Tiu. A system of interaction and structure II: The need for deep inference. Logical Methods\nin Computer Science, 2(2:4):1–24, 2006.\nhttp://arxiv.org/pdf/cs.LO/0512036.\n[TS96]\nA.S. Troelstra and H. Schwichtenberg. Basic Proof Theory, volume 43 of Cambridge Tracts in The-\noretical Computer Science. Cambridge University Press, 1996.\nUniversity of Bath, Bath BA2 7AY, UK,\nhttp://cs.bath.ac.uk/pb/ and\nhttp://alessio.guglielmi.name/res"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"ξ{a ∧B} ≡b ∧[(a ∧B) ∨c]. The size of ξ{ } is defined as |ξ{ }| = |ξ{a}| −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"{a1/b1, a2/b2, . . .}; we use ρ for renamings. A renaming of α by ρ = {a1/b1, a2/b2, . . .}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"An instance of α by σ = {A1/β1, A2/β2, . . . } is indicated by ασ and is obtained by si-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"α ∨f = α","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"α ∧t = α","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"t ∨t = t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"f ∧f = f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"if α = β then ξ{α} = ξ{β}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"Figure 1. Equality = on formulae.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"Definition 2.8. The equality relation = on formulae is defined by closing the equations in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"the inference rule = as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"= β , where α = β.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"The following remark helps in assessing how much complexity is hidden in =.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"Remark 2.9. It is possible to decide α = β in polynomial time by reducing α and β to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"some canonical form and comparing the canonical forms. A canonical form under = of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"steps, where n = |α| + |β|. Notice that at each step the size of the formula stays the same or","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"such that ν1, . . ., νk is a sequence of inference rules that alternate between the = rule and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"generated by =. The size |Φ| of derivation Φ is the number of unit, atom, and variable","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"Definition 2.13. CoS proof systems KSg = {i↓, w↓, c↓, s}, SKSg = KSg ∪{i↑, w↑,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"c↑}, KS = {ai↓, aw↓, ac↓, s, m} and SKS = KS ∪{ai↑, aw↑, ac↑} are defined in Figures 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"= γ ∨[(((α ∧d) ∧c) ∧[α ∨ ̄α]) ∨δ]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"= [ ̄α ∨γ] ∨[((α ∧c) ∧(α ∧d)) ∨δ]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"= A ∧[f ∨t]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"= A ∧t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"= [c ∨(a ∧b)] ∨[c ∨(a ∧b)]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"where n = |α|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"consists of an instance of = and in the latter the instance of i↑is also an instance of ai↑. We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"and we obtain a derivation whose length is O(n), and so its size is O(n2), where n = |α|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"where n = |α|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"= f ∧[f ∨t]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"= f ∨γ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"= f ∧f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"n = |α|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"where n = |α|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"n = |α|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"translate a Gentzen multiset φ = α1, . . ., αh into α1 ∨· · · ∨αh.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"= φ ∨f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"= rule have been omitted (they depend on the exact translation of Gentzen multisets into","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"= [B ∨ψ] ∧[A ∨φ]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"= φ ∨(A ∧[B ∨ψ])","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"= [ ̄A ∨ψ] ∧[A ∨φ]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"= φ ∨(A ∧[ ̄A ∨ψ])","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"instances of the = rule have been omitted, and Φ1 ∧Φ2 stands for the derivation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"i=n αi ≡αn ∧βn−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"Remark 3.9. |Sn| = 2 + 2 Pn−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"k=1 |γn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"k| + 2 = 2 Pn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"k=2(|βn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"k| + 1) + 4 = 2n2 + 2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"previous derivation are the largest. Since |αn+1| = 2 and |βn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"k| = 2(n −k + 1), their size is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"k=2 |βn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"k|)+2 = 2n2+8n+2. Each i↓macro inference step involves one s","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"and two ai↓steps in KS, for a total of six steps, including = ones; each c↓macro inference","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"2n · 6 + 2n · 2 + (2n −1) · 6 = 28n −6, and so |Φ| ≤(28n −6)(2n2 + 8n + 2) ∈O(n3).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"= ̄A ∨[ ̄B ∨(A ∧B)]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"= (A ∧(B ∧ ̄C)) ∨[ ̄A ∨[( ̄B ∧t) ∨C]]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"= (A ∧(B ∧ ̄C)) ∨[(A ∧ ̄B) ∨[[ ̄A ∨ ̄A] ∨C]]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"We proceed by induction on the length of Υ = β1, . . ., βk, β and we prove the existence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":". The base case k = 0 is as follows: 1) if β is a premiss,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"then Φ′ = β; 2) if β ≡Fiσ, for some i and σ, then Φ′ = Φiσ. For the inductive step,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"given Υk = β1, . . . , βk and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"• if β is a premiss, then Φ′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"• if β ≡Fiσ, for some i and substitution σ, then Φ′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"= (β1 ∧· · · ∧βk) ∧t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"size of a formula appearing in Υ, and so |Φ| ∈O(n2), where |Υ| = n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"‘amount of inference’ of =-rule instances. The next two lemmas take care of these two","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"m = |ξ{ }| and n = |ξ{α} →ξ{β}|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"Υ1 = α →β, Υ′{A/α, B/β,C/γ1}, α ∨γ1 →β ∨γ1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"Lemma 4.9. For every SKS formulae α and β such that α = β there is a Frege derivation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"with premiss α, conclusion β, length O(n3), and size O(n4), where n = |α| + |β|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"and size O(m′), where m′ = |γ| + |γ′|. By Lemma 4.8, for every ξ{ } there is a derivation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"where m = |ξ{γ} →ξ{γ′}|. By concatenating the proof and derivation so obtained, we can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"derivation with premiss α, conclusion β, length O(n), and size O(n2), where n = |α| + |β|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"σh = Ah/βh, ..., σ1 = A1/β1, and by prepending to it proofs of the tautologies β1 ↔β1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"other proper rules of system SKSg, so its instances are interleaved with =-rule instances.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"= [(A1 ∧ ̄β1) ∨(β1 ∧ ̄A1)] ∨α","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"We define the set DT = { f(n) | n > 0 }.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"condition of relation =.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":83170,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}