| {"paper_meta":{"paper_id":"arxiv:0706.3412","title":"0706.3412","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:0706.3412v1 [cs.CC] 22 Jun 2007\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n1\nOn Canonical Forms of Complete Problems via First-order\nProjections\nNerio Borges\nDepartamento de Matem ́aticas\nUniversidad Sim ́on Bol ́ıvar\nCaracas, Venezuela\nnborges@usb.ve\nBlai Bonet\nDepartamento de Computaci ́on\nUniversidad Sim ́on Bol ́ıvar\nCaracas, Venezuela\nbonet@ldc.usb.ve\nOctober 25, 2018\nAbstract\nThe class of problems complete for NP via first-order reductions is known to be\ncharacterized by existential second-order sentences of a fixed form. All such sentences\nare built around the so-called generalized IS-form of the sentence that defines Inde-\npendentSet. This result can also be understood as that every sentence that defines\na NP-complete problem P can be decomposed in two disjuncts such that the first one\ncharacterizes a fragment of P as hard as IndependentSet and the second the rest of\nP. That is, a decomposition that divides every such sentence into a a “quotient and\nresidue” modulo IndependentSet.\nIn this paper, we show that this result can be generalized over a wide collection of\ncomplexity classes, including the so-called nice classes. Moreover, we show that such\ndecomposition can be done for any complete problem with respect to the given class, and\nthat two such decompositions are non-equivalent in general. Interestingly, our results\nare based on simple and well-known properties of first-order reductions.\nKeywords: Finite Model Theory, Complexity Theory, First-Order Reductions, Canonical\nForms\n1\nIntroduction\nDescriptive complexity studies the interplay between complexity theory, finite model theory\nand mathematical logic. Since its inception in 1974 [3], descriptive complexity has been\nable to characterize all major complexity classes in term of logical languages independent of\nany computational model, thus suggesting that the computational complexity of languages\nis a property intrinsic to them and not an accidental consequence of our choice for the\ncomputational model.\nIn descriptive complexity, problems are understood as sets of (finite) models which\nare described by logical formulae over given vocabularies. Reductions between problems\ncorrespond to logical relations between the set of models that characterize the problems.\nAs important as the notion of polynomial many-one reductions in structural complexity,\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n2\nthere is the notion of first-order reductions in descriptive complexity, and among such, the\nfirst-order projections (fops). A fop is a very weak type of polynomial-time reduction whose\nstudy have provided interesting results such as that common NP-complete problems like\nSat, Clique and others remain complete via fop reductions, and that such NP-complete\nproblems can be described by logical sentences in a canonical form [2, 5].\nIn this paper we continue the study of the syntactic aspects of complete problems via fop\nreductions extending the work of Medina and Immerman [7, 6]. In particular, we provide a\ngeneral characterization of complete problems via fops for a large collection of complexity\nclasses that cover well beyond just NP, including classes like P, PSPACE, Σp\nn and Πp\nn, and\nothers. Interestingly, our results rely on very general assumptions and tools already known\nin the field.\nThe paper is organized as follows. In Sect. 2, we give standard definitions and known\nresults which provide the theoretical framework of the paper and make it self contained.\nSect. 3 contains our main result, namely the generalization of the Medina-Immerman result,\ntogether with relevant remarks and some examples. Later, Sect. 4 shows a general result\nabout the existence of non-isomorphic problems via fop reductions, which implies that our\ncanonical form is indeed minimal. Finally, Sect. 5 concludes with a brief summary and\ndirections for future work.\n2\nPreliminaries\n2.1\nLogics, Finite Models, and Decision Problems\nA logical vocabulary is a tuple τ = ⟨Ra1\n1 , . . . , Rar\nr , c1, . . . , cs, f r1\n1 , . . . , f rt\nt ⟩where the Rjs are\nrelational symbols of arity aj, cis are constant symbols, and the fks are rk-ary functional\nsymbols. A structure for τ, also referred as τ-structure or just structure if τ is clear from\ncontext, is a tuple A = ⟨|A|, RA\n1 , . . . , RA\nr , cA\n1 , . . . , cA\ns , f A\n1 , . . . , f A\nt ⟩where\n• |A| is the universe (or domain) of A,\n• RA\nj ⊆|A|aj is a aj-ary relation over |A|,\n• cj ∈|A| is an element of the universe, and\n• f A\nk : |A|rk →|A| is a total rk-ary function over |A|.\nFor vocabulary τ, STRUC[τ] denotes the class of all finite structures, i.e.\nthose whose\nuniverse is an initial segment {0, 1, . . . , n −1} of N.\nIn addition to above logical symbols, we also have the numerical relational symbols ‘=’,\n‘≤’, ‘BIT’ and ‘suc’, and constants ‘0’ and ‘max’, which are assumed to belong to each\nvocabulary, and have fixed interpretations on every structure A:\n• = and ≤are interpreted as the usual equality and order on N,\n• A ⊨BIT(i, j) iffthe j-th bit in the binary representation of i is 1,\n• A ⊨suc(x, y) iffy is the successor of x in the usual order on N, and\n• 0 and max denote the least and greatest element in |A|.\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n3\nIf L denotes a logic, the language L[τ] is the set of all well-formed formulae of L over the\nvocabulary τ. A numerical formula in L[τ] is a formula with only numerical symbols. For\nexample, SO∃[τ] is the set of all second-order formulae of form ∃Q1 · · · ∃QnΦ where the Qis\nare relational variables and Φ is a first-order formula over over vocabulary τ. As usual, FO\ndenotes first-order logic and SO denotes second-order logic.\nA formula with no free variables is a sentence. For sentence φ ∈L[τ], the class of all finite\nmodels that satisfy φ is denoted as MOD[φ]. For fixed τ, it is possible to code every finite\nτ-structures into a sequence of bits, i.e. a binary string, using a map MOD[τ] ⇝{0, 1}∗.\nHence, a collection of finite models can be represented as a collection of strings, or language.\nIn descriptive complexity, a decision problem P is characterized by a subset of models\nfrom STRUC[τ] for some fixed τ. For example, the problem Clique can be characterized\nby structures A = ⟨|A|, EA, kA⟩over the vocabulary τ = ⟨E2, k⟩, where E is a binary\nrelational symbol and k is a constant, such that G = (|A|, EA) makes up an undirected\ngraph and kA ∈{0, . . . , |A|−1} denotes the size of a clique in G. Such models are typically\ncharacterized by a sentence Ψ over some fragment L. The problem Clique, for example,\ncan be characterized with a SO∃sentence over τ [3]; see below.\n2.2\nFirst-Order Queries, Fops, and Duals\nLet τ and σ be two vocabularies where σ = ⟨Ra1\n1 , . . . , Rar\nr , c1, . . . , cs⟩has no functional\nsymbols (from now on, we only consider vocabularies with no functional symbols). Let\nk ≥1 and consider the tuple I = ⟨φ0, . . . , φr, ψ1, . . . , ψs⟩of r + s + 1 first-order formulae in\nFO[τ] of form φ0(x1, . . . , xk), φi(x1, . . . , xkai) and ψj(x1, . . . , xk) = (x1 = c′\nj1∧· · ·∧xk = c′\njk)\nwhere the c′\njis are constant symbols from τ (possibly with repetitions). That is, φ0 has at\nmost k free variables among x1, . . . , xk, φi has at most kai free variables among x1, . . . , xkai,\nand ψj denotes a tuple in {c′ : c′ ∈τ}k.\nSuch tuple defines a mapping A ⇝I(A), called a first-order query of arity k, from\nτ-structures into σ-structures given by:\n• the universe |I(A)| .= {(u1, . . . , uk) ∈|A|k : A ⊨φ0(u1, . . . , uk)} is ordered lexico-\ngraphically,\n• the relations are RI(A)\ni\n.= {( ̄u1, . . . , ̄uai) ∈|A|kai : A ⊨φi( ̄u1, . . . , ̄uai)},\n• the constants are cI(A)\nj\n.= ̄u for the unique ̄u with A ⊨φ0( ̄u) ∧ψj( ̄u).\nFurthermore, if for T ⊆STRUC[τ] and S ⊆STRUC[σ], it is true that A ∈T iffI(A) ∈S,\nthen I is called a first-order reduction from T to S.\nA first-order query is called a first-order projection (fop) if φ0 is numerical and each φi\nhas form:\nφj( ̄x) ≡α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x))\nwhere the αis are numerical and mutually exclusive, and each λi is a τ-literal. Projections\nare typically denoted by the letter p. If p is a reduction from Π to Ψ, we write p : Π ≤fop Ψ,\nand if Π is complete via ≤fop reductions we say that Π is ≤fop-complete.\nProjections have interesting properties. For example, projections are a special case of\nValiant’s non-uniform projections [10]. For our purposes, we are interested in the fact that\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n4\nfor each projection p there is a first-order sentence βp ∈FO[σ] that characterizes the image\nof p, i.e. B ⊨βp iffB = p(A) for some A ∈STRUC[τ]. The sentence βp is called the\ncharacteristic sentence of p [1].\nFinally, there is a syntactic operator associated to a first-order query that plays a fun-\ndamental role in our results. For a first-order query I : STRUC[τ] →STRUC[σ], the dual\noperator bI maps formulae in L[σ] to formulae in L[τ] in such a way that\nA ⊨bI(θ)\nif and only if\nI(A) ⊨θ\nfor all θ ∈L[σ] and A ∈STRUC[τ] [5, Sect. 3.2].\n2.3\nComplexity Classes\nFor a family F of proper complexity functions [9], we consider the complexity classes\nTIME(F) = ∪f∈FTIME(f), and similarly for non-deterministic time and space. A com-\nplexity class C defined by F is nice if it has a universal problem of the form\nUC = {⟨Mi, ω, 1t⟩: Mi accepts ω within fi(t) resources}\nwhere L(Mi) ∈C and fi ∈F bounds the resources of Mi. Some well-known classes that are\nnice are L, NL, P, NP and PSPACE. Allender et al. [1] showed that if Π is ≤fop-complete\nfor a nice class C, then it is complete via injective fops of arity at least 2. The following\nproperties are shown easily:\nProposition 1 Let C be a complexity class defined by family F.\nThen, (a) if C is a\ndeterministic class and F is closed under sums, then C is closed under finite unions; (b)\nif C is a nondeterministic class and F is such that for every f, g ∈F there is h ∈F with\nf, g ≤h, then C is closed under finite unions; (c) if C is closed under finite unions and C\nis captured by logic L, then L is closed under disjunctions.\nThe nice classes L, NL, P, NP and PSPACE are known to be characterized by SO-\nDetKrom, SO-Krom, SO-Horn, SO∃and SO+TC respectively [4, 5]. Additionally, Σp\nk and\nΠp\nk are characterized by SO∃∀· · · Qk and SO∀∃· · · Q′\nk sentences where Qk = ∃, Q′\nk = ∀if k\nis odd, and Qk = ∀, Q′\nk = ∃if k is even. Thus, by the proposition, all these logical fragments\nare closed under disjunctions, and also under conjunctions with first-order formulae. We\nwill make use of these facts later.\n3\nCanonical Forms of Complete Problems\nMedina and Immerman characterized ≤fop-complete problems for NP syntactically using\nthe IndependentSet problem. This problem consists of checking whether an input graph\nG has an independent set of size k. IndependentSet is known to be complete for NP\nunder different notions of reductions, and in particular, under fop reductions [7]. Indepen-\ndentSet in characterized by the following SO∃[τ] sentence, for τ = ⟨E2, k⟩:\nΨIS = (∃f ∈Inj)(∀x, y)\n \nx ̸= y ∧fx ≤k ∧fy ≤k →¬E(x, y)\n \n(1)\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n5\nwhere ‘f ∈Inj’ means that f is a total and 1-1 function, i.e. an ordering of the elements of\nthe universe, and fx denotes f(x). Although it seems that (1) quantifies over a functional\nvariable, f is indeed a relational variable such that fx is the unique element such that\nf(x, fx). The condition f ∈Inj is easily defined in first-order logic. Observe that the only\nsecond-order variable in (1) is f which is existentially quantified.\nTheorem 2 ([7]) Let L ⊆STRUC[σ] be a NP problem characterized by Ψ ∈L[σ] where\nσ = ⟨Q1⟩is the vocabulary of binary strings. Then, a problem L is NP-complete via ≤fop\nreductions iffthere is an injective fop p : STRUC[⟨E2, k⟩] →STRUC[σ] such that\nΨ ≡(βp ∧ΥIS) ∨(¬βp ∧Λ)\n(2)\nwhere βp ∈FO[σ] is the characteristic sentence of p, ΥIS ∈SO∃[σ] is a generalized IS-form\n[7], and Λ is a SO∃[σ] sentence.\nIntuitively, this result says that if sentence Ψ characterizes a ≤fop-complete problem L\nfor NP, then it can be decomposed in two disjuncts Ψ = ΨIS ∨Ψrest such that MOD[ΨIS]\nis ≤fop-complete for NP and MOD[Ψrest] equals the “rest” of L which is not necessarily\ncomplete.\nOur main contribution is to show that above result can be generalized over a wide\ncollection of complexity classes, including the nice classes, and that such decomposition can\nbe done modulo any ≤fop-complete problem for the given class. Moreover, we also show\ntwo such decompositions are not in general equivalent.\nThe main obstacle for such generalization is to take care of the sentence ΥIS for classes\ndifferent than NP. As it will be shown, we do not have to consider each different class\nin isolation, since the corresponding Υ sentences will be the duals of the sentence Ψ that\ncharacterize the complete problem.\nLet us first define the relation ∼=Π over STRUC[τ] with respect to a given problem\nΠ ⊆STRUC[τ]. For structures A and B, define\nA ∼=Π B\niff\n(A ∈Π ⇔B ∈Π) .\n(3)\nClearly, ∼=Π is an equivalence relation that partitions STRUC[τ] into Π and its complement.\nBy using dual operators and the equivalence relation, we are able to show the following\ngeneralization of Theorem 2. In the following, τ and σ refer to any two vocabularies.\nTheorem 3 (Main) Let C be a complexity class captured by fragment L closed under\ndisjunctions and closed under conjunctions with FO. Let Π ⊆STRUC[τ] be a ≤fop-complete\nproblem for C characterized by Ψ ∈L[τ], and B a problem over vocabulary σ. Then, B is\n≤fop-complete for C if and only if there is a fop p : STRUC[τ] →STRUC[σ] such that for\nall B ∈STRUC[σ]:\nB ∈B\niff\nB ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ)\n(4)\nwhere\n(a) βp ∈FO[σ] is the characteristic of p, i.e. B ⊨βp iffB ∈p(STRUC[τ]),\n(b) Λ ∈L[σ], and\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n6\n(c) I : STRUC[σ] →STRUC[τ] is a first-order query such that for all A ∈STRUC[τ],\nI(p(A)) ∼=Π A.\nProof: For the necessity, assume that B is ≤fop-complete for C; i.e. B is characterized by\nsome sentence Λ ∈L[σ] and there is p : Π ≤fop B. For B ∈B we consider the two cases\nwhether B ̸∈p(STRUC[τ]) or not. For the first case, B ⊨¬βp ∧Λ. For the second case,\nB ⊨βp and\nB = p(A)\n(for some A ∈STRUC[τ] by (a))\n=⇒\nA ∈Π\n(since p is reduction)\n=⇒\nA ⊨Ψ\n(Ψ characterizes Π)\n=⇒\nI(p(A)) ⊨Ψ\n(by condition (c))\n=⇒\np(A) ⊨bI(Ψ)\n(def. of dual of I) .\nTherefore, B ∈B =⇒B ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). Now, let B ∈STRUC[σ] be such that\nB ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). If B ⊨Λ, then B ∈B. Otherwise,\nB ⊨βp ∧bI(Ψ)\n=⇒\nB = p(A) and p(A) ⊨bI(Ψ)\n(for some A ∈STRUC[τ])\n=⇒\nI(p(A)) ⊨Ψ\n(def. of dual)\n=⇒\nA ⊨Ψ\n(by (c))\n=⇒\nA ∈Π\n(Ψ characterizes Π)\n=⇒\nB ∈B\n(since p is reduction) .\nIt remains to show that there are first-order queries satisfying (c). Since Π is complete,\nthere is a fop I : STRUC[σ] →STRUC[τ] that reduces p(Π) to Π. Note that p(Π) ⊆B\nsince p is also a reduction. For A ∈STRUC[τ], observe\nA ∈Π ⇒p(A) ∈p(Π) ⇒I(p(A)) ∈Π ,\nI(p(A)) ∈Π ⇒p(A) ∈p(Π) ⇒p(A) ∈B ⇒A ∈Π .\nThus, I : p(Π) ≤fop Π satisfies A ∈Π iffI(p(A)) ∈Π; i.e. A ∼=Π I(p(A)).\nFor the sufficiency, assume there is a fop p : STRUC[τ] →STRUC[σ] such that (4) holds\nfor all B ∈STRUC[σ]. We need to show that B is complete for C. The inclusion B ∈C\nis direct from the closure properties on L. For the hardness, we show that p is indeed a\nreduction from Π to B. For A ∈STRUC[τ], we have p(A) ⊨βp. If A ∈Π, then\nA ⊨Ψ ⇒I(p(A)) ⊨Ψ ⇒p(A) ⊨bI(Ψ) ⇒p(A) ∈B .\nOn the other hand, if p(A) ∈B, then\np(A) ⊨βp ⇒p(A) ⊨bI(Ψ) ⇒I(p(A)) ⊨Ψ ⇒A ⊨Ψ ⇒A ∈Π .\nThus, A ∈Π iffp(A) ∈B, p is a reduction, and B is complete.\n□\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n7\nCorollary 4 The theorem holds if the first-order query I is the reduction I : p(Π) ≤fop Π\nwhich exists since Π is complete.\nMoreover, a first-order query J satisfying (c) is essentially equivalent (with respect to\nΨ) to the reduction I : p(Π) ≤fop Π. Indeed, for such J and a finite σ-structure B = p(A)\nfor A ∈STRUC[τ],\nB ⊨bJ(Ψ) ⇐⇒J(B) ⊨Ψ ⇐⇒A ⊨Ψ ⇐⇒I(B) ⊨Ψ ⇐⇒B ⊨bI(Ψ) .\nIf we consider nice complexity classes, then the fop p can be assumed to be injective by\na result of Allender et. al [1].\nCorollary 5 For nice classes, the fop p : STRUC[τ] →STRUC[σ] can be assumed to be\ninjective.\nTo see that Theorem 2 is equivalent to Corollary 5 when C = NP, let τ = ⟨E2, k⟩and\nσ = ⟨Q1⟩be the vocabularies for graphs and binary strings respectively, and consider a\nproblem L ⊆STRUC[σ] complete for NP characterized by ΨL. According to Theorem 2,\nΨL ≡(βp ∧ΥIS) ∨(¬βp ∧Λ)\nwhere p : IndependentSet →L is a first-order projection and Λ is a SO∃sentence. On\nthe other hand, according to Corollary 5, ΨL also satisfies\nΨL ≡(βp ∧bI(ΨIS)) ∨(¬βp ∧Λ′) .\nAs shown before, ΥIS and bI(ΨIS) are equivalent on p(STRUC[τ]), and thus Λ and Λ′ must\nbe equivalent on STRUC[σ] ∩MOD[¬β].\n3.1\nExamples\nConsider Clique ⊆STRUC[τ = ⟨E2, k⟩] characterized by the SO∃sentence\nΨCL = (∃f ∈Inj)(∀x, y)\n \nx ̸= y ∧fx ≤k ∧fy ≤k →E(x, y)\n \n.\nFor σ = τ, it is not hard to see that IndependentSet can be reduced to Clique using\nthe fop p = λxy⟨φ0, φ1, ψ⟩, of arity 1, where\nφ0(x) = true ,\nφ1(x, y) = ¬E(x, y) ,\nψ(x) = (x = k) .\nClearly, if A = ⟨|A|, EA, kA⟩, then |p(A)| = |A|, Ep(A) = |A|2 \\ EA and kp(A) = kA.\nTherefore, p(p(A)) = A for all A ∈STRUC[τ], and hence\np(p(A)) ∈IndependentSet\niff\nA ∈IndependentSet.\nFurthermore, βp = true and since Clique is also known to be NP-complete with respect\nto ≤fop reductions, we have\nΨCL ≡(βp ∧bp(ΨIS)) ∨(¬βp ∧Γ) = bp(ΨIS) .\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n8\nConversely, beginning with the observation tha βp = true and bp(ΨIS) = ΨCL we can\nconclude, by Theorem 3, that Clique is ≤fop-complete for NP. We call this formulation\nof Clique as its canonical form with respect to IndependentSet. In this example, the\nformula ΨCL was already in its canonical form with respect to IndependentSet.\nFor a second example, consider the problem SubGraphIso defined by tuples ⟨G, G′⟩\nsuch that the graph G contains a subgraph isomorphic to graph G′. Such tuples can be\nexpressed with the vocabulary σ = ⟨F 2, H2, k⟩where F and H define the edges of G\nand G′, and the constant k defines the initial segment {0, . . . , k −1} for the edges of G′.\nAmong other things, instances of SubGraphIso are identified with structures B in which\nHB ⊆{0, . . . , k −1}2. SubGraphIso is defined by the SO∃sentence ΨSG\n(∃f ∈Inj)(∀x, y)\n \nx ̸= y ∧fx < k ∧fy < k →(H(fx, fy) →F(x, y))\n \n.\nA fop reduction p from Clique into SubGraphIso outputs ⟨G, Kk, k⟩on input ⟨G, k⟩. The\nfop is p = ⟨φ0, φ1, φ2, ψ⟩given by\nφ0 = true , φ1 = E(x, y) , φ2 = (x < k ∧y < k) , ψ = (x = k) .\nThe characteristic sentence of p is\nβp = x < k ∧y < k →F(x, y) .\nThe reduction I : p(Clique) ≤fop Clique given by I = ⟨φ0 = true, φ1 = F(x, y)⟩satisfies\nB ∈p(Clique) if and only if I(B) ∈Clique for all B. Since ΨSG is equivalent to (βp ∧\nbI(ΨCL)) ∨(¬βp ∧ΨSG), then, by Corollary 5, SubGraphIso is complete for NP via ≤fop\nreductions.\nFinally, other classes that satisfies the conditions of Corollary 5 are L, NL, P, PSPACE,\nand all Σp\nk and Πp\nk.\n4\nNon-Isomorphic Complete Problems for Nice Classes\nThe next result is a more general version of one already known for NP [7]. The proof is\nanalogous to the NP case. Among other things. it implies that we cannot get rid of the\ndisjunction in Corollary 5.\nTheorem 6 If C is a nice complexity class, then there are two C-complete problems that\nare not fop-isomorphic.\nProof: Let Γ ⊆{0, 1}∗be a ≤fop-complete problem for C, and define Γ′ = {ω0, ω1 : ω ∈Γ}.\nIt is easy to see that Γ′ is complete via fops; e.g. define the projection p : STRUC[τ =\n⟨S1⟩] →STRUC[σ = ⟨T 1⟩], of arity 2, as p = ⟨φ0(x, y), φ1(x, y)⟩where φ0(x, y) = (x =\n0)∨(x = 1∧y = 0) gives the domain of p(A) and φ1(x, y) = (x = 0∧S(y))∨(x = 1∧y = 0)\ngives T p(A). Thus, for A with domain |A| = {0, . . . , n −1}, φ0 defines\n|p(A)| = {(0, y) : 0 ≤y < n} ∪{(1, 0)} .\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n9\nFormula φ1 identifies the n bits of A with the tuples (0, x) and assigns “value”1 to the tuple\n(1, 0). Observe that the order induced in p(A) is (0, 0) < (0, 1) < · · · < (0, n −1) < (1, 0).\nTherefore, ω ∈Γ iffp(ω) ∈Γ′ which shows that Γ′ is complete.\nSince C is a nice complexity class, there is a fop p : STRUC[τ] →STRUC[σ] that is\ninjective, of arity k ≥2, that reduces Γ to Γ′. We will show that p cannot be onto by\nshowing that if ω ∈Γ, then either ω0 ̸∈p(Γ) or ω1 ̸∈p(Γ).\nConsider the formula φ( ̄x) that defines the interpretation of T in the structure p(A) of\nform\nφ( ̄x) = α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x)) .\nWe are going to show w0 ∈p(Γ) =⇒w1 ̸∈p(Γ). Suppose that |ω0| = n + 1 and that\nω0 = p(ω′) for some ω′ ∈Γ represented by the structure A. Each bit in ω0 corresponds to a\nk-tuple in p(A), i.e. ω0 ∼ ̄u0 ̄u2 . . . ̄un where ̄uj is 1 iffω′ ⊨φ( ̄uj). Since ̄un ∼0, ω′ ⊭α0( ̄un).\nConsider the two cases whether ω′ ⊨αl( ̄un) for some 1 ≤l≤r, or not.\nIn the latter case, we can conclude that ω′′ ⊭αl( ̄un) for every ω′′ ∈{0, 1}|w| and\n1 ≤l≤r since αl, being a numerical formula, obtains a value that only depends on the\nsize of its input; thus, ω1 ̸∈p(Γ).\nIn the former case, ω′ ⊨αl( ̄un), for some unique l, and ω′ ⊭λl( ̄un) since ̄un ∼0. Thus,\nsince λl( ̄un) is a literal, some bit of ω′ determines the value 0 for ̄un. On the other hand,\nobserve that\nω′ ∈Γ ⇐⇒p(ω′) = ω0 ∈Γ′ ⇐⇒ω ∈Γ\nwhere the first equivalence follows since p is a reduction, and the second by construction\nof Γ′. Furthermore, being p injective, implies that each bit in ω′ determines one bit in ω.\nTherefore, there is a bit in ω′ that determines two bits in ω0: one bit in ω and the rightmost\n0. If ω1 were in p(Γ), then the same bit in the preimage of ω1 would determine the same\nbit in ω and the rightmost 1, this time in an inconsistent manner. Therefore, ω1 ̸∈p(Γ). □\n5\nConclusions\nWe have extended the canonical form proposed by Medina and Immerman to all complexity\nclasses characterized by fragments L closed under disjunctions, and under conjunctions with\nFO. Although, Medina and Immerman’s method could be generalized to other nice classes\nbeyond NP, it requires the formulation of “generalized” sentences.\nOur method, on the\nother hand, circumvent this problem by considering the dual operator. Additionally, it is\nnot clear how Medina and Immerman’s method could be used to find canonical forms with\nrespect to problems that are not “graph” problems, or on classes that do not have complete\nproblems based on explicit graphs, e.g. PSPACE.\nAs for the near future, we are currently working on syntactic operators that preserve\ncompleteness via fops for general complexity classes.\nThis subject is also addressed by\nMedina [6] where syntactic operators I : L[τ] →L[σ], that map formulae into formulae,\nare defined such that if Ψ characterizes a NP-complete problem, then so is I(Ψ). We think\nthat as inverse images play a fundamental role in (mathematical) analysis, inverse images\nof syntactic transformations are worth to explore. In our case, we look for operators I such\nthat if I(Ψ) defines a complete problem, then Ψ also defines a complete problem; Nijjar also\n\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n10\nmention that such transformations are worth exploring [8]. We believe that such operators\ncould be use to establish completeness of problems in an easier way.\nReferences\n[1] E. Allender, J. Balc ́azar, and N. Immerman. A first-order isomorphism theorem. SIAM\nJournal of Computing, 26(2):555–567, 1997.\n[2] E. Dahlhaus. Reduction to NP-complete problems by interpretations. Lecture Notes\nin Computer Science, pages 357–365, 1984.\n[3] R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In\nR. Karp, editor, SIAM-AMS Proc. 7, pages 27–41, 1974.\n[4] E. Gradel. Capturing complexity classes by fragments of second-order logic. Theoretical\nComputer Science, 101:35–57, 1992.\n[5] N. Immerman. Descriptive Complexity. Springer, 1998.\n[6] J. A. Medina. A Descriptive Approach to The Class NP. PhD thesis, University of\nMassachusetts, Amherst, 1997.\n[7] J. A. Medina and N. Immerman. A syntactic characterization of NP-completeness. In\nProc. IEEE Symp. Logic in Computer Science, pages 241–250, 1994.\n[8] P. Nijjar. An attempt to automate np-hardness reductions via SO∃logic. Master’s\nthesis, Waterloo University, Ontario, Canada, 2004.\n[9] C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.\n[10] L. Valiant. Reducibility by algebraic projections. L’Ensignment Math ́ematique, 28:253–\n268, 1982.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0706.3412v1 [cs.CC] 22 Jun 2007\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n1\nOn Canonical Forms of Complete Problems via First-order\nProjections\nNerio Borges\nDepartamento de Matem ́aticas\nUniversidad Sim ́on Bol ́ıvar\nCaracas, Venezuela\nnborges@usb.ve\nBlai Bonet\nDepartamento de Computaci ́on\nUniversidad Sim ́on Bol ́ıvar\nCaracas, Venezuela\nbonet@ldc.usb.ve\nOctober 25, 2018\nAbstract\nThe class of problems complete for NP via first-order reductions is known to be\ncharacterized by existential second-order sentences of a fixed form. All such sentences\nare built around the so-called generalized IS-form of the sentence that defines Inde-\npendentSet. This result can also be understood as that every sentence that defines\na NP-complete problem P can be decomposed in two disjuncts such that the first one\ncharacterizes a fragment of P as hard as IndependentSet and the second the rest of\nP. That is, a decomposition that divides every such sentence into a a “quotient and\nresidue” modulo IndependentSet.\nIn this paper, we show that this result can be generalized over a wide collection of\ncomplexity classes, including the so-called nice classes. Moreover, we show that such\ndecomposition can be done for any complete problem with respect to the given class, and\nthat two such decompositions are non-equivalent in general. Interestingly, our results\nare based on simple and well-known properties of first-order reductions.\nKeywords: Finite Model Theory, Complexity Theory, First-Order Reductions, Canonical\nForms\n1\nIntroduction\nDescriptive complexity studies the interplay between complexity theory, finite model theory\nand mathematical logic. Since its inception in 1974 [3], descriptive complexity has been\nable to characterize all major complexity classes in term of logical languages independent of\nany computational model, thus suggesting that the computational complexity of languages\nis a property intrinsic to them and not an accidental consequence of our choice for the\ncomputational model.\nIn descriptive complexity, problems are understood as sets of (finite) models which\nare described by logical formulae over given vocabularies. Reductions between problems\ncorrespond to logical relations between the set of models that characterize the problems.\nAs important as the notion of polynomial many-one reductions in structural complexity,"},{"paragraph_id":"p2","order":2,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n2\nthere is the notion of first-order reductions in descriptive complexity, and among such, the\nfirst-order projections (fops). A fop is a very weak type of polynomial-time reduction whose\nstudy have provided interesting results such as that common NP-complete problems like\nSat, Clique and others remain complete via fop reductions, and that such NP-complete\nproblems can be described by logical sentences in a canonical form [2, 5].\nIn this paper we continue the study of the syntactic aspects of complete problems via fop\nreductions extending the work of Medina and Immerman [7, 6]. In particular, we provide a\ngeneral characterization of complete problems via fops for a large collection of complexity\nclasses that cover well beyond just NP, including classes like P, PSPACE, Σp\nn and Πp\nn, and\nothers. Interestingly, our results rely on very general assumptions and tools already known\nin the field.\nThe paper is organized as follows. In Sect. 2, we give standard definitions and known\nresults which provide the theoretical framework of the paper and make it self contained.\nSect. 3 contains our main result, namely the generalization of the Medina-Immerman result,\ntogether with relevant remarks and some examples. Later, Sect. 4 shows a general result\nabout the existence of non-isomorphic problems via fop reductions, which implies that our\ncanonical form is indeed minimal. Finally, Sect. 5 concludes with a brief summary and\ndirections for future work.\n2\nPreliminaries\n2.1\nLogics, Finite Models, and Decision Problems\nA logical vocabulary is a tuple τ = ⟨Ra1\n1 , . . . , Rar\nr , c1, . . . , cs, f r1\n1 , . . . , f rt\nt ⟩where the Rjs are\nrelational symbols of arity aj, cis are constant symbols, and the fks are rk-ary functional\nsymbols. A structure for τ, also referred as τ-structure or just structure if τ is clear from\ncontext, is a tuple A = ⟨|A|, RA\n1 , . . . , RA\nr , cA\n1 , . . . , cA\ns , f A\n1 , . . . , f A\nt ⟩where\n• |A| is the universe (or domain) of A,\n• RA\nj ⊆|A|aj is a aj-ary relation over |A|,\n• cj ∈|A| is an element of the universe, and\n• f A\nk : |A|rk →|A| is a total rk-ary function over |A|.\nFor vocabulary τ, STRUC[τ] denotes the class of all finite structures, i.e.\nthose whose\nuniverse is an initial segment {0, 1, . . . , n −1} of N.\nIn addition to above logical symbols, we also have the numerical relational symbols ‘=’,\n‘≤’, ‘BIT’ and ‘suc’, and constants ‘0’ and ‘max’, which are assumed to belong to each\nvocabulary, and have fixed interpretations on every structure A:\n• = and ≤are interpreted as the usual equality and order on N,\n• A ⊨BIT(i, j) iffthe j-th bit in the binary representation of i is 1,\n• A ⊨suc(x, y) iffy is the successor of x in the usual order on N, and\n• 0 and max denote the least and greatest element in |A|."},{"paragraph_id":"p3","order":3,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n3\nIf L denotes a logic, the language L[τ] is the set of all well-formed formulae of L over the\nvocabulary τ. A numerical formula in L[τ] is a formula with only numerical symbols. For\nexample, SO∃[τ] is the set of all second-order formulae of form ∃Q1 · · · ∃QnΦ where the Qis\nare relational variables and Φ is a first-order formula over over vocabulary τ. As usual, FO\ndenotes first-order logic and SO denotes second-order logic.\nA formula with no free variables is a sentence. For sentence φ ∈L[τ], the class of all finite\nmodels that satisfy φ is denoted as MOD[φ]. For fixed τ, it is possible to code every finite\nτ-structures into a sequence of bits, i.e. a binary string, using a map MOD[τ] ⇝{0, 1}∗.\nHence, a collection of finite models can be represented as a collection of strings, or language.\nIn descriptive complexity, a decision problem P is characterized by a subset of models\nfrom STRUC[τ] for some fixed τ. For example, the problem Clique can be characterized\nby structures A = ⟨|A|, EA, kA⟩over the vocabulary τ = ⟨E2, k⟩, where E is a binary\nrelational symbol and k is a constant, such that G = (|A|, EA) makes up an undirected\ngraph and kA ∈{0, . . . , |A|−1} denotes the size of a clique in G. Such models are typically\ncharacterized by a sentence Ψ over some fragment L. The problem Clique, for example,\ncan be characterized with a SO∃sentence over τ [3]; see below.\n2.2\nFirst-Order Queries, Fops, and Duals\nLet τ and σ be two vocabularies where σ = ⟨Ra1\n1 , . . . , Rar\nr , c1, . . . , cs⟩has no functional\nsymbols (from now on, we only consider vocabularies with no functional symbols). Let\nk ≥1 and consider the tuple I = ⟨φ0, . . . , φr, ψ1, . . . , ψs⟩of r + s + 1 first-order formulae in\nFO[τ] of form φ0(x1, . . . , xk), φi(x1, . . . , xkai) and ψj(x1, . . . , xk) = (x1 = c′\nj1∧· · ·∧xk = c′\njk)\nwhere the c′\njis are constant symbols from τ (possibly with repetitions). That is, φ0 has at\nmost k free variables among x1, . . . , xk, φi has at most kai free variables among x1, . . . , xkai,\nand ψj denotes a tuple in {c′ : c′ ∈τ}k.\nSuch tuple defines a mapping A ⇝I(A), called a first-order query of arity k, from\nτ-structures into σ-structures given by:\n• the universe |I(A)| .= {(u1, . . . , uk) ∈|A|k : A ⊨φ0(u1, . . . , uk)} is ordered lexico-\ngraphically,\n• the relations are RI(A)\ni\n.= {( ̄u1, . . . , ̄uai) ∈|A|kai : A ⊨φi( ̄u1, . . . , ̄uai)},\n• the constants are cI(A)\nj\n.= ̄u for the unique ̄u with A ⊨φ0( ̄u) ∧ψj( ̄u).\nFurthermore, if for T ⊆STRUC[τ] and S ⊆STRUC[σ], it is true that A ∈T iffI(A) ∈S,\nthen I is called a first-order reduction from T to S.\nA first-order query is called a first-order projection (fop) if φ0 is numerical and each φi\nhas form:\nφj( ̄x) ≡α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x))\nwhere the αis are numerical and mutually exclusive, and each λi is a τ-literal. Projections\nare typically denoted by the letter p. If p is a reduction from Π to Ψ, we write p : Π ≤fop Ψ,\nand if Π is complete via ≤fop reductions we say that Π is ≤fop-complete.\nProjections have interesting properties. For example, projections are a special case of\nValiant’s non-uniform projections [10]. For our purposes, we are interested in the fact that"},{"paragraph_id":"p4","order":4,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n4\nfor each projection p there is a first-order sentence βp ∈FO[σ] that characterizes the image\nof p, i.e. B ⊨βp iffB = p(A) for some A ∈STRUC[τ]. The sentence βp is called the\ncharacteristic sentence of p [1].\nFinally, there is a syntactic operator associated to a first-order query that plays a fun-\ndamental role in our results. For a first-order query I : STRUC[τ] →STRUC[σ], the dual\noperator bI maps formulae in L[σ] to formulae in L[τ] in such a way that\nA ⊨bI(θ)\nif and only if\nI(A) ⊨θ\nfor all θ ∈L[σ] and A ∈STRUC[τ] [5, Sect. 3.2].\n2.3\nComplexity Classes\nFor a family F of proper complexity functions [9], we consider the complexity classes\nTIME(F) = ∪f∈FTIME(f), and similarly for non-deterministic time and space. A com-\nplexity class C defined by F is nice if it has a universal problem of the form\nUC = {⟨Mi, ω, 1t⟩: Mi accepts ω within fi(t) resources}\nwhere L(Mi) ∈C and fi ∈F bounds the resources of Mi. Some well-known classes that are\nnice are L, NL, P, NP and PSPACE. Allender et al. [1] showed that if Π is ≤fop-complete\nfor a nice class C, then it is complete via injective fops of arity at least 2. The following\nproperties are shown easily:\nProposition 1 Let C be a complexity class defined by family F.\nThen, (a) if C is a\ndeterministic class and F is closed under sums, then C is closed under finite unions; (b)\nif C is a nondeterministic class and F is such that for every f, g ∈F there is h ∈F with\nf, g ≤h, then C is closed under finite unions; (c) if C is closed under finite unions and C\nis captured by logic L, then L is closed under disjunctions.\nThe nice classes L, NL, P, NP and PSPACE are known to be characterized by SO-\nDetKrom, SO-Krom, SO-Horn, SO∃and SO+TC respectively [4, 5]. Additionally, Σp\nk and\nΠp\nk are characterized by SO∃∀· · · Qk and SO∀∃· · · Q′\nk sentences where Qk = ∃, Q′\nk = ∀if k\nis odd, and Qk = ∀, Q′\nk = ∃if k is even. Thus, by the proposition, all these logical fragments\nare closed under disjunctions, and also under conjunctions with first-order formulae. We\nwill make use of these facts later.\n3\nCanonical Forms of Complete Problems\nMedina and Immerman characterized ≤fop-complete problems for NP syntactically using\nthe IndependentSet problem. This problem consists of checking whether an input graph\nG has an independent set of size k. IndependentSet is known to be complete for NP\nunder different notions of reductions, and in particular, under fop reductions [7]. Indepen-\ndentSet in characterized by the following SO∃[τ] sentence, for τ = ⟨E2, k⟩:\nΨIS = (∃f ∈Inj)(∀x, y)"},{"paragraph_id":"p5","order":5,"text":"x ̸= y ∧fx ≤k ∧fy ≤k →¬E(x, y)"},{"paragraph_id":"p6","order":6,"text":"(1)"},{"paragraph_id":"p7","order":7,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n5\nwhere ‘f ∈Inj’ means that f is a total and 1-1 function, i.e. an ordering of the elements of\nthe universe, and fx denotes f(x). Although it seems that (1) quantifies over a functional\nvariable, f is indeed a relational variable such that fx is the unique element such that\nf(x, fx). The condition f ∈Inj is easily defined in first-order logic. Observe that the only\nsecond-order variable in (1) is f which is existentially quantified.\nTheorem 2 ([7]) Let L ⊆STRUC[σ] be a NP problem characterized by Ψ ∈L[σ] where\nσ = ⟨Q1⟩is the vocabulary of binary strings. Then, a problem L is NP-complete via ≤fop\nreductions iffthere is an injective fop p : STRUC[⟨E2, k⟩] →STRUC[σ] such that\nΨ ≡(βp ∧ΥIS) ∨(¬βp ∧Λ)\n(2)\nwhere βp ∈FO[σ] is the characteristic sentence of p, ΥIS ∈SO∃[σ] is a generalized IS-form\n[7], and Λ is a SO∃[σ] sentence.\nIntuitively, this result says that if sentence Ψ characterizes a ≤fop-complete problem L\nfor NP, then it can be decomposed in two disjuncts Ψ = ΨIS ∨Ψrest such that MOD[ΨIS]\nis ≤fop-complete for NP and MOD[Ψrest] equals the “rest” of L which is not necessarily\ncomplete.\nOur main contribution is to show that above result can be generalized over a wide\ncollection of complexity classes, including the nice classes, and that such decomposition can\nbe done modulo any ≤fop-complete problem for the given class. Moreover, we also show\ntwo such decompositions are not in general equivalent.\nThe main obstacle for such generalization is to take care of the sentence ΥIS for classes\ndifferent than NP. As it will be shown, we do not have to consider each different class\nin isolation, since the corresponding Υ sentences will be the duals of the sentence Ψ that\ncharacterize the complete problem.\nLet us first define the relation ∼=Π over STRUC[τ] with respect to a given problem\nΠ ⊆STRUC[τ]. For structures A and B, define\nA ∼=Π B\niff\n(A ∈Π ⇔B ∈Π) .\n(3)\nClearly, ∼=Π is an equivalence relation that partitions STRUC[τ] into Π and its complement.\nBy using dual operators and the equivalence relation, we are able to show the following\ngeneralization of Theorem 2. In the following, τ and σ refer to any two vocabularies.\nTheorem 3 (Main) Let C be a complexity class captured by fragment L closed under\ndisjunctions and closed under conjunctions with FO. Let Π ⊆STRUC[τ] be a ≤fop-complete\nproblem for C characterized by Ψ ∈L[τ], and B a problem over vocabulary σ. Then, B is\n≤fop-complete for C if and only if there is a fop p : STRUC[τ] →STRUC[σ] such that for\nall B ∈STRUC[σ]:\nB ∈B\niff\nB ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ)\n(4)\nwhere\n(a) βp ∈FO[σ] is the characteristic of p, i.e. B ⊨βp iffB ∈p(STRUC[τ]),\n(b) Λ ∈L[σ], and"},{"paragraph_id":"p8","order":8,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n6\n(c) I : STRUC[σ] →STRUC[τ] is a first-order query such that for all A ∈STRUC[τ],\nI(p(A)) ∼=Π A.\nProof: For the necessity, assume that B is ≤fop-complete for C; i.e. B is characterized by\nsome sentence Λ ∈L[σ] and there is p : Π ≤fop B. For B ∈B we consider the two cases\nwhether B ̸∈p(STRUC[τ]) or not. For the first case, B ⊨¬βp ∧Λ. For the second case,\nB ⊨βp and\nB = p(A)\n(for some A ∈STRUC[τ] by (a))\n=⇒\nA ∈Π\n(since p is reduction)\n=⇒\nA ⊨Ψ\n(Ψ characterizes Π)\n=⇒\nI(p(A)) ⊨Ψ\n(by condition (c))\n=⇒\np(A) ⊨bI(Ψ)\n(def. of dual of I) .\nTherefore, B ∈B =⇒B ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). Now, let B ∈STRUC[σ] be such that\nB ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). If B ⊨Λ, then B ∈B. Otherwise,\nB ⊨βp ∧bI(Ψ)\n=⇒\nB = p(A) and p(A) ⊨bI(Ψ)\n(for some A ∈STRUC[τ])\n=⇒\nI(p(A)) ⊨Ψ\n(def. of dual)\n=⇒\nA ⊨Ψ\n(by (c))\n=⇒\nA ∈Π\n(Ψ characterizes Π)\n=⇒\nB ∈B\n(since p is reduction) .\nIt remains to show that there are first-order queries satisfying (c). Since Π is complete,\nthere is a fop I : STRUC[σ] →STRUC[τ] that reduces p(Π) to Π. Note that p(Π) ⊆B\nsince p is also a reduction. For A ∈STRUC[τ], observe\nA ∈Π ⇒p(A) ∈p(Π) ⇒I(p(A)) ∈Π ,\nI(p(A)) ∈Π ⇒p(A) ∈p(Π) ⇒p(A) ∈B ⇒A ∈Π .\nThus, I : p(Π) ≤fop Π satisfies A ∈Π iffI(p(A)) ∈Π; i.e. A ∼=Π I(p(A)).\nFor the sufficiency, assume there is a fop p : STRUC[τ] →STRUC[σ] such that (4) holds\nfor all B ∈STRUC[σ]. We need to show that B is complete for C. The inclusion B ∈C\nis direct from the closure properties on L. For the hardness, we show that p is indeed a\nreduction from Π to B. For A ∈STRUC[τ], we have p(A) ⊨βp. If A ∈Π, then\nA ⊨Ψ ⇒I(p(A)) ⊨Ψ ⇒p(A) ⊨bI(Ψ) ⇒p(A) ∈B .\nOn the other hand, if p(A) ∈B, then\np(A) ⊨βp ⇒p(A) ⊨bI(Ψ) ⇒I(p(A)) ⊨Ψ ⇒A ⊨Ψ ⇒A ∈Π .\nThus, A ∈Π iffp(A) ∈B, p is a reduction, and B is complete.\n□"},{"paragraph_id":"p9","order":9,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n7\nCorollary 4 The theorem holds if the first-order query I is the reduction I : p(Π) ≤fop Π\nwhich exists since Π is complete.\nMoreover, a first-order query J satisfying (c) is essentially equivalent (with respect to\nΨ) to the reduction I : p(Π) ≤fop Π. Indeed, for such J and a finite σ-structure B = p(A)\nfor A ∈STRUC[τ],\nB ⊨bJ(Ψ) ⇐⇒J(B) ⊨Ψ ⇐⇒A ⊨Ψ ⇐⇒I(B) ⊨Ψ ⇐⇒B ⊨bI(Ψ) .\nIf we consider nice complexity classes, then the fop p can be assumed to be injective by\na result of Allender et. al [1].\nCorollary 5 For nice classes, the fop p : STRUC[τ] →STRUC[σ] can be assumed to be\ninjective.\nTo see that Theorem 2 is equivalent to Corollary 5 when C = NP, let τ = ⟨E2, k⟩and\nσ = ⟨Q1⟩be the vocabularies for graphs and binary strings respectively, and consider a\nproblem L ⊆STRUC[σ] complete for NP characterized by ΨL. According to Theorem 2,\nΨL ≡(βp ∧ΥIS) ∨(¬βp ∧Λ)\nwhere p : IndependentSet →L is a first-order projection and Λ is a SO∃sentence. On\nthe other hand, according to Corollary 5, ΨL also satisfies\nΨL ≡(βp ∧bI(ΨIS)) ∨(¬βp ∧Λ′) .\nAs shown before, ΥIS and bI(ΨIS) are equivalent on p(STRUC[τ]), and thus Λ and Λ′ must\nbe equivalent on STRUC[σ] ∩MOD[¬β].\n3.1\nExamples\nConsider Clique ⊆STRUC[τ = ⟨E2, k⟩] characterized by the SO∃sentence\nΨCL = (∃f ∈Inj)(∀x, y)"},{"paragraph_id":"p10","order":10,"text":"x ̸= y ∧fx ≤k ∧fy ≤k →E(x, y)"},{"paragraph_id":"p11","order":11,"text":".\nFor σ = τ, it is not hard to see that IndependentSet can be reduced to Clique using\nthe fop p = λxy⟨φ0, φ1, ψ⟩, of arity 1, where\nφ0(x) = true ,\nφ1(x, y) = ¬E(x, y) ,\nψ(x) = (x = k) .\nClearly, if A = ⟨|A|, EA, kA⟩, then |p(A)| = |A|, Ep(A) = |A|2 \\ EA and kp(A) = kA.\nTherefore, p(p(A)) = A for all A ∈STRUC[τ], and hence\np(p(A)) ∈IndependentSet\niff\nA ∈IndependentSet.\nFurthermore, βp = true and since Clique is also known to be NP-complete with respect\nto ≤fop reductions, we have\nΨCL ≡(βp ∧bp(ΨIS)) ∨(¬βp ∧Γ) = bp(ΨIS) ."},{"paragraph_id":"p12","order":12,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n8\nConversely, beginning with the observation tha βp = true and bp(ΨIS) = ΨCL we can\nconclude, by Theorem 3, that Clique is ≤fop-complete for NP. We call this formulation\nof Clique as its canonical form with respect to IndependentSet. In this example, the\nformula ΨCL was already in its canonical form with respect to IndependentSet.\nFor a second example, consider the problem SubGraphIso defined by tuples ⟨G, G′⟩\nsuch that the graph G contains a subgraph isomorphic to graph G′. Such tuples can be\nexpressed with the vocabulary σ = ⟨F 2, H2, k⟩where F and H define the edges of G\nand G′, and the constant k defines the initial segment {0, . . . , k −1} for the edges of G′.\nAmong other things, instances of SubGraphIso are identified with structures B in which\nHB ⊆{0, . . . , k −1}2. SubGraphIso is defined by the SO∃sentence ΨSG\n(∃f ∈Inj)(∀x, y)"},{"paragraph_id":"p13","order":13,"text":"x ̸= y ∧fx < k ∧fy < k →(H(fx, fy) →F(x, y))"},{"paragraph_id":"p14","order":14,"text":".\nA fop reduction p from Clique into SubGraphIso outputs ⟨G, Kk, k⟩on input ⟨G, k⟩. The\nfop is p = ⟨φ0, φ1, φ2, ψ⟩given by\nφ0 = true , φ1 = E(x, y) , φ2 = (x < k ∧y < k) , ψ = (x = k) .\nThe characteristic sentence of p is\nβp = x < k ∧y < k →F(x, y) .\nThe reduction I : p(Clique) ≤fop Clique given by I = ⟨φ0 = true, φ1 = F(x, y)⟩satisfies\nB ∈p(Clique) if and only if I(B) ∈Clique for all B. Since ΨSG is equivalent to (βp ∧\nbI(ΨCL)) ∨(¬βp ∧ΨSG), then, by Corollary 5, SubGraphIso is complete for NP via ≤fop\nreductions.\nFinally, other classes that satisfies the conditions of Corollary 5 are L, NL, P, PSPACE,\nand all Σp\nk and Πp\nk.\n4\nNon-Isomorphic Complete Problems for Nice Classes\nThe next result is a more general version of one already known for NP [7]. The proof is\nanalogous to the NP case. Among other things. it implies that we cannot get rid of the\ndisjunction in Corollary 5.\nTheorem 6 If C is a nice complexity class, then there are two C-complete problems that\nare not fop-isomorphic.\nProof: Let Γ ⊆{0, 1}∗be a ≤fop-complete problem for C, and define Γ′ = {ω0, ω1 : ω ∈Γ}.\nIt is easy to see that Γ′ is complete via fops; e.g. define the projection p : STRUC[τ =\n⟨S1⟩] →STRUC[σ = ⟨T 1⟩], of arity 2, as p = ⟨φ0(x, y), φ1(x, y)⟩where φ0(x, y) = (x =\n0)∨(x = 1∧y = 0) gives the domain of p(A) and φ1(x, y) = (x = 0∧S(y))∨(x = 1∧y = 0)\ngives T p(A). Thus, for A with domain |A| = {0, . . . , n −1}, φ0 defines\n|p(A)| = {(0, y) : 0 ≤y < n} ∪{(1, 0)} ."},{"paragraph_id":"p15","order":15,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n9\nFormula φ1 identifies the n bits of A with the tuples (0, x) and assigns “value”1 to the tuple\n(1, 0). Observe that the order induced in p(A) is (0, 0) < (0, 1) < · · · < (0, n −1) < (1, 0).\nTherefore, ω ∈Γ iffp(ω) ∈Γ′ which shows that Γ′ is complete.\nSince C is a nice complexity class, there is a fop p : STRUC[τ] →STRUC[σ] that is\ninjective, of arity k ≥2, that reduces Γ to Γ′. We will show that p cannot be onto by\nshowing that if ω ∈Γ, then either ω0 ̸∈p(Γ) or ω1 ̸∈p(Γ).\nConsider the formula φ( ̄x) that defines the interpretation of T in the structure p(A) of\nform\nφ( ̄x) = α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x)) .\nWe are going to show w0 ∈p(Γ) =⇒w1 ̸∈p(Γ). Suppose that |ω0| = n + 1 and that\nω0 = p(ω′) for some ω′ ∈Γ represented by the structure A. Each bit in ω0 corresponds to a\nk-tuple in p(A), i.e. ω0 ∼ ̄u0 ̄u2 . . . ̄un where ̄uj is 1 iffω′ ⊨φ( ̄uj). Since ̄un ∼0, ω′ ⊭α0( ̄un).\nConsider the two cases whether ω′ ⊨αl( ̄un) for some 1 ≤l≤r, or not.\nIn the latter case, we can conclude that ω′′ ⊭αl( ̄un) for every ω′′ ∈{0, 1}|w| and\n1 ≤l≤r since αl, being a numerical formula, obtains a value that only depends on the\nsize of its input; thus, ω1 ̸∈p(Γ).\nIn the former case, ω′ ⊨αl( ̄un), for some unique l, and ω′ ⊭λl( ̄un) since ̄un ∼0. Thus,\nsince λl( ̄un) is a literal, some bit of ω′ determines the value 0 for ̄un. On the other hand,\nobserve that\nω′ ∈Γ ⇐⇒p(ω′) = ω0 ∈Γ′ ⇐⇒ω ∈Γ\nwhere the first equivalence follows since p is a reduction, and the second by construction\nof Γ′. Furthermore, being p injective, implies that each bit in ω′ determines one bit in ω.\nTherefore, there is a bit in ω′ that determines two bits in ω0: one bit in ω and the rightmost\n0. If ω1 were in p(Γ), then the same bit in the preimage of ω1 would determine the same\nbit in ω and the rightmost 1, this time in an inconsistent manner. Therefore, ω1 ̸∈p(Γ). □\n5\nConclusions\nWe have extended the canonical form proposed by Medina and Immerman to all complexity\nclasses characterized by fragments L closed under disjunctions, and under conjunctions with\nFO. Although, Medina and Immerman’s method could be generalized to other nice classes\nbeyond NP, it requires the formulation of “generalized” sentences.\nOur method, on the\nother hand, circumvent this problem by considering the dual operator. Additionally, it is\nnot clear how Medina and Immerman’s method could be used to find canonical forms with\nrespect to problems that are not “graph” problems, or on classes that do not have complete\nproblems based on explicit graphs, e.g. PSPACE.\nAs for the near future, we are currently working on syntactic operators that preserve\ncompleteness via fops for general complexity classes.\nThis subject is also addressed by\nMedina [6] where syntactic operators I : L[τ] →L[σ], that map formulae into formulae,\nare defined such that if Ψ characterizes a NP-complete problem, then so is I(Ψ). We think\nthat as inverse images play a fundamental role in (mathematical) analysis, inverse images\nof syntactic transformations are worth to explore. In our case, we look for operators I such\nthat if I(Ψ) defines a complete problem, then Ψ also defines a complete problem; Nijjar also"},{"paragraph_id":"p16","order":16,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n10\nmention that such transformations are worth exploring [8]. We believe that such operators\ncould be use to establish completeness of problems in an easier way.\nReferences\n[1] E. Allender, J. Balc ́azar, and N. Immerman. A first-order isomorphism theorem. SIAM\nJournal of Computing, 26(2):555–567, 1997.\n[2] E. Dahlhaus. Reduction to NP-complete problems by interpretations. Lecture Notes\nin Computer Science, pages 357–365, 1984.\n[3] R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In\nR. Karp, editor, SIAM-AMS Proc. 7, pages 27–41, 1974.\n[4] E. Gradel. Capturing complexity classes by fragments of second-order logic. Theoretical\nComputer Science, 101:35–57, 1992.\n[5] N. Immerman. Descriptive Complexity. Springer, 1998.\n[6] J. A. Medina. A Descriptive Approach to The Class NP. PhD thesis, University of\nMassachusetts, Amherst, 1997.\n[7] J. A. Medina and N. Immerman. A syntactic characterization of NP-completeness. In\nProc. IEEE Symp. Logic in Computer Science, pages 241–250, 1994.\n[8] P. Nijjar. An attempt to automate np-hardness reductions via SO∃logic. Master’s\nthesis, Waterloo University, Ontario, Canada, 2004.\n[9] C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.\n[10] L. Valiant. Reducibility by algebraic projections. L’Ensignment Math ́ematique, 28:253–\n268, 1982."}],"pages":[{"page":1,"text":"arXiv:0706.3412v1 [cs.CC] 22 Jun 2007\nLogic and Computation Complexity, Wroc law, Poland, July 15, 2007\n1\nOn Canonical Forms of Complete Problems via First-order\nProjections\nNerio Borges\nDepartamento de Matem ́aticas\nUniversidad Sim ́on Bol ́ıvar\nCaracas, Venezuela\nnborges@usb.ve\nBlai Bonet\nDepartamento de Computaci ́on\nUniversidad Sim ́on Bol ́ıvar\nCaracas, Venezuela\nbonet@ldc.usb.ve\nOctober 25, 2018\nAbstract\nThe class of problems complete for NP via first-order reductions is known to be\ncharacterized by existential second-order sentences of a fixed form. All such sentences\nare built around the so-called generalized IS-form of the sentence that defines Inde-\npendentSet. This result can also be understood as that every sentence that defines\na NP-complete problem P can be decomposed in two disjuncts such that the first one\ncharacterizes a fragment of P as hard as IndependentSet and the second the rest of\nP. That is, a decomposition that divides every such sentence into a a “quotient and\nresidue” modulo IndependentSet.\nIn this paper, we show that this result can be generalized over a wide collection of\ncomplexity classes, including the so-called nice classes. Moreover, we show that such\ndecomposition can be done for any complete problem with respect to the given class, and\nthat two such decompositions are non-equivalent in general. Interestingly, our results\nare based on simple and well-known properties of first-order reductions.\nKeywords: Finite Model Theory, Complexity Theory, First-Order Reductions, Canonical\nForms\n1\nIntroduction\nDescriptive complexity studies the interplay between complexity theory, finite model theory\nand mathematical logic. Since its inception in 1974 [3], descriptive complexity has been\nable to characterize all major complexity classes in term of logical languages independent of\nany computational model, thus suggesting that the computational complexity of languages\nis a property intrinsic to them and not an accidental consequence of our choice for the\ncomputational model.\nIn descriptive complexity, problems are understood as sets of (finite) models which\nare described by logical formulae over given vocabularies. Reductions between problems\ncorrespond to logical relations between the set of models that characterize the problems.\nAs important as the notion of polynomial many-one reductions in structural complexity,"},{"page":2,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n2\nthere is the notion of first-order reductions in descriptive complexity, and among such, the\nfirst-order projections (fops). A fop is a very weak type of polynomial-time reduction whose\nstudy have provided interesting results such as that common NP-complete problems like\nSat, Clique and others remain complete via fop reductions, and that such NP-complete\nproblems can be described by logical sentences in a canonical form [2, 5].\nIn this paper we continue the study of the syntactic aspects of complete problems via fop\nreductions extending the work of Medina and Immerman [7, 6]. In particular, we provide a\ngeneral characterization of complete problems via fops for a large collection of complexity\nclasses that cover well beyond just NP, including classes like P, PSPACE, Σp\nn and Πp\nn, and\nothers. Interestingly, our results rely on very general assumptions and tools already known\nin the field.\nThe paper is organized as follows. In Sect. 2, we give standard definitions and known\nresults which provide the theoretical framework of the paper and make it self contained.\nSect. 3 contains our main result, namely the generalization of the Medina-Immerman result,\ntogether with relevant remarks and some examples. Later, Sect. 4 shows a general result\nabout the existence of non-isomorphic problems via fop reductions, which implies that our\ncanonical form is indeed minimal. Finally, Sect. 5 concludes with a brief summary and\ndirections for future work.\n2\nPreliminaries\n2.1\nLogics, Finite Models, and Decision Problems\nA logical vocabulary is a tuple τ = ⟨Ra1\n1 , . . . , Rar\nr , c1, . . . , cs, f r1\n1 , . . . , f rt\nt ⟩where the Rjs are\nrelational symbols of arity aj, cis are constant symbols, and the fks are rk-ary functional\nsymbols. A structure for τ, also referred as τ-structure or just structure if τ is clear from\ncontext, is a tuple A = ⟨|A|, RA\n1 , . . . , RA\nr , cA\n1 , . . . , cA\ns , f A\n1 , . . . , f A\nt ⟩where\n• |A| is the universe (or domain) of A,\n• RA\nj ⊆|A|aj is a aj-ary relation over |A|,\n• cj ∈|A| is an element of the universe, and\n• f A\nk : |A|rk →|A| is a total rk-ary function over |A|.\nFor vocabulary τ, STRUC[τ] denotes the class of all finite structures, i.e.\nthose whose\nuniverse is an initial segment {0, 1, . . . , n −1} of N.\nIn addition to above logical symbols, we also have the numerical relational symbols ‘=’,\n‘≤’, ‘BIT’ and ‘suc’, and constants ‘0’ and ‘max’, which are assumed to belong to each\nvocabulary, and have fixed interpretations on every structure A:\n• = and ≤are interpreted as the usual equality and order on N,\n• A ⊨BIT(i, j) iffthe j-th bit in the binary representation of i is 1,\n• A ⊨suc(x, y) iffy is the successor of x in the usual order on N, and\n• 0 and max denote the least and greatest element in |A|."},{"page":3,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n3\nIf L denotes a logic, the language L[τ] is the set of all well-formed formulae of L over the\nvocabulary τ. A numerical formula in L[τ] is a formula with only numerical symbols. For\nexample, SO∃[τ] is the set of all second-order formulae of form ∃Q1 · · · ∃QnΦ where the Qis\nare relational variables and Φ is a first-order formula over over vocabulary τ. As usual, FO\ndenotes first-order logic and SO denotes second-order logic.\nA formula with no free variables is a sentence. For sentence φ ∈L[τ], the class of all finite\nmodels that satisfy φ is denoted as MOD[φ]. For fixed τ, it is possible to code every finite\nτ-structures into a sequence of bits, i.e. a binary string, using a map MOD[τ] ⇝{0, 1}∗.\nHence, a collection of finite models can be represented as a collection of strings, or language.\nIn descriptive complexity, a decision problem P is characterized by a subset of models\nfrom STRUC[τ] for some fixed τ. For example, the problem Clique can be characterized\nby structures A = ⟨|A|, EA, kA⟩over the vocabulary τ = ⟨E2, k⟩, where E is a binary\nrelational symbol and k is a constant, such that G = (|A|, EA) makes up an undirected\ngraph and kA ∈{0, . . . , |A|−1} denotes the size of a clique in G. Such models are typically\ncharacterized by a sentence Ψ over some fragment L. The problem Clique, for example,\ncan be characterized with a SO∃sentence over τ [3]; see below.\n2.2\nFirst-Order Queries, Fops, and Duals\nLet τ and σ be two vocabularies where σ = ⟨Ra1\n1 , . . . , Rar\nr , c1, . . . , cs⟩has no functional\nsymbols (from now on, we only consider vocabularies with no functional symbols). Let\nk ≥1 and consider the tuple I = ⟨φ0, . . . , φr, ψ1, . . . , ψs⟩of r + s + 1 first-order formulae in\nFO[τ] of form φ0(x1, . . . , xk), φi(x1, . . . , xkai) and ψj(x1, . . . , xk) = (x1 = c′\nj1∧· · ·∧xk = c′\njk)\nwhere the c′\njis are constant symbols from τ (possibly with repetitions). That is, φ0 has at\nmost k free variables among x1, . . . , xk, φi has at most kai free variables among x1, . . . , xkai,\nand ψj denotes a tuple in {c′ : c′ ∈τ}k.\nSuch tuple defines a mapping A ⇝I(A), called a first-order query of arity k, from\nτ-structures into σ-structures given by:\n• the universe |I(A)| .= {(u1, . . . , uk) ∈|A|k : A ⊨φ0(u1, . . . , uk)} is ordered lexico-\ngraphically,\n• the relations are RI(A)\ni\n.= {( ̄u1, . . . , ̄uai) ∈|A|kai : A ⊨φi( ̄u1, . . . , ̄uai)},\n• the constants are cI(A)\nj\n.= ̄u for the unique ̄u with A ⊨φ0( ̄u) ∧ψj( ̄u).\nFurthermore, if for T ⊆STRUC[τ] and S ⊆STRUC[σ], it is true that A ∈T iffI(A) ∈S,\nthen I is called a first-order reduction from T to S.\nA first-order query is called a first-order projection (fop) if φ0 is numerical and each φi\nhas form:\nφj( ̄x) ≡α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x))\nwhere the αis are numerical and mutually exclusive, and each λi is a τ-literal. Projections\nare typically denoted by the letter p. If p is a reduction from Π to Ψ, we write p : Π ≤fop Ψ,\nand if Π is complete via ≤fop reductions we say that Π is ≤fop-complete.\nProjections have interesting properties. For example, projections are a special case of\nValiant’s non-uniform projections [10]. For our purposes, we are interested in the fact that"},{"page":4,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n4\nfor each projection p there is a first-order sentence βp ∈FO[σ] that characterizes the image\nof p, i.e. B ⊨βp iffB = p(A) for some A ∈STRUC[τ]. The sentence βp is called the\ncharacteristic sentence of p [1].\nFinally, there is a syntactic operator associated to a first-order query that plays a fun-\ndamental role in our results. For a first-order query I : STRUC[τ] →STRUC[σ], the dual\noperator bI maps formulae in L[σ] to formulae in L[τ] in such a way that\nA ⊨bI(θ)\nif and only if\nI(A) ⊨θ\nfor all θ ∈L[σ] and A ∈STRUC[τ] [5, Sect. 3.2].\n2.3\nComplexity Classes\nFor a family F of proper complexity functions [9], we consider the complexity classes\nTIME(F) = ∪f∈FTIME(f), and similarly for non-deterministic time and space. A com-\nplexity class C defined by F is nice if it has a universal problem of the form\nUC = {⟨Mi, ω, 1t⟩: Mi accepts ω within fi(t) resources}\nwhere L(Mi) ∈C and fi ∈F bounds the resources of Mi. Some well-known classes that are\nnice are L, NL, P, NP and PSPACE. Allender et al. [1] showed that if Π is ≤fop-complete\nfor a nice class C, then it is complete via injective fops of arity at least 2. The following\nproperties are shown easily:\nProposition 1 Let C be a complexity class defined by family F.\nThen, (a) if C is a\ndeterministic class and F is closed under sums, then C is closed under finite unions; (b)\nif C is a nondeterministic class and F is such that for every f, g ∈F there is h ∈F with\nf, g ≤h, then C is closed under finite unions; (c) if C is closed under finite unions and C\nis captured by logic L, then L is closed under disjunctions.\nThe nice classes L, NL, P, NP and PSPACE are known to be characterized by SO-\nDetKrom, SO-Krom, SO-Horn, SO∃and SO+TC respectively [4, 5]. Additionally, Σp\nk and\nΠp\nk are characterized by SO∃∀· · · Qk and SO∀∃· · · Q′\nk sentences where Qk = ∃, Q′\nk = ∀if k\nis odd, and Qk = ∀, Q′\nk = ∃if k is even. Thus, by the proposition, all these logical fragments\nare closed under disjunctions, and also under conjunctions with first-order formulae. We\nwill make use of these facts later.\n3\nCanonical Forms of Complete Problems\nMedina and Immerman characterized ≤fop-complete problems for NP syntactically using\nthe IndependentSet problem. This problem consists of checking whether an input graph\nG has an independent set of size k. IndependentSet is known to be complete for NP\nunder different notions of reductions, and in particular, under fop reductions [7]. Indepen-\ndentSet in characterized by the following SO∃[τ] sentence, for τ = ⟨E2, k⟩:\nΨIS = (∃f ∈Inj)(∀x, y)\n \nx ̸= y ∧fx ≤k ∧fy ≤k →¬E(x, y)\n \n(1)"},{"page":5,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n5\nwhere ‘f ∈Inj’ means that f is a total and 1-1 function, i.e. an ordering of the elements of\nthe universe, and fx denotes f(x). Although it seems that (1) quantifies over a functional\nvariable, f is indeed a relational variable such that fx is the unique element such that\nf(x, fx). The condition f ∈Inj is easily defined in first-order logic. Observe that the only\nsecond-order variable in (1) is f which is existentially quantified.\nTheorem 2 ([7]) Let L ⊆STRUC[σ] be a NP problem characterized by Ψ ∈L[σ] where\nσ = ⟨Q1⟩is the vocabulary of binary strings. Then, a problem L is NP-complete via ≤fop\nreductions iffthere is an injective fop p : STRUC[⟨E2, k⟩] →STRUC[σ] such that\nΨ ≡(βp ∧ΥIS) ∨(¬βp ∧Λ)\n(2)\nwhere βp ∈FO[σ] is the characteristic sentence of p, ΥIS ∈SO∃[σ] is a generalized IS-form\n[7], and Λ is a SO∃[σ] sentence.\nIntuitively, this result says that if sentence Ψ characterizes a ≤fop-complete problem L\nfor NP, then it can be decomposed in two disjuncts Ψ = ΨIS ∨Ψrest such that MOD[ΨIS]\nis ≤fop-complete for NP and MOD[Ψrest] equals the “rest” of L which is not necessarily\ncomplete.\nOur main contribution is to show that above result can be generalized over a wide\ncollection of complexity classes, including the nice classes, and that such decomposition can\nbe done modulo any ≤fop-complete problem for the given class. Moreover, we also show\ntwo such decompositions are not in general equivalent.\nThe main obstacle for such generalization is to take care of the sentence ΥIS for classes\ndifferent than NP. As it will be shown, we do not have to consider each different class\nin isolation, since the corresponding Υ sentences will be the duals of the sentence Ψ that\ncharacterize the complete problem.\nLet us first define the relation ∼=Π over STRUC[τ] with respect to a given problem\nΠ ⊆STRUC[τ]. For structures A and B, define\nA ∼=Π B\niff\n(A ∈Π ⇔B ∈Π) .\n(3)\nClearly, ∼=Π is an equivalence relation that partitions STRUC[τ] into Π and its complement.\nBy using dual operators and the equivalence relation, we are able to show the following\ngeneralization of Theorem 2. In the following, τ and σ refer to any two vocabularies.\nTheorem 3 (Main) Let C be a complexity class captured by fragment L closed under\ndisjunctions and closed under conjunctions with FO. Let Π ⊆STRUC[τ] be a ≤fop-complete\nproblem for C characterized by Ψ ∈L[τ], and B a problem over vocabulary σ. Then, B is\n≤fop-complete for C if and only if there is a fop p : STRUC[τ] →STRUC[σ] such that for\nall B ∈STRUC[σ]:\nB ∈B\niff\nB ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ)\n(4)\nwhere\n(a) βp ∈FO[σ] is the characteristic of p, i.e. B ⊨βp iffB ∈p(STRUC[τ]),\n(b) Λ ∈L[σ], and"},{"page":6,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n6\n(c) I : STRUC[σ] →STRUC[τ] is a first-order query such that for all A ∈STRUC[τ],\nI(p(A)) ∼=Π A.\nProof: For the necessity, assume that B is ≤fop-complete for C; i.e. B is characterized by\nsome sentence Λ ∈L[σ] and there is p : Π ≤fop B. For B ∈B we consider the two cases\nwhether B ̸∈p(STRUC[τ]) or not. For the first case, B ⊨¬βp ∧Λ. For the second case,\nB ⊨βp and\nB = p(A)\n(for some A ∈STRUC[τ] by (a))\n=⇒\nA ∈Π\n(since p is reduction)\n=⇒\nA ⊨Ψ\n(Ψ characterizes Π)\n=⇒\nI(p(A)) ⊨Ψ\n(by condition (c))\n=⇒\np(A) ⊨bI(Ψ)\n(def. of dual of I) .\nTherefore, B ∈B =⇒B ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). Now, let B ∈STRUC[σ] be such that\nB ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). If B ⊨Λ, then B ∈B. Otherwise,\nB ⊨βp ∧bI(Ψ)\n=⇒\nB = p(A) and p(A) ⊨bI(Ψ)\n(for some A ∈STRUC[τ])\n=⇒\nI(p(A)) ⊨Ψ\n(def. of dual)\n=⇒\nA ⊨Ψ\n(by (c))\n=⇒\nA ∈Π\n(Ψ characterizes Π)\n=⇒\nB ∈B\n(since p is reduction) .\nIt remains to show that there are first-order queries satisfying (c). Since Π is complete,\nthere is a fop I : STRUC[σ] →STRUC[τ] that reduces p(Π) to Π. Note that p(Π) ⊆B\nsince p is also a reduction. For A ∈STRUC[τ], observe\nA ∈Π ⇒p(A) ∈p(Π) ⇒I(p(A)) ∈Π ,\nI(p(A)) ∈Π ⇒p(A) ∈p(Π) ⇒p(A) ∈B ⇒A ∈Π .\nThus, I : p(Π) ≤fop Π satisfies A ∈Π iffI(p(A)) ∈Π; i.e. A ∼=Π I(p(A)).\nFor the sufficiency, assume there is a fop p : STRUC[τ] →STRUC[σ] such that (4) holds\nfor all B ∈STRUC[σ]. We need to show that B is complete for C. The inclusion B ∈C\nis direct from the closure properties on L. For the hardness, we show that p is indeed a\nreduction from Π to B. For A ∈STRUC[τ], we have p(A) ⊨βp. If A ∈Π, then\nA ⊨Ψ ⇒I(p(A)) ⊨Ψ ⇒p(A) ⊨bI(Ψ) ⇒p(A) ∈B .\nOn the other hand, if p(A) ∈B, then\np(A) ⊨βp ⇒p(A) ⊨bI(Ψ) ⇒I(p(A)) ⊨Ψ ⇒A ⊨Ψ ⇒A ∈Π .\nThus, A ∈Π iffp(A) ∈B, p is a reduction, and B is complete.\n□"},{"page":7,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n7\nCorollary 4 The theorem holds if the first-order query I is the reduction I : p(Π) ≤fop Π\nwhich exists since Π is complete.\nMoreover, a first-order query J satisfying (c) is essentially equivalent (with respect to\nΨ) to the reduction I : p(Π) ≤fop Π. Indeed, for such J and a finite σ-structure B = p(A)\nfor A ∈STRUC[τ],\nB ⊨bJ(Ψ) ⇐⇒J(B) ⊨Ψ ⇐⇒A ⊨Ψ ⇐⇒I(B) ⊨Ψ ⇐⇒B ⊨bI(Ψ) .\nIf we consider nice complexity classes, then the fop p can be assumed to be injective by\na result of Allender et. al [1].\nCorollary 5 For nice classes, the fop p : STRUC[τ] →STRUC[σ] can be assumed to be\ninjective.\nTo see that Theorem 2 is equivalent to Corollary 5 when C = NP, let τ = ⟨E2, k⟩and\nσ = ⟨Q1⟩be the vocabularies for graphs and binary strings respectively, and consider a\nproblem L ⊆STRUC[σ] complete for NP characterized by ΨL. According to Theorem 2,\nΨL ≡(βp ∧ΥIS) ∨(¬βp ∧Λ)\nwhere p : IndependentSet →L is a first-order projection and Λ is a SO∃sentence. On\nthe other hand, according to Corollary 5, ΨL also satisfies\nΨL ≡(βp ∧bI(ΨIS)) ∨(¬βp ∧Λ′) .\nAs shown before, ΥIS and bI(ΨIS) are equivalent on p(STRUC[τ]), and thus Λ and Λ′ must\nbe equivalent on STRUC[σ] ∩MOD[¬β].\n3.1\nExamples\nConsider Clique ⊆STRUC[τ = ⟨E2, k⟩] characterized by the SO∃sentence\nΨCL = (∃f ∈Inj)(∀x, y)\n \nx ̸= y ∧fx ≤k ∧fy ≤k →E(x, y)\n \n.\nFor σ = τ, it is not hard to see that IndependentSet can be reduced to Clique using\nthe fop p = λxy⟨φ0, φ1, ψ⟩, of arity 1, where\nφ0(x) = true ,\nφ1(x, y) = ¬E(x, y) ,\nψ(x) = (x = k) .\nClearly, if A = ⟨|A|, EA, kA⟩, then |p(A)| = |A|, Ep(A) = |A|2 \\ EA and kp(A) = kA.\nTherefore, p(p(A)) = A for all A ∈STRUC[τ], and hence\np(p(A)) ∈IndependentSet\niff\nA ∈IndependentSet.\nFurthermore, βp = true and since Clique is also known to be NP-complete with respect\nto ≤fop reductions, we have\nΨCL ≡(βp ∧bp(ΨIS)) ∨(¬βp ∧Γ) = bp(ΨIS) ."},{"page":8,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n8\nConversely, beginning with the observation tha βp = true and bp(ΨIS) = ΨCL we can\nconclude, by Theorem 3, that Clique is ≤fop-complete for NP. We call this formulation\nof Clique as its canonical form with respect to IndependentSet. In this example, the\nformula ΨCL was already in its canonical form with respect to IndependentSet.\nFor a second example, consider the problem SubGraphIso defined by tuples ⟨G, G′⟩\nsuch that the graph G contains a subgraph isomorphic to graph G′. Such tuples can be\nexpressed with the vocabulary σ = ⟨F 2, H2, k⟩where F and H define the edges of G\nand G′, and the constant k defines the initial segment {0, . . . , k −1} for the edges of G′.\nAmong other things, instances of SubGraphIso are identified with structures B in which\nHB ⊆{0, . . . , k −1}2. SubGraphIso is defined by the SO∃sentence ΨSG\n(∃f ∈Inj)(∀x, y)\n \nx ̸= y ∧fx < k ∧fy < k →(H(fx, fy) →F(x, y))\n \n.\nA fop reduction p from Clique into SubGraphIso outputs ⟨G, Kk, k⟩on input ⟨G, k⟩. The\nfop is p = ⟨φ0, φ1, φ2, ψ⟩given by\nφ0 = true , φ1 = E(x, y) , φ2 = (x < k ∧y < k) , ψ = (x = k) .\nThe characteristic sentence of p is\nβp = x < k ∧y < k →F(x, y) .\nThe reduction I : p(Clique) ≤fop Clique given by I = ⟨φ0 = true, φ1 = F(x, y)⟩satisfies\nB ∈p(Clique) if and only if I(B) ∈Clique for all B. Since ΨSG is equivalent to (βp ∧\nbI(ΨCL)) ∨(¬βp ∧ΨSG), then, by Corollary 5, SubGraphIso is complete for NP via ≤fop\nreductions.\nFinally, other classes that satisfies the conditions of Corollary 5 are L, NL, P, PSPACE,\nand all Σp\nk and Πp\nk.\n4\nNon-Isomorphic Complete Problems for Nice Classes\nThe next result is a more general version of one already known for NP [7]. The proof is\nanalogous to the NP case. Among other things. it implies that we cannot get rid of the\ndisjunction in Corollary 5.\nTheorem 6 If C is a nice complexity class, then there are two C-complete problems that\nare not fop-isomorphic.\nProof: Let Γ ⊆{0, 1}∗be a ≤fop-complete problem for C, and define Γ′ = {ω0, ω1 : ω ∈Γ}.\nIt is easy to see that Γ′ is complete via fops; e.g. define the projection p : STRUC[τ =\n⟨S1⟩] →STRUC[σ = ⟨T 1⟩], of arity 2, as p = ⟨φ0(x, y), φ1(x, y)⟩where φ0(x, y) = (x =\n0)∨(x = 1∧y = 0) gives the domain of p(A) and φ1(x, y) = (x = 0∧S(y))∨(x = 1∧y = 0)\ngives T p(A). Thus, for A with domain |A| = {0, . . . , n −1}, φ0 defines\n|p(A)| = {(0, y) : 0 ≤y < n} ∪{(1, 0)} ."},{"page":9,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n9\nFormula φ1 identifies the n bits of A with the tuples (0, x) and assigns “value”1 to the tuple\n(1, 0). Observe that the order induced in p(A) is (0, 0) < (0, 1) < · · · < (0, n −1) < (1, 0).\nTherefore, ω ∈Γ iffp(ω) ∈Γ′ which shows that Γ′ is complete.\nSince C is a nice complexity class, there is a fop p : STRUC[τ] →STRUC[σ] that is\ninjective, of arity k ≥2, that reduces Γ to Γ′. We will show that p cannot be onto by\nshowing that if ω ∈Γ, then either ω0 ̸∈p(Γ) or ω1 ̸∈p(Γ).\nConsider the formula φ( ̄x) that defines the interpretation of T in the structure p(A) of\nform\nφ( ̄x) = α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x)) .\nWe are going to show w0 ∈p(Γ) =⇒w1 ̸∈p(Γ). Suppose that |ω0| = n + 1 and that\nω0 = p(ω′) for some ω′ ∈Γ represented by the structure A. Each bit in ω0 corresponds to a\nk-tuple in p(A), i.e. ω0 ∼ ̄u0 ̄u2 . . . ̄un where ̄uj is 1 iffω′ ⊨φ( ̄uj). Since ̄un ∼0, ω′ ⊭α0( ̄un).\nConsider the two cases whether ω′ ⊨αl( ̄un) for some 1 ≤l≤r, or not.\nIn the latter case, we can conclude that ω′′ ⊭αl( ̄un) for every ω′′ ∈{0, 1}|w| and\n1 ≤l≤r since αl, being a numerical formula, obtains a value that only depends on the\nsize of its input; thus, ω1 ̸∈p(Γ).\nIn the former case, ω′ ⊨αl( ̄un), for some unique l, and ω′ ⊭λl( ̄un) since ̄un ∼0. Thus,\nsince λl( ̄un) is a literal, some bit of ω′ determines the value 0 for ̄un. On the other hand,\nobserve that\nω′ ∈Γ ⇐⇒p(ω′) = ω0 ∈Γ′ ⇐⇒ω ∈Γ\nwhere the first equivalence follows since p is a reduction, and the second by construction\nof Γ′. Furthermore, being p injective, implies that each bit in ω′ determines one bit in ω.\nTherefore, there is a bit in ω′ that determines two bits in ω0: one bit in ω and the rightmost\n0. If ω1 were in p(Γ), then the same bit in the preimage of ω1 would determine the same\nbit in ω and the rightmost 1, this time in an inconsistent manner. Therefore, ω1 ̸∈p(Γ). □\n5\nConclusions\nWe have extended the canonical form proposed by Medina and Immerman to all complexity\nclasses characterized by fragments L closed under disjunctions, and under conjunctions with\nFO. Although, Medina and Immerman’s method could be generalized to other nice classes\nbeyond NP, it requires the formulation of “generalized” sentences.\nOur method, on the\nother hand, circumvent this problem by considering the dual operator. Additionally, it is\nnot clear how Medina and Immerman’s method could be used to find canonical forms with\nrespect to problems that are not “graph” problems, or on classes that do not have complete\nproblems based on explicit graphs, e.g. PSPACE.\nAs for the near future, we are currently working on syntactic operators that preserve\ncompleteness via fops for general complexity classes.\nThis subject is also addressed by\nMedina [6] where syntactic operators I : L[τ] →L[σ], that map formulae into formulae,\nare defined such that if Ψ characterizes a NP-complete problem, then so is I(Ψ). We think\nthat as inverse images play a fundamental role in (mathematical) analysis, inverse images\nof syntactic transformations are worth to explore. In our case, we look for operators I such\nthat if I(Ψ) defines a complete problem, then Ψ also defines a complete problem; Nijjar also"},{"page":10,"text":"Logic and Computation Complexity, Wroc law, Poland, July 15, 2007\n10\nmention that such transformations are worth exploring [8]. We believe that such operators\ncould be use to establish completeness of problems in an easier way.\nReferences\n[1] E. Allender, J. Balc ́azar, and N. Immerman. A first-order isomorphism theorem. SIAM\nJournal of Computing, 26(2):555–567, 1997.\n[2] E. Dahlhaus. Reduction to NP-complete problems by interpretations. Lecture Notes\nin Computer Science, pages 357–365, 1984.\n[3] R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In\nR. Karp, editor, SIAM-AMS Proc. 7, pages 27–41, 1974.\n[4] E. Gradel. Capturing complexity classes by fragments of second-order logic. Theoretical\nComputer Science, 101:35–57, 1992.\n[5] N. Immerman. Descriptive Complexity. Springer, 1998.\n[6] J. A. Medina. A Descriptive Approach to The Class NP. PhD thesis, University of\nMassachusetts, Amherst, 1997.\n[7] J. A. Medina and N. Immerman. A syntactic characterization of NP-completeness. In\nProc. IEEE Symp. Logic in Computer Science, pages 241–250, 1994.\n[8] P. Nijjar. An attempt to automate np-hardness reductions via SO∃logic. Master’s\nthesis, Waterloo University, Ontario, Canada, 2004.\n[9] C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.\n[10] L. Valiant. Reducibility by algebraic projections. L’Ensignment Math ́ematique, 28:253–\n268, 1982."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"A logical vocabulary is a tuple τ = ⟨Ra1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"context, is a tuple A = ⟨|A|, RA","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"In addition to above logical symbols, we also have the numerical relational symbols ‘=’,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"• = and ≤are interpreted as the usual equality and order on N,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"by structures A = ⟨|A|, EA, kA⟩over the vocabulary τ = ⟨E2, k⟩, where E is a binary","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"relational symbol and k is a constant, such that G = (|A|, EA) makes up an undirected","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"Let τ and σ be two vocabularies where σ = ⟨Ra1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"k ≥1 and consider the tuple I = ⟨φ0, . . . , φr, ψ1, . . . , ψs⟩of r + s + 1 first-order formulae in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"FO[τ] of form φ0(x1, . . . , xk), φi(x1, . . . , xkai) and ψj(x1, . . . , xk) = (x1 = c′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"j1∧· · ·∧xk = c′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"• the universe |I(A)| .= {(u1, . . . , uk) ∈|A|k : A ⊨φ0(u1, . . . , uk)} is ordered lexico-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":".= {( ̄u1, . . . , ̄uai) ∈|A|kai : A ⊨φi( ̄u1, . . . , ̄uai)},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":".= ̄u for the unique ̄u with A ⊨φ0( ̄u) ∧ψj( ̄u).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"of p, i.e. B ⊨βp iffB = p(A) for some A ∈STRUC[τ]. The sentence βp is called the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"TIME(F) = ∪f∈FTIME(f), and similarly for non-deterministic time and space. A com-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"UC = {⟨Mi, ω, 1t⟩: Mi accepts ω within fi(t) resources}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"k sentences where Qk = ∃, Q′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"k = ∀if k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"is odd, and Qk = ∀, Q′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"k = ∃if k is even. Thus, by the proposition, all these logical fragments","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"dentSet in characterized by the following SO∃[τ] sentence, for τ = ⟨E2, k⟩:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"ΨIS = (∃f ∈Inj)(∀x, y)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"x ̸= y ∧fx ≤k ∧fy ≤k →¬E(x, y)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"σ = ⟨Q1⟩is the vocabulary of binary strings. Then, a problem L is NP-complete via ≤fop","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"for NP, then it can be decomposed in two disjuncts Ψ = ΨIS ∨Ψrest such that MOD[ΨIS]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"Let us first define the relation ∼=Π over STRUC[τ] with respect to a given problem","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"A ∼=Π B","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"Clearly, ∼=Π is an equivalence relation that partitions STRUC[τ] into Π and its complement.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"I(p(A)) ∼=Π A.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"B = p(A)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"Therefore, B ∈B =⇒B ⊨(βp ∧bI(Ψ)) ∨(¬βp ∧Λ). Now, let B ∈STRUC[σ] be such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"B = p(A) and p(A) ⊨bI(Ψ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"Thus, I : p(Π) ≤fop Π satisfies A ∈Π iffI(p(A)) ∈Π; i.e. A ∼=Π I(p(A)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"Ψ) to the reduction I : p(Π) ≤fop Π. Indeed, for such J and a finite σ-structure B = p(A)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"To see that Theorem 2 is equivalent to Corollary 5 when C = NP, let τ = ⟨E2, k⟩and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"σ = ⟨Q1⟩be the vocabularies for graphs and binary strings respectively, and consider a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"Consider Clique ⊆STRUC[τ = ⟨E2, k⟩] characterized by the SO∃sentence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"ΨCL = (∃f ∈Inj)(∀x, y)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"x ̸= y ∧fx ≤k ∧fy ≤k →E(x, y)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"For σ = τ, it is not hard to see that IndependentSet can be reduced to Clique using","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"the fop p = λxy⟨φ0, φ1, ψ⟩, of arity 1, where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"φ0(x) = true ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"φ1(x, y) = ¬E(x, y) ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"ψ(x) = (x = k) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"Clearly, if A = ⟨|A|, EA, kA⟩, then |p(A)| = |A|, Ep(A) = |A|2 \\ EA and kp(A) = kA.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"Therefore, p(p(A)) = A for all A ∈STRUC[τ], and hence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"Furthermore, βp = true and since Clique is also known to be NP-complete with respect","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"ΨCL ≡(βp ∧bp(ΨIS)) ∨(¬βp ∧Γ) = bp(ΨIS) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"Conversely, beginning with the observation tha βp = true and bp(ΨIS) = ΨCL we can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"expressed with the vocabulary σ = ⟨F 2, H2, k⟩where F and H define the edges of G","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"x ̸= y ∧fx < k ∧fy < k →(H(fx, fy) →F(x, y))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"fop is p = ⟨φ0, φ1, φ2, ψ⟩given by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"φ0 = true , φ1 = E(x, y) , φ2 = (x < k ∧y < k) , ψ = (x = k) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"βp = x < k ∧y < k →F(x, y) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"The reduction I : p(Clique) ≤fop Clique given by I = ⟨φ0 = true, φ1 = F(x, y)⟩satisfies","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"Proof: Let Γ ⊆{0, 1}∗be a ≤fop-complete problem for C, and define Γ′ = {ω0, ω1 : ω ∈Γ}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"It is easy to see that Γ′ is complete via fops; e.g. define the projection p : STRUC[τ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"⟨S1⟩] →STRUC[σ = ⟨T 1⟩], of arity 2, as p = ⟨φ0(x, y), φ1(x, y)⟩where φ0(x, y) = (x =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"0)∨(x = 1∧y = 0) gives the domain of p(A) and φ1(x, y) = (x = 0∧S(y))∨(x = 1∧y = 0)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"gives T p(A). Thus, for A with domain |A| = {0, . . . , n −1}, φ0 defines","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"|p(A)| = {(0, y) : 0 ≤y < n} ∪{(1, 0)} .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"φ( ̄x) = α0( ̄x) ∨(α1( ̄x) ∧λ1( ̄x)) ∨· · · ∨(αr( ̄x) ∧λr( ̄x)) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"We are going to show w0 ∈p(Γ) =⇒w1 ̸∈p(Γ). Suppose that |ω0| = n + 1 and that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"ω0 = p(ω′) for some ω′ ∈Γ represented by the structure A. Each bit in ω0 corresponds to a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"ω′ ∈Γ ⇐⇒p(ω′) = ω0 ∈Γ′ ⇐⇒ω ∈Γ","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":24587,"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}} |