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{"paper_meta":{"paper_id":"arxiv:0704.0301","title":"0704.0301","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:0704.0301v1 [cs.CC] 3 Apr 2007\nDifferential recursion and\ndifferentially algebraic functions∗\nAkitoshi Kawamura\nDepartment of Computer Science\nUniversity of Toronto\nMoore introduced a class of real-valued “recursive” functions by analogy with\nKleene’s formulation of the standard recursive functions. While his concise def-\ninition inspired a new line of research on analog computation, it contains some\ntechnical inaccuracies. Focusing on his “primitive recursive” functions, we pin\ndown what is problematic and discuss possible attempts to remove the ambiguity\nregarding the behavior of the differential recursion operator on partial functions.\nIt turns out that in any case the purported relation to differentially algebraic\nfunctions, and hence to Shannon’s model of analog computation, fails.\n1. Introduction\nThere are several different kinds of theoretical models that talk about “computability” and\n“complexity” of real functions. Computable Analysis [19] and some other equivalent models\nuse approximation in one way or another to bring real numbers into the framework of the\nstandard Computability Theory that deals with discrete data in discrete time. Another well-\nknown model is the Blum–Shub–Smale model [1] in which continuous quantities are treated\nas an entity in themselves but the machine still works with discrete clock ticks.\nA third approach is analog computation in which not only are the data real-valued, but\nalso the transition takes place in continuous time [15].\nOne of the oldest and the best-\nstudied models of such computation is Shannon’s General Purpose Analog Computer [18]\nthat models the Differential Analyzer [3], a computing device built and put to use during\nthe thirties through the fifties. The GPAC, after some refinements [8, 12, 16], was shown\ncapable of generating (in a sense) all and only the differentially algebraic functions. We will\nexplore this class in Section 2 and show that it can be characterized in many different ways.\nLittle is known about how such analog models relate to the standard (digital) computabil-\nity. Moore [13] addressed this question for his new function classes that also try to express\n∗Presented at the Second Conference on Computability in Europe (CiE 2006), Swansea, Wales, UK,\nJuly 2006. Supported in part by Research Fellowship (DC1, 18-11700) of the Japan Society for Promotion\nof Science while the author was at Tokyo Institute of Technology.\n1\n\nthe power of GPAC-like computation. In imitation of Kleene’s characterization of the usual\nrecursive functions, these classes are defined as the closures under certain operators that\nare supposedly real-number versions of primitive recursion and minimization. He makes the\nfollowing claims, among others, that relate his classes of real primitive recursive and real re-\ncursive functions to analog and digital computation, respectively [13, Propositions 9 and 13].\nClaim 1. Real primitive recursive functions are differentially algebraic.1\nClaim 2. Each (partial) recursive function on the nonnegative integers (in the standard\nsense) is a restriction of some real recursive function.\nDespite its impact on the subsequent study on the classes and their variants [2, 4, 5, 8, 14],\nhis work lacked formality in some ways, as already pointed out [6, 7]. In fact, the definition of\nthe classes suffers from ambiguity. In Section 3 of this paper, we reformulate Moore’s theory\nup to the real primitive recursive functions in a mathematically sound way. With the aid of\nthe preparation in Section 2, we show that, even though our formulation of the class seems\nthe most restrictive possible, Claim 1 fails. In section 4, we discuss some issues about classes\nother than the real primitive recursive functions, including Claim 2.\nThroughout the paper, we write N, Z, R for the sets of nonnegative integers (including\n0), integers and real numbers, respectively.\nPartial functions\nIn this paper, a function f :⊆Rm →Rn may be partial, as opposed to\ntotal; that is, the set dom f of x ∈Rm for which the value f x ∈Rn is defined is allowed to\nbe a proper subset of Rm. By the restriction of f to a set J ⊆Rm we mean the function g\nwith dom g = J ∩dom f such that gx = f x for every x ∈dom g. When dom f is open, f\nis said to be (real) analytic if for every a = (a0, . . . , am−1) ∈dom f there are an open set\nJ ⊆dom f containing a and a family (cp)p∈Nm of n-tuples of real numbers such that the sum\nP\np=(p0,...,pm−1)∈Nm cp · (x0 −a0)p0 · · · (xm−1 −am−1)pm−1\n(1)\nconverges to f x for each x = (x0, . . . , xm−1) ∈J (regardless of the ordering of summation).\nSee Krantz and Parks [10, Chapters 1 and 2] for well-known properties of analytic functions.\nWhen f is analytic, we write D(a0,...,am−1)f (and not ∂a0+···+am−1f/∂ta0\n0 · · ·∂tam−1\nm−1 ) for the\nmixed partial derivative of f of order ai along the ith place (which is known to exist).\nMoore [13] does not explicitly deal with partial functions. We believe that this is responsible\nfor ambiguous and erroneous statements made in his seminal work as well as in some of the\nsubsequent works by other authors. Although there are some situations in mathematical\nanalysis where we can pretend that there are no partial functions (namely, when we are only\ndiscussing properties defined locally, such as continuity or analyticity), this is not the case\nwith the notions we want to discuss here. If, say, the above Claim 2 is to make any nontrivial\nsense, it is clearly inappropriate to talk about “real recursiveness at x,” as there is a real\nfunction which is simple locally but the restriction of which to N is highly complicated in the\nrecursion-theoretic sense. We therefore emphasize that partial functions must be dealt with\nseriously, and devote this paper to accordingly reformulating the theory wherever possible.\n1Moore writes M0 for the class of real primitive recursion functions. Claim 1 was later replaced by a similar\nclaim [6, Proposition 2] for a more “restricted” class G than M0, but its definition is again unclear.\n2\n\n2. Differentially algebraic functions\nWe show some facts about single- and multi-place differentially algebraic functions.\nTheorem 3. Let m, n and i < m be nonnegative integers and f :⊆Rm →Rn be an analytic\nfunction with open domain. Let (i), (ii), (iii) and (iv) be the following statements:\n(i) for any open connected set J ⊆dom f, there is a Zn-coefficient nonzero polynomial P\nsuch that\nP\n f x, Deif x, D2·eif x, . . . , D(arity P −1)·eif x\n \n= 0\n(2)\nfor all x ∈J, where ei ∈Nm is the vector whose ith component is 1 and others are 0;\n(ii) for each x0 ∈dom f, there are a Zn-coefficient nonzero polynomial P and an open set J\ncontaining x0 such that we have (2) for all x ∈J;\n(iii) for each x0 ∈dom f, there are a Zn-coefficient nonzero polynomial P and an open\ninterval J containing the ith component of x0 such that we have (2) for all x whose ith\ncomponent is in J and whose other components equal those of x0;\n(iv) for each x ∈dom f, there is a Zn-coefficient nonzero polynomial P satisfying (2).\nLet (iR), (iiR) and (iiiR) be the statements obtained by replacing Z by R in (i), (ii) and (iii),\nrespectively. Then (i), (ii), (iii), (iv), (iR), (iiR) and (iiiR) are equivalent.\nProof. The implications (i) ⇒(ii) ⇒(iii) ⇒(iv) and (i) ⇒(iR) ⇒(iiR) ⇒(iiiR) are obvious.\nIt has been known that (iiiR) ⇒(iii), see Theorem 16. To see (iv) ⇒(i), consider, for each Zn-\ncoefficient polynomial P, the set JP of all x ∈J satisfying (2). Since by (iv) these countably\nmany closed sets JP cover the open set J, one of them must have nonempty interior by Baire\nCategory Theorem 14. This JP must then equal J by the Identity Theorem 15.\nDefinition 4. Let m and n be nonnegative integers. An analytic function f :⊆Rm →Rn is\ndifferentially algebraic2 if for each i < m it satisfies one (or all) of the clauses in Theorem 3.\nNote that f need not be the unique solution of (2). For example, every function (with\nopen domain) that is constant on each connected component of its domain is differentially\nalgebraic because of the single set of equations Deif x = 0.\nThe clauses (ii)–(iv) show that being differentially algebraic is a “local” property.\nWhen dom f is connected, (i) reduces to the following statement:\n(i′) there is a Zn-coefficient nonzero polynomial P such that we have (2) for all x ∈dom f.\nHence, the clause (iii) shows that, as long as dom f is connected, our definition is equivalent\nto that of many authors, including Moore [13], who first state the definition for m = 1 by (i′)\nand then extend it to m > 1 by saying that a function is differentially algebraic when it is so\nas a unary function of each argument when all other arguments are held fixed.3 We proved\nTheorem 3 in order to use (i) to present a counterexample to Claim 1 later.\n2Also termed algebraic transcendental or hypotranscendental. Functions without this property is said to be\ntranscendentally transcendental or hypertranscendental.\n3Their definition for the case m = 1 is slightly different from (i′) in that it replaces (2) by P (x, f x, Deif x, . . . ,\nD(arity P −2)·eif x) = 0. But the proof of (a) ⇒(b) of Lemma 5 shows that this difference is superficial.\n3\n\nLet us characterize differentially algebraic functions in yet another way for the case n = 1.\nFor a field E, its subfield F and a set B ⊆E, we write F(B), agreeing tacitly on E, for the\nsmallest subfield of E that includes F and B. We write F for the algebraic closure of F, that\nis, the set of those elements of E that annuls some F-coefficient unary nonzero polynomial.\nLet J ⊆Rm be an open set and consider the ring Cω[J] of analytic functions g :⊆Rm →R\nwith dom g = J. Note that R is embedded into this ring by regarding each x ∈R as the\nconstant function taking the value x. To assert (2) for all x ∈J is to say that\nP\n f, Deif, D2·eif, . . . , D(arity P −1)·eif\n \n= 0\n(3)\nin Cω[J]. If J is connected, Cω[J] has a quotient field by the Identity Theorem 15, so the\nnotation R(D f) in the following lemma makes sense. We write D f = { Daf | a ∈Nm }.\nLemma 5. Let J ⊆Rm be open and connected. For f ∈Cω[J], the following are equivalent:\n(a) f is differentially algebraic;\n(b) D f ⊆R(B) for some finite set B ⊆D f;\n(c) D f ⊆R(B) for some finite set B ⊆R(D f).\nProof. The implication (b) ⇒(c) is trivial. The Transcendence Degree Theorem 17 shows\n(c) ⇒(a). For (a) ⇒(b), assume that for each i we have an R-coefficient polynomial Pi\nwith\nPi(f, Deif, D2·eif, . . . , DNi·eif) = 0,\n(4)\nwhere Ni = arity Pi −1. By choosing Ni to be smallest and then the degree of Pi in the last\nplace to be smallest, we may assume that\nΞ = (D(0,...,0,1)Pi)(f, Deif, D2·eif, . . . , DNi·eif)\n(5)\nis nonzero. Consider the order ≤on Nm defined by setting a ≤b when a + c = b for some\nc ∈Nm. We will show by induction on a ∈Nm that Daf ∈R({ Dbf | b ≤(N0, . . . , Nm−1) }).\nThe case a ≤(N0, . . . , Nm−1) being trivial, assume that a ≥(Ni + 1) · ei for some i. Apply\nDa−Ni·ei to both sides of (4) and calculate using chain rules to obtain\nΨ + Ξ · Daf = 0,\n(6)\nwhere Ψ can be written as a sum of products of several derivatives of f of order ≤a and\n̸= a, which hence enjoy the induction hypothesis.\nApart from purely theoretical interest, the significance of differentially algebraic functions\nlies in their relation to the General Purpose Analog Computer, an analog computation model\nintroduced by Shannon [18] and later refined by Pour-El [16]. More precisely, if a function\nf :⊆R →R with nonempty domain is differentially algebraic, then the restriction of f to\nsome nonempty subset of dom f is GPAC generable [16, Theorem 4]; conversely, if a function\nf :⊆R →R with nonempty domain is GPAC generable, then the restriction of f to some\nnonempty subset of dom f is differentially algebraic [12, Theorem 2]. Gra ̧ca later considered\nthe Polynomial GPAC, a simpler refinement than Pour-El’s, and proved analogous results [8].\n4\n\n3. Real primitive recursive functions\nThe class of real primitive recursive functions is defined [13] as the smallest class containing\nsome basic functions and closed under the operators specified below.\nUnfortunately, the\noriginal definition contains ambiguity, resulting in some inconsistent claims about the class.\nTo remedy this, we shall revisit the definitions carefully in Sections 3.1 and 3.2. Section 3.3\ndiscusses an alternative approach by Campagnolo. Section 3.4 disproves Claim 1.\n3.1. Two basic operators\nThe real primitive recursive functions are defined through three operators: juxtaposition,\ncomposition and differential recursion. The first two are very simple.\nDefinition 6. Given g0, . . . , gn−1 :⊆Rm →R, define their juxtaposition jx (g0, . . . , gn−1) :⊆\nRm →Rn by setting jx (g0, . . . , gn−1)x = (g0x, . . . , gn−1x) whenever x ∈dom g0 ∩· · · ∩\ndom gn−1. Given f :⊆Rm →Rn and g :⊆Rl →Rm, define their composition cm (f, g) :⊆\nRl →Rn by setting cm (f, g)x = f (gx) whenever x ∈dom g and gx ∈dom f.\nWe write f ◦g for cm (f, g). As remarked in Section 1, it is important to define precisely\nwhat the operators do on partial functions. Note how Definition 6 specifies the domain of\nthe functions constructed. If gx is not defined, neither is (f ◦g)x, even if, say, f is a constant\nfunction defined everywhere. We thus work in the following (informal) general principle.\nPrinciple 7. For the value of an expression to be defined, the value of each of its subexpres-\nsions has to be defined.\nWe remark that this was not explicitly intended by Moore. In fact, he presents an example\nto the contrary when he claims [13, Section 6] that the total function inv :⊆R →R given\nby\ninvx =\n(\n0\nif x = 0\n1/x\nif x ̸= 0\n(7)\ncan be obtained by composing the binary multiplication with jx (zero?, g), where zero? is\nfrom (30) and g is the restriction of inv to R\\{0}. Some authors point this out [4, p. 22] and\ncriticize it [7, p. 47]. Without discussing which definition is more “natural,” we adopt our\nrestrictive Definition 6, simply because it is not clear how to formulate a general definition\nthat would admit this construction of inv.\nThe operators jx and cm preserve analyticity [10, Proposition 2.2.8].\nTheorem 8. The property of being differentially algebraic is preserved by jx and cm.\nProof. This is trivial for jx. For cm, it suffices to show that if f :⊆Rm →R and g0, . . . ,\ngm−1 :⊆Rl →R are differentially algebraic, so is f ◦g, where g = jx (g0, . . . , gm−1). We\nmay assume that dom g and J = dom (f ◦g) are connected. We use the characterization (b)\nof Lemma 5. Calculate each element of D (f ◦g) by the chain rule to see that it belongs to\nR\n { d ◦g | d ∈D f } ∪\nm−1\nS\ni=0\n{ q↾J | q ∈D gi }\n \n,\n(8)\nwhere q↾J means the restriction of q to J. By the assumption, there are finite subsets A ⊆D f\nand Bi ⊆D gi with D f ⊆R(A) and D gi ⊆R(Bi) for each i = 0, . . . , m −1. This implies\nthat (8) stays unchanged by replacing D f by A and D gi by Bi.\n5\n\ninitial value\nt\nx\nx = h0 t\nx = h1 t\nFigure 1: When the equation is satisfied by both h0 and h1 (as well as their restriction to\neach interval containing the origin), how do we say that the shaded interval is where\nthere is a “unique solution”?\n3.2. The differential recursion operator\nTo formulate the third operator, we need a notion of unique solution of an integral equation\nof the form (9) below, where h is the unknown. For example, it sounds natural to say that\nthe tangent function restricted to (−π/2, π/2) uniquely solves ht =\nR t\n0\n 1+(hτ)2 \ndτ. But as\nwe are talking about partial functions, the word “unique” should be used carefully, because\nthe restriction to any subinterval J ⊆(−π/2, π/2) containing 0 also satisfies the equation on\nJ. Thus, out of the set H of all solutions, we need to pick one function that deserves to be\ncalled the unique solution defined on the largest possible interval (Figure 1). Though Moore\ndid not discuss this, it is not hard to formulate this intuition: for a set H of functions of a\ntype, we say that a function h ∈H is unique in H if the restriction of any function in H to\ndom h is a restriction of h.\nDefinition 9. Let f :⊆Rm →Rn and g :⊆Rm+1+n →Rn. For each v ∈Rm, let Hv be the\nset of all functions h :⊆R →Rn such that\n(a) dom h is either the empty set or a possibly unbounded interval containing 0,\n(b) v ∈dom f if dom h is nonempty,\n(c) (v, τ, hτ) ∈dom g for each τ ∈dom h, and\n(d) every t ∈dom h satisfies\nht = f v +\nZ t\n0\ng(v, τ, hτ) dτ.\n(9)\nLet Kv be the set of functions unique in Hv. By Lemma 18 in the appendix, Kv has an\nelement hv of which all functions in Kv is a restriction. Define dr (f, g) :⊆Rm+1 →Rn by\ndom\n dr (f, g)\n \n= { (v, t) ∈Rm+1 | t ∈dom hv } and dr (f, g)(v, t) = hv t.\nDefinition 10. The class of real primitive recursive functions is the smallest class containing\nthe nullary functions 00→1, 10→1, −10→1 and closed under jx, cm and dr.\n6\n\nLemma 11. The following functions are real primitive recursive: for each n ∈N, the n-ary\nconstants 0n→1, 1n→1, −1n→1; for n ∈N and i = 0, . . . , n −1, the n-ary projection idn→1\ni\nto\nthe ith component; binary add and mul; the functions inv+ (mapping x > 0 to 1/x), sqrt+\n(mapping x > 0 to √x) and ln (natural logarithm) defined on (0, ∞); the total functions sin,\ncos and exp; the circle ratio π as a nullary function.\nProof. The constant 0n→1 is built by 0n→1 = 00→1 ◦jx ( ); similarly for 1n→1 and −1n→1.\nThen inductively define idi+1→1\ni\n= dr (0i→1, 1i+2→1) and idn+1→1\ni\n= dr (idn→1\ni\n, 0n+2→1). Using\nthese, let add = dr (id1→1\n0\n, 13→1) and mul = dr (01→1, id3→1\n0\n). For inv+, define\nf = dr\n 10→1, mul ◦jx\n −11→1, mul ◦jx (id1→1\n0\n, id1→1\n0\n)\n \n◦id2→1\n1\n \n,\n(10)\ninv+ = f ◦\n add ◦jx (id1→1\n0\n, −11→1)\n \n,\n(11)\nor, more colloquially,\nf t = 1 −\nZ t\n0\n(f τ)2 dτ,\ninv+t = f (t −1).\n(12)\nSquare root is defined analogously by\nf t = 1 +\nZ t\n0\ninv+(2 · f τ) dτ,\nsqrt+t = f (t −1).\n(13)\nLogarithm and exponentiation are analogous, using suitable integral equations.\nFor the\ntrigonometric functions, let sin = id2→1\n0\n◦trig and cos = id2→1\n1\n◦trig, where\ntrig = dr\n jx (00→1, 10→1), jx\n id3→1\n2\n,\n mul ◦jx (−13→1, id3→1\n1\n)\n \n,\n(14)\nwhich is to say,\n \nsint\ncost\n \n=\n \n0\n1\n \n+\nZ t\n0\n \ncosτ\n−sinτ\n \ndτ.\n(15)\nThe circle ratio is π( ) = 4·Arctan1, with Arctan defined by a suitable integral equation.\nSome authors say “the function 1/x is real primitive recursive” to mean that inv+ is. It is\nnot clear how such assertions without specification of domain can be justified.\nThe reader may have felt uncomfortable with the unwieldy process of Definition 9 in picking\nthe right solution hv out of Hv.\nThis can be simplified if we discuss only real primitive\nrecursive functions, because of the following facts that result from the Uniqueness Theorem\nfor initial value problems [11] and the Cauchy–Kowalewsky Theorem [10, Section 2.4].\nTheorem 12. Let f, g, v, Hv and hv be as in Definition 9.\n(a) If g is an analytic4 function with open domain, Hv is the set of all restrictions of hv.\n(b) If f and g are analytic functions with open domain, so is dr (f, g).\nThe fact (a) says that a solution of (9) may diverge to infinity at some point but can never\n“branch” as in Figure 1, provided g is smooth enough. We therefore could have dispensed\nwith Lemma 18 and simply let hv be the (graph) union of Hv, so far as real primitive recursive\nfunctions are concerned, because they are analytic by (b).\n4This fact is often stated with a weaker assumption that g be Lipschitz continuous.\n7\n\nt\nx\n1\nx = gt\nIntegrate?\ny = kt\nt\ny\n1\nFigure 2: Integrand with a singularity.\n3.3. Campagnolo’s differential recursion\nThe clauses (a)–(c) of Definition 9 guarantee that the integral equation (9) makes sense for all\nt ∈dom h. The clause (c), however, could be slightly relaxed, since a small set of singularities\nin the integrand does not affect the integration. Define drC by replacing (c) with\n(c′) (v, τ, hτ) ∈dom g for any τ ∈dom h \\ S, where S is a countable set of isolated points.\nThis is due to Campagnolo [4, Definition 2.4.2], though he does not present a precise spec-\nification of the “unique” solution as we noted in the Section 3.2. The choice between (c)\nand (c′) is somewhat similar to the discussion regarding Principle 7. The issue is whether\ng(v, τ, hτ), where τ ∈[0, t], is a “subexpression” of the right-hand side of the equation (9).\nWithout going into the philosophical discussion to ask which is “natural,” we point out some\ndifferences this choice incurs.\nTheorem 12 (b) fails if we replace dr by drC, as the following example shows (Figure 2).\nThe function g :⊆R →R defined by dom g = R\\{1} and gt = inv+\n sqrt+\n sqrt+(t−1)2 \n=\n1/\np\n|t −1| is real primitive recursive by Lemma 11. But k = drC (−20→1, g ◦id2→1\n0\n) :⊆R →\nR, where −20→1 is the constant function with value −2, is the total function given by\nkt =\n(\n+2 · √t −1\nif t ⩾1,\n−2 · √−t + 1\nif t < 1,\n(16)\nwhich is not differentiable at 1. Note that dr (−20→1, g ◦id2→1\n0\n) is its restriction to (−∞, 1)\nand thus analytic. For a subtler example, recall the equation (13) for sqrt+; with drC, the\nsame equation produces the square root function defined on [0, ∞), rather than on (0, ∞).\nThis breaks the assumption of Theorem 12 (a) and thus gives rise to incomparable functions\nin Hv when, say, f = 10→1 and g = k ◦id2→1\n1\n, with k from (16); that is, the equation\nht = 1 +\nZ t\n0\nk(hτ) dτ\n(17)\nhas two solutions that take different values at a point.\nKeeping the class analytic also conforms to Moore’s intention [13, Definition 9] to make\nthe equation (9) equivalent to\nh0 = f v,\nD1ht = g(v, t, ht),\n(18)\nwhich would not make sense for non-differentiable h.\n8\n\n3.4. A primitive recursive but not differentially algebraic function\nClaim 1 would not make sense if we adopted drC in defining real primitive recursive functions,\nbecause there would then arise non-analytic functions, as we noted above. We now show that,\neven under our restrictive definition with the analyticity-preserving dr, the claim fails.\nDefine ˇΓ :⊆R2 →R by dom Γ = (0, ∞)2 and\nˇΓ (R, x) =\nZ R\n1/R\nexp\n (x −1) · ln t −t\n \ndt.\n(19)\nDefine Euler’s gamma function Γ :⊆R →R by dom Γ = (0, ∞) and\nΓ x = lim\nR→∞\nˇΓ (R, x).\n(20)\nIt can be verified that this value converges and satisfies\nDnΓ x = lim\nR→∞D(0,n) ˇΓ (R, x)\n(21)\nfor each n ∈N and x ∈(0, ∞). H ̈older showed that Γ is not differentially algebraic [9].\nWe do not know if Γ is real primitive recursive, but ˇΓ is easily shown real primitive\nrecursive, using Lemma 11. However, contrary to Claim 1, it is not differentially algebraic.\nFor assume that it were. We would then have a nonzero polynomial P such that\nP\n ˇΓ (R, x), D(0,1) ˇΓ (R, x), . . . , D(0,arity P −1) ˇΓ (R, x)\n \n= 0\n(22)\nfor each (R, x) ∈(0, ∞)2. Note that we used the characterization (i) of Theorem 3 in order\nto take P independent of R. We take the limit of (22) as R →∞, which by (21) yields\nP (Γ x, D1Γ x, . . . , Darity P −1Γ x) = 0,\n(23)\ncontradicting H ̈older.\n4. Other classes and related works\nThis section discusses some other operators introduced by Moore and other authors.\n4.1. Minimization and Moore’s real recursive functions\nFor a function f :⊆Rm+1 →R, Moore defines mn f :⊆Rm →R by\nmn f v =\n(\nt+ = inf { t ≥0 | f (v, t) = 0 }\nif t+ < −t−,\nt−= sup { t ≤0 | f (v, t) = 0 }\notherwise.\n(24)\nThe class of real recursive functions5 is the smallest class containing all real primitive recursive\nfunctions and closed under jx, cm, dr and mn.\n5This “recursiveness” of Moore’s should not be confused with the same word also used in the context of\nComputable Analysis. As we see in Appendix C, Moore’s real recursive functions can even be discontinuous.\n9\n\nMoore states the definition of mn in a way that leaves ambiguous whether (24) has a value\nwhen, say, dom f = Rm × [1, ∞) and f (v, t) = 2 −t for all t ≥1. Should it have the value 2,\nor be left undefined because “the zero-searching program gets stuck”?\nIt turns out that, whichever definition we choose, Moore’s claim about iteration [13, Propo-\nsition 11] remains true, in the following modified form. Since the original proof again forgets\npartial functions, we present a new proof in Appendix C.\nLemma 13. If f :⊆Rm →Rm is real recursive, there is a real recursive function g :⊆\nRm+1 →Rm that extends the function g′ defined by dom g′ =\n \n(v, k) ∈R × (N \\ {0})\n v ∈\ndom f k \nand g′(v, k) = f kv for all (v, k) ∈dom g′, where f k = f ◦· · · ◦f\n|\n{z\n}\nk\n.\nWe have to note, however, that the class of real recursive functions is probably not well-\nbehaved, since, with mn producing non-smooth functions, the class no longer enjoys Theo-\nrem 12. We therefore doubt the significance of Claim 2, although it could be justified by\nusing Lemma 13 to simulate Turing machines as Moore did.\n4.2. Linear differential recursion\nWe have seen that many of the problems in Moore’s original work were caused by failure to\ndeal with partial functions properly. Some authors avoid this trouble by studying only oper-\nators preserving totality, so that partial functions never come into discussion. Campagnolo\nand Moore [5] take this path by considering linear differential recursion in place of dr. For\nclasses defined by this operator, some relationships with digital computation are known [2, 4].\n4.3. Open problems\nClaim 1 has been the main rationale for calling variants of Moore’s classes a model of analog\ncomputation. Now that we have lost it, an important challenge is the following.\nOpen Problem. Find a subclass of our real primitive recursive functions, preferably with an\nequally simple definition, that has a close relationship to the differentially algebraic functions.\nAnother direction would be to reformulate further the rest of Moore’s work, as well as other\nauthors’ works that also suffer from the same kind of ambiguity. For example, it may be\ninteresting to work out Mycka and Costa’s class arising from the operator of taking limits [14].\nAcknowledgement\nThe author thanks Mariko Yasugi at Kyoto Sangyo University for the discussion that led\nto this work. Comments by J. F. Costa at Instituto Superior T ́ecnico helped improve the\npresentation of Section 3.2.\n10\n\nReferences\n[1] Blum, L., Cucker, F., Shub, M. and Smale, S. Complexity and Real Computation,\nSpringer-Verlag (1997).\n[2] Bournez, O. and Hainry, E. An Analog Characterization of Elementarily Computable Func-\ntions over the Real Numbers, Proceedings of the Thirty-First International Colloquium on\nAutomata, Languages and Programming, Springer (2004), Lecture Notes in Computer Science\n3142.\n[3] Bush, V. The Differential Analyzer: A New Machine for Solving Differential Equations, Jour-\nnal of the Franklin Institute, 212 (1931), 447–488.\n[4] Campagnolo, M. L. Computational Complexity of Real Valued Recursive Functions and Ana-\nlog Circuits, PhD thesis, Instituto Superior T ́ecnico (July 2001).\n[5] Campagnolo, M. L. and Moore, C. Upper and Lower Bounds on Continuous-Time Com-\nputation, Second International Conference on Unconventional Models of Computation (2001).\n[6] Campagnolo, M. L., Moore, C. and Costa, J. F. Iteration, Inequalities, and Differentia-\nbility in Analog Computers, Journal of Complexity, 16, 4 (December 2000), 642–660.\n[7] Grac ̧a, D. The General Purpose Analog Computer and Recursive Functions Over the Reals,\nMaster’s thesis, Instituto Superior T ́ecnico (July 2002).\n[8] Grac ̧a, D. Some recent developments on Shannon’s General Purpose Analog Computer, Math-\nematical Logic Quarterly, 50 (April 2004), 473–485.\n[9] H ̈older, O. L. Ueber die Eigenschaft der Gammafunction keiner algebraischen Differential-\ngleichung zu gen ̈ugen, Mathematische Annalen, 28 (1886), 1–13.\n[10] Krantz, S. G. and Parks, H. R. A Primer of Real Analytic Functions, Birkh ̈auser Advanced\nTexts, Birkh ̈auser Boston, second edition (June 2002).\n[11] Lang, S. Real and Functional Analysis, Vol. 142 of Graduate Texts in Mathematics, Springer-\nVerlag, third edition (1993).\n[12] Lipshitz, L. and Rubel, L. A. A Differentially Algebraic Replacement Theorem, and Analog\nComputability, Proceedings of the American Mathematical Society, 99, 2 (February 1987),\n367–372.\n[13] Moore, C. Recursion Theory on the Reals and Continuous-Time Computation, Theoretical\nComputer Science, 162 (1996), 23–44.\n[14] Mycka, J. and Costa, J. F. Real Recursive Functions and Their Hierarchy, Journal of\nComplexity, 20, 6 (December 2004), 835–857.\n[15] Orponen, P. A Survey of Continuous-Time Computation Theory, Advances in Algorithms,\nLanguages, and Complexity, Kluwer Academic Publishers (1997), 209–224.\n[16] Pour-El, M. B. Abstract Computability and its Relation to the General Purpose Analog\nComputer (Some Connections Between Logic, Differential Equations and Analog Computers),\nTransactions of the American Mathematical Society, 199 (1974), 1–28.\n[17] Ritt, J. F. and Gourin, E. An Assemblage-Theoretic Proof of the Existence of Transcen-\ndentally Transcendental Functions, Bulletin of the American Mathematical Society, 33 (1927),\n182–184.\n[18] Shannon, C. E. Mathematical Theory of the Differential Analyzer, Journal of Mathematics\nand Physics, 20, 4 (1941), 337–354.\n[19] Weihrauch, K. Computable Analysis: An Introduction, Texts in Theoretical Computer Sci-\nence, Springer-Verlag (2000).\n11\n\nA. Old results\nWe list some known theorems that we used in Section 2.\nThe following Baire Category Theorem is used in the proof Theorem 3.\nTheorem 14. Let J be a subset of Rm. The union of countably many closed subsets of J with\nempty interior has empty interior.\nProof. Let J0, J1, . . . be closed subsets of J with empty interior, and U be any nonempty open\nsubset of J. We will show that U \\ S\nP ∈N JP is nonempty. For each P ∈N, we take xP ∈Rm\nand εP ∈R as follows. Write B(x, ε) for the open set of points in J whose distance from x is less\nthan ε. Let x0 ∈U and ε0 ∈(0, 1) be such that B(x0, ε0) ⊆U. For each P ∈N, let xP +1 ∈U\nand εP +1 ∈(0, 2−P −1) be such that B(xP +1, εP +1) ⊆B(xP , εP ) \\ JP . This is possible because\nB(xP , εP ) \\ JP is open and nonempty, since JP is closed and has empty interior. As P tends to\ninfinity, xP converges to a point in U \\ S\nP ∈N JP .\nThe proof of Theorem 3 also uses the following Identity Theorem (for real analytic functions\nof several variables), also known as the Principle of Analytic Continuation. It can be proved by\nstraightforwardly generalizing the same assertion for unary functions [10, Section 1.2].\nTheorem 15. An analytic function with open connected domain that vanishes on an open set\nvanishes everywhere.\nLet J ⊆R be an open interval. It is well known that functions u0, . . . , uk−1 ∈Cω[J] are linearly\ndependent if and only if the determinant\n (Diuj)i,j=0,...,k−1\n , called their Wronskian, is zero. Using\nthis fact, Ritt and Gourin [17] showed (iiiR) ⇒(iii) of Theorem 3.\nTheorem 16. Let J ⊆R be an open interval and let f ∈Cω[J]. If we have\nP (f, D1f, D2f, . . . , Darity P −1f) = 0\n(25)\nfor some R-coefficient nonzero polynomial P, then we have (25) for some Z-coefficient nonzero\npolynomial P.\nProof. By the assumption, there is a finite set B ⊆Narity P such that the functions\nf ν0 · (Df)ν1 · · · (Darity P −1f)νarity P −1,\nfor (ν0, . . . , νarity P −1) ∈B,\n(26)\nare linearly dependent. The Wronskian of (26) thus vanishes, which is a Z-coefficient polynomial\nin f, D1f, . . . , Darity P +|B|−1f. This polynomial is nonzero, since otherwise (26) would be linearly\ndependent for arbitrary f, which is absurd.\nOne direction of Lemma 5 uses the following Transcendence Degree Theorem.\nTheorem 17. Let F be a subfield of a field E and D be a subset of E. If D ⊆F(B) for some finite\nset B ⊆E, then D ⊆F(C) for some finite set C ⊆D.\nProof. For each d ∈D, the assumption gives\ndl =\nl−1\nP\nj=0\nβj · dj\n(27)\n12\n\nfor some l ∈N \\ {0} and βj ∈F(B). Suppose that for some d = d0 ∈D, this equation contains\nsome b ∈B \\ D, since otherwise we are done. Then we can rewrite (27) as\nbk =\nk−1\nP\ni=0\nαi · bi\n(28)\nfor some k ∈N \\ {0} and αi ∈F(B′), where B′ = (B \\ {b}) ∪{d0}.\nFor each d ∈D and t ∈N, we can substitute (27) and (28) repeatedly in dt to write\ndt =\nk−1\nP\ni=0\nl−1\nP\nj=0\nγi,j · ci · dj\n(29)\nfor some γi,j ∈F(B′). The k · l + 1 elements 1, d, d2, . . . , dk·l are hence linearly dependent over\nF(B′). We have thus found a set B′ with D ⊆F(B′) such that B′ \\ D has strictly less elements\nthan B \\ D. Repeat.\nB. Maximal unique function\nThis section shows that, from a set K of functions with a certain property, we can choose a function\nof which all functions in K is a restriction. This was used to justify Definition 9 in the presence of\nnon-analytic functions where Theorem 12 (a) does not apply.\nWe say that a set I ⊆R is 0-convex if it is either the empty set or a possibly unbounded interval\ncontaining 0. Note that the union of 0-convex sets is 0-convex.\nWe say that a set K of functions from R is consistent if for any t ∈R, the set { gt | g ∈K }\nhas at most one element. In this case, the union of K means the unique function k such that\ndom h = S\ng∈K dom g and for each t ∈dom h, there is some g ∈K with ht = gt.\nLemma 18. Let H be a set of functions from R with 0-convex domain. Then the set K of functions\nunique in H is consistent. Moreover, if its union belongs to H, it belongs to K.\nProof. For the first claim, suppose otherwise.\nThen there are functions k0, k1 ∈K and t ∈\ndom k0 ∩dom k1 such that k0t ̸= k1t. This contradicts the fact that k0 is unique in H.\nFor the second claim, suppose that the union k of K is not unique in H. That is, there are a\nfunction g ∈H and t ∈dom k ∩dom g such that gt ̸= kt. There is k0 ∈K for which t ∈dom k0.\nWe have k0t = kt ̸= gt, contradicting the fact that k0 is unique in H.\nThis lemma can be applied to H = Hv in the situation of Definition 9, because there the union\nof any consistent subset of Hv belongs to Hv.\nC. Iteration\nAs we noted, the definition (24) of the operator mn is ambiguous, as it contains a subexpression\nf (v, t) that may be undefined for some (v, t). So when is mn f v defined? Possible answers include:\n(a) When t+ and t−are defined.\n(b) When at least either t+ or t−is defined; the condition t+ < −t−will be used only when both\nare defined.\nAnd when is t+ (resp. t−) defined? Possible answers include:\n13\n\nx = clk t\n1\n2\n1\nt\nx\nx = zigzag t\n1\n2\n1\nt\nx\nfv\nf 2v\nf 3v\nx = g(v, t)\nx = h(v, t)\n1\n2\nv\nt\nx\nFigure 3: Simulating iteration f kv by real recursive functions.\n(i) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈R.\n(ii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[−t, t]\n(resp. [t, −t]).\n(iii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[0, t]\n(resp. [t, 0]).\n(iv) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0.\nFor (i), (ii) and (iii), we may also consider adding the phrase “except for some countably many\nisolated τ” (compare (c′) in Section 3.3).\nMoore’s informal explanation by a programming language [13, Section 7] seems to suggest (b)\nand (ii). However, without discussing which is the “right” definition of mn, we show that, whichever\nwe choose, Lemma 13 holds. The following proof is consistent with any of the above 2 × 7 possible\ndefinitions.\nProof of Lemma 13. Denote mn f v by μt. f (v, t). Let\nzero?x = μy. (x2 + y2) · (1 −y),\n(30)\ninteger?x = zero?\n sin (π · x)\n \n,\n(31)\nroundx = x −μr. integer?(x −r),\n(32)\nso that (32) is the unique integer in (x −1/2, x + 1/2]. We get inv of (7) by\ninvx = μt. x · (x · t −1).\n(33)\nThe above four functions are total. Let\ndigit(x, b, i) = round\n x\nbi −1\n2\n \n−b · round\n x\nbi+1 −1\n2\n \n(34)\nfor b > 0, where bi = exp(i · ln b). When b > 1 and i are integers, digit(x, b, i) is the digit in bi’s\nplace when x is written in base-b notation. Define\nclkt = digit(t, 2, −1),\nzigzagt = 0 +\nZ t\n0\n(2 −4 · clkτ) dτ,\n(35)\n g(v, t)\nh(v, t)\n \n=\n v\nv\n \n+\nZ t\n0\n 2 · (1 −clkτ) ·\n f\n h(v, τ) −clkτ · (h(v, τ) −v)\n \n−h(v, τ)\n \n2 · clkτ ·\n h(v, τ) −g(v, τ)\n \n· inv(zigzagτ)\n \ndτ,\n(36)\nas in Figure 3. We have f kv = g(v, k −1/2) for k ∈N \\ {0}.\nNote that clkτ · (h(v, τ) −v) in (36) cannot be dropped, because of Principle 7.\n14","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0704.0301v1 [cs.CC] 3 Apr 2007\nDifferential recursion and\ndifferentially algebraic functions∗\nAkitoshi Kawamura\nDepartment of Computer Science\nUniversity of Toronto\nMoore introduced a class of real-valued “recursive” functions by analogy with\nKleene’s formulation of the standard recursive functions. While his concise def-\ninition inspired a new line of research on analog computation, it contains some\ntechnical inaccuracies. Focusing on his “primitive recursive” functions, we pin\ndown what is problematic and discuss possible attempts to remove the ambiguity\nregarding the behavior of the differential recursion operator on partial functions.\nIt turns out that in any case the purported relation to differentially algebraic\nfunctions, and hence to Shannon’s model of analog computation, fails.\n1. Introduction\nThere are several different kinds of theoretical models that talk about “computability” and\n“complexity” of real functions. Computable Analysis [19] and some other equivalent models\nuse approximation in one way or another to bring real numbers into the framework of the\nstandard Computability Theory that deals with discrete data in discrete time. Another well-\nknown model is the Blum–Shub–Smale model [1] in which continuous quantities are treated\nas an entity in themselves but the machine still works with discrete clock ticks.\nA third approach is analog computation in which not only are the data real-valued, but\nalso the transition takes place in continuous time [15].\nOne of the oldest and the best-\nstudied models of such computation is Shannon’s General Purpose Analog Computer [18]\nthat models the Differential Analyzer [3], a computing device built and put to use during\nthe thirties through the fifties. The GPAC, after some refinements [8, 12, 16], was shown\ncapable of generating (in a sense) all and only the differentially algebraic functions. We will\nexplore this class in Section 2 and show that it can be characterized in many different ways.\nLittle is known about how such analog models relate to the standard (digital) computabil-\nity. Moore [13] addressed this question for his new function classes that also try to express\n∗Presented at the Second Conference on Computability in Europe (CiE 2006), Swansea, Wales, UK,\nJuly 2006. Supported in part by Research Fellowship (DC1, 18-11700) of the Japan Society for Promotion\nof Science while the author was at Tokyo Institute of Technology.\n1"},{"paragraph_id":"p2","order":2,"text":"the power of GPAC-like computation. In imitation of Kleene’s characterization of the usual\nrecursive functions, these classes are defined as the closures under certain operators that\nare supposedly real-number versions of primitive recursion and minimization. He makes the\nfollowing claims, among others, that relate his classes of real primitive recursive and real re-\ncursive functions to analog and digital computation, respectively [13, Propositions 9 and 13].\nClaim 1. Real primitive recursive functions are differentially algebraic.1\nClaim 2. Each (partial) recursive function on the nonnegative integers (in the standard\nsense) is a restriction of some real recursive function.\nDespite its impact on the subsequent study on the classes and their variants [2, 4, 5, 8, 14],\nhis work lacked formality in some ways, as already pointed out [6, 7]. In fact, the definition of\nthe classes suffers from ambiguity. In Section 3 of this paper, we reformulate Moore’s theory\nup to the real primitive recursive functions in a mathematically sound way. With the aid of\nthe preparation in Section 2, we show that, even though our formulation of the class seems\nthe most restrictive possible, Claim 1 fails. In section 4, we discuss some issues about classes\nother than the real primitive recursive functions, including Claim 2.\nThroughout the paper, we write N, Z, R for the sets of nonnegative integers (including\n0), integers and real numbers, respectively.\nPartial functions\nIn this paper, a function f :⊆Rm →Rn may be partial, as opposed to\ntotal; that is, the set dom f of x ∈Rm for which the value f x ∈Rn is defined is allowed to\nbe a proper subset of Rm. By the restriction of f to a set J ⊆Rm we mean the function g\nwith dom g = J ∩dom f such that gx = f x for every x ∈dom g. When dom f is open, f\nis said to be (real) analytic if for every a = (a0, . . . , am−1) ∈dom f there are an open set\nJ ⊆dom f containing a and a family (cp)p∈Nm of n-tuples of real numbers such that the sum\nP\np=(p0,...,pm−1)∈Nm cp · (x0 −a0)p0 · · · (xm−1 −am−1)pm−1\n(1)\nconverges to f x for each x = (x0, . . . , xm−1) ∈J (regardless of the ordering of summation).\nSee Krantz and Parks [10, Chapters 1 and 2] for well-known properties of analytic functions.\nWhen f is analytic, we write D(a0,...,am−1)f (and not ∂a0+···+am−1f/∂ta0\n0 · · ·∂tam−1\nm−1 ) for the\nmixed partial derivative of f of order ai along the ith place (which is known to exist).\nMoore [13] does not explicitly deal with partial functions. We believe that this is responsible\nfor ambiguous and erroneous statements made in his seminal work as well as in some of the\nsubsequent works by other authors. Although there are some situations in mathematical\nanalysis where we can pretend that there are no partial functions (namely, when we are only\ndiscussing properties defined locally, such as continuity or analyticity), this is not the case\nwith the notions we want to discuss here. If, say, the above Claim 2 is to make any nontrivial\nsense, it is clearly inappropriate to talk about “real recursiveness at x,” as there is a real\nfunction which is simple locally but the restriction of which to N is highly complicated in the\nrecursion-theoretic sense. We therefore emphasize that partial functions must be dealt with\nseriously, and devote this paper to accordingly reformulating the theory wherever possible.\n1Moore writes M0 for the class of real primitive recursion functions. Claim 1 was later replaced by a similar\nclaim [6, Proposition 2] for a more “restricted” class G than M0, but its definition is again unclear.\n2"},{"paragraph_id":"p3","order":3,"text":"2. Differentially algebraic functions\nWe show some facts about single- and multi-place differentially algebraic functions.\nTheorem 3. Let m, n and i < m be nonnegative integers and f :⊆Rm →Rn be an analytic\nfunction with open domain. Let (i), (ii), (iii) and (iv) be the following statements:\n(i) for any open connected set J ⊆dom f, there is a Zn-coefficient nonzero polynomial P\nsuch that\nP\n f x, Deif x, D2·eif x, . . . , D(arity P −1)·eif x"},{"paragraph_id":"p4","order":4,"text":"= 0\n(2)\nfor all x ∈J, where ei ∈Nm is the vector whose ith component is 1 and others are 0;\n(ii) for each x0 ∈dom f, there are a Zn-coefficient nonzero polynomial P and an open set J\ncontaining x0 such that we have (2) for all x ∈J;\n(iii) for each x0 ∈dom f, there are a Zn-coefficient nonzero polynomial P and an open\ninterval J containing the ith component of x0 such that we have (2) for all x whose ith\ncomponent is in J and whose other components equal those of x0;\n(iv) for each x ∈dom f, there is a Zn-coefficient nonzero polynomial P satisfying (2).\nLet (iR), (iiR) and (iiiR) be the statements obtained by replacing Z by R in (i), (ii) and (iii),\nrespectively. Then (i), (ii), (iii), (iv), (iR), (iiR) and (iiiR) are equivalent.\nProof. The implications (i) ⇒(ii) ⇒(iii) ⇒(iv) and (i) ⇒(iR) ⇒(iiR) ⇒(iiiR) are obvious.\nIt has been known that (iiiR) ⇒(iii), see Theorem 16. To see (iv) ⇒(i), consider, for each Zn-\ncoefficient polynomial P, the set JP of all x ∈J satisfying (2). Since by (iv) these countably\nmany closed sets JP cover the open set J, one of them must have nonempty interior by Baire\nCategory Theorem 14. This JP must then equal J by the Identity Theorem 15.\nDefinition 4. Let m and n be nonnegative integers. An analytic function f :⊆Rm →Rn is\ndifferentially algebraic2 if for each i < m it satisfies one (or all) of the clauses in Theorem 3.\nNote that f need not be the unique solution of (2). For example, every function (with\nopen domain) that is constant on each connected component of its domain is differentially\nalgebraic because of the single set of equations Deif x = 0.\nThe clauses (ii)–(iv) show that being differentially algebraic is a “local” property.\nWhen dom f is connected, (i) reduces to the following statement:\n(i′) there is a Zn-coefficient nonzero polynomial P such that we have (2) for all x ∈dom f.\nHence, the clause (iii) shows that, as long as dom f is connected, our definition is equivalent\nto that of many authors, including Moore [13], who first state the definition for m = 1 by (i′)\nand then extend it to m > 1 by saying that a function is differentially algebraic when it is so\nas a unary function of each argument when all other arguments are held fixed.3 We proved\nTheorem 3 in order to use (i) to present a counterexample to Claim 1 later.\n2Also termed algebraic transcendental or hypotranscendental. Functions without this property is said to be\ntranscendentally transcendental or hypertranscendental.\n3Their definition for the case m = 1 is slightly different from (i′) in that it replaces (2) by P (x, f x, Deif x, . . . ,\nD(arity P −2)·eif x) = 0. But the proof of (a) ⇒(b) of Lemma 5 shows that this difference is superficial.\n3"},{"paragraph_id":"p5","order":5,"text":"Let us characterize differentially algebraic functions in yet another way for the case n = 1.\nFor a field E, its subfield F and a set B ⊆E, we write F(B), agreeing tacitly on E, for the\nsmallest subfield of E that includes F and B. We write F for the algebraic closure of F, that\nis, the set of those elements of E that annuls some F-coefficient unary nonzero polynomial.\nLet J ⊆Rm be an open set and consider the ring Cω[J] of analytic functions g :⊆Rm →R\nwith dom g = J. Note that R is embedded into this ring by regarding each x ∈R as the\nconstant function taking the value x. To assert (2) for all x ∈J is to say that\nP\n f, Deif, D2·eif, . . . , D(arity P −1)·eif"},{"paragraph_id":"p6","order":6,"text":"= 0\n(3)\nin Cω[J]. If J is connected, Cω[J] has a quotient field by the Identity Theorem 15, so the\nnotation R(D f) in the following lemma makes sense. We write D f = { Daf | a ∈Nm }.\nLemma 5. Let J ⊆Rm be open and connected. For f ∈Cω[J], the following are equivalent:\n(a) f is differentially algebraic;\n(b) D f ⊆R(B) for some finite set B ⊆D f;\n(c) D f ⊆R(B) for some finite set B ⊆R(D f).\nProof. The implication (b) ⇒(c) is trivial. The Transcendence Degree Theorem 17 shows\n(c) ⇒(a). For (a) ⇒(b), assume that for each i we have an R-coefficient polynomial Pi\nwith\nPi(f, Deif, D2·eif, . . . , DNi·eif) = 0,\n(4)\nwhere Ni = arity Pi −1. By choosing Ni to be smallest and then the degree of Pi in the last\nplace to be smallest, we may assume that\nΞ = (D(0,...,0,1)Pi)(f, Deif, D2·eif, . . . , DNi·eif)\n(5)\nis nonzero. Consider the order ≤on Nm defined by setting a ≤b when a + c = b for some\nc ∈Nm. We will show by induction on a ∈Nm that Daf ∈R({ Dbf | b ≤(N0, . . . , Nm−1) }).\nThe case a ≤(N0, . . . , Nm−1) being trivial, assume that a ≥(Ni + 1) · ei for some i. Apply\nDa−Ni·ei to both sides of (4) and calculate using chain rules to obtain\nΨ + Ξ · Daf = 0,\n(6)\nwhere Ψ can be written as a sum of products of several derivatives of f of order ≤a and\n̸= a, which hence enjoy the induction hypothesis.\nApart from purely theoretical interest, the significance of differentially algebraic functions\nlies in their relation to the General Purpose Analog Computer, an analog computation model\nintroduced by Shannon [18] and later refined by Pour-El [16]. More precisely, if a function\nf :⊆R →R with nonempty domain is differentially algebraic, then the restriction of f to\nsome nonempty subset of dom f is GPAC generable [16, Theorem 4]; conversely, if a function\nf :⊆R →R with nonempty domain is GPAC generable, then the restriction of f to some\nnonempty subset of dom f is differentially algebraic [12, Theorem 2]. Gra ̧ca later considered\nthe Polynomial GPAC, a simpler refinement than Pour-El’s, and proved analogous results [8].\n4"},{"paragraph_id":"p7","order":7,"text":"3. Real primitive recursive functions\nThe class of real primitive recursive functions is defined [13] as the smallest class containing\nsome basic functions and closed under the operators specified below.\nUnfortunately, the\noriginal definition contains ambiguity, resulting in some inconsistent claims about the class.\nTo remedy this, we shall revisit the definitions carefully in Sections 3.1 and 3.2. Section 3.3\ndiscusses an alternative approach by Campagnolo. Section 3.4 disproves Claim 1.\n3.1. Two basic operators\nThe real primitive recursive functions are defined through three operators: juxtaposition,\ncomposition and differential recursion. The first two are very simple.\nDefinition 6. Given g0, . . . , gn−1 :⊆Rm →R, define their juxtaposition jx (g0, . . . , gn−1) :⊆\nRm →Rn by setting jx (g0, . . . , gn−1)x = (g0x, . . . , gn−1x) whenever x ∈dom g0 ∩· · · ∩\ndom gn−1. Given f :⊆Rm →Rn and g :⊆Rl →Rm, define their composition cm (f, g) :⊆\nRl →Rn by setting cm (f, g)x = f (gx) whenever x ∈dom g and gx ∈dom f.\nWe write f ◦g for cm (f, g). As remarked in Section 1, it is important to define precisely\nwhat the operators do on partial functions. Note how Definition 6 specifies the domain of\nthe functions constructed. If gx is not defined, neither is (f ◦g)x, even if, say, f is a constant\nfunction defined everywhere. We thus work in the following (informal) general principle.\nPrinciple 7. For the value of an expression to be defined, the value of each of its subexpres-\nsions has to be defined.\nWe remark that this was not explicitly intended by Moore. In fact, he presents an example\nto the contrary when he claims [13, Section 6] that the total function inv :⊆R →R given\nby\ninvx =\n(\n0\nif x = 0\n1/x\nif x ̸= 0\n(7)\ncan be obtained by composing the binary multiplication with jx (zero?, g), where zero? is\nfrom (30) and g is the restriction of inv to R\\{0}. Some authors point this out [4, p. 22] and\ncriticize it [7, p. 47]. Without discussing which definition is more “natural,” we adopt our\nrestrictive Definition 6, simply because it is not clear how to formulate a general definition\nthat would admit this construction of inv.\nThe operators jx and cm preserve analyticity [10, Proposition 2.2.8].\nTheorem 8. The property of being differentially algebraic is preserved by jx and cm.\nProof. This is trivial for jx. For cm, it suffices to show that if f :⊆Rm →R and g0, . . . ,\ngm−1 :⊆Rl →R are differentially algebraic, so is f ◦g, where g = jx (g0, . . . , gm−1). We\nmay assume that dom g and J = dom (f ◦g) are connected. We use the characterization (b)\nof Lemma 5. Calculate each element of D (f ◦g) by the chain rule to see that it belongs to\nR\n { d ◦g | d ∈D f } ∪\nm−1\nS\ni=0\n{ q↾J | q ∈D gi }"},{"paragraph_id":"p8","order":8,"text":",\n(8)\nwhere q↾J means the restriction of q to J. By the assumption, there are finite subsets A ⊆D f\nand Bi ⊆D gi with D f ⊆R(A) and D gi ⊆R(Bi) for each i = 0, . . . , m −1. This implies\nthat (8) stays unchanged by replacing D f by A and D gi by Bi.\n5"},{"paragraph_id":"p9","order":9,"text":"initial value\nt\nx\nx = h0 t\nx = h1 t\nFigure 1: When the equation is satisfied by both h0 and h1 (as well as their restriction to\neach interval containing the origin), how do we say that the shaded interval is where\nthere is a “unique solution”?\n3.2. The differential recursion operator\nTo formulate the third operator, we need a notion of unique solution of an integral equation\nof the form (9) below, where h is the unknown. For example, it sounds natural to say that\nthe tangent function restricted to (−π/2, π/2) uniquely solves ht =\nR t\n0\n 1+(hτ)2 \ndτ. But as\nwe are talking about partial functions, the word “unique” should be used carefully, because\nthe restriction to any subinterval J ⊆(−π/2, π/2) containing 0 also satisfies the equation on\nJ. Thus, out of the set H of all solutions, we need to pick one function that deserves to be\ncalled the unique solution defined on the largest possible interval (Figure 1). Though Moore\ndid not discuss this, it is not hard to formulate this intuition: for a set H of functions of a\ntype, we say that a function h ∈H is unique in H if the restriction of any function in H to\ndom h is a restriction of h.\nDefinition 9. Let f :⊆Rm →Rn and g :⊆Rm+1+n →Rn. For each v ∈Rm, let Hv be the\nset of all functions h :⊆R →Rn such that\n(a) dom h is either the empty set or a possibly unbounded interval containing 0,\n(b) v ∈dom f if dom h is nonempty,\n(c) (v, τ, hτ) ∈dom g for each τ ∈dom h, and\n(d) every t ∈dom h satisfies\nht = f v +\nZ t\n0\ng(v, τ, hτ) dτ.\n(9)\nLet Kv be the set of functions unique in Hv. By Lemma 18 in the appendix, Kv has an\nelement hv of which all functions in Kv is a restriction. Define dr (f, g) :⊆Rm+1 →Rn by\ndom\n dr (f, g)"},{"paragraph_id":"p10","order":10,"text":"= { (v, t) ∈Rm+1 | t ∈dom hv } and dr (f, g)(v, t) = hv t.\nDefinition 10. The class of real primitive recursive functions is the smallest class containing\nthe nullary functions 00→1, 10→1, −10→1 and closed under jx, cm and dr.\n6"},{"paragraph_id":"p11","order":11,"text":"Lemma 11. The following functions are real primitive recursive: for each n ∈N, the n-ary\nconstants 0n→1, 1n→1, −1n→1; for n ∈N and i = 0, . . . , n −1, the n-ary projection idn→1\ni\nto\nthe ith component; binary add and mul; the functions inv+ (mapping x > 0 to 1/x), sqrt+\n(mapping x > 0 to √x) and ln (natural logarithm) defined on (0, ∞); the total functions sin,\ncos and exp; the circle ratio π as a nullary function.\nProof. The constant 0n→1 is built by 0n→1 = 00→1 ◦jx ( ); similarly for 1n→1 and −1n→1.\nThen inductively define idi+1→1\ni\n= dr (0i→1, 1i+2→1) and idn+1→1\ni\n= dr (idn→1\ni\n, 0n+2→1). Using\nthese, let add = dr (id1→1\n0\n, 13→1) and mul = dr (01→1, id3→1\n0\n). For inv+, define\nf = dr\n 10→1, mul ◦jx\n −11→1, mul ◦jx (id1→1\n0\n, id1→1\n0\n)"},{"paragraph_id":"p12","order":12,"text":"◦id2→1\n1"},{"paragraph_id":"p13","order":13,"text":",\n(10)\ninv+ = f ◦\n add ◦jx (id1→1\n0\n, −11→1)"},{"paragraph_id":"p14","order":14,"text":",\n(11)\nor, more colloquially,\nf t = 1 −\nZ t\n0\n(f τ)2 dτ,\ninv+t = f (t −1).\n(12)\nSquare root is defined analogously by\nf t = 1 +\nZ t\n0\ninv+(2 · f τ) dτ,\nsqrt+t = f (t −1).\n(13)\nLogarithm and exponentiation are analogous, using suitable integral equations.\nFor the\ntrigonometric functions, let sin = id2→1\n0\n◦trig and cos = id2→1\n1\n◦trig, where\ntrig = dr\n jx (00→1, 10→1), jx\n id3→1\n2\n,\n mul ◦jx (−13→1, id3→1\n1\n)"},{"paragraph_id":"p15","order":15,"text":",\n(14)\nwhich is to say,"},{"paragraph_id":"p16","order":16,"text":"sint\ncost"},{"paragraph_id":"p17","order":17,"text":"="},{"paragraph_id":"p18","order":18,"text":"0\n1"},{"paragraph_id":"p19","order":19,"text":"+\nZ t\n0"},{"paragraph_id":"p20","order":20,"text":"cosτ\n−sinτ"},{"paragraph_id":"p21","order":21,"text":"dτ.\n(15)\nThe circle ratio is π( ) = 4·Arctan1, with Arctan defined by a suitable integral equation.\nSome authors say “the function 1/x is real primitive recursive” to mean that inv+ is. It is\nnot clear how such assertions without specification of domain can be justified.\nThe reader may have felt uncomfortable with the unwieldy process of Definition 9 in picking\nthe right solution hv out of Hv.\nThis can be simplified if we discuss only real primitive\nrecursive functions, because of the following facts that result from the Uniqueness Theorem\nfor initial value problems [11] and the Cauchy–Kowalewsky Theorem [10, Section 2.4].\nTheorem 12. Let f, g, v, Hv and hv be as in Definition 9.\n(a) If g is an analytic4 function with open domain, Hv is the set of all restrictions of hv.\n(b) If f and g are analytic functions with open domain, so is dr (f, g).\nThe fact (a) says that a solution of (9) may diverge to infinity at some point but can never\n“branch” as in Figure 1, provided g is smooth enough. We therefore could have dispensed\nwith Lemma 18 and simply let hv be the (graph) union of Hv, so far as real primitive recursive\nfunctions are concerned, because they are analytic by (b).\n4This fact is often stated with a weaker assumption that g be Lipschitz continuous.\n7"},{"paragraph_id":"p22","order":22,"text":"t\nx\n1\nx = gt\nIntegrate?\ny = kt\nt\ny\n1\nFigure 2: Integrand with a singularity.\n3.3. Campagnolo’s differential recursion\nThe clauses (a)–(c) of Definition 9 guarantee that the integral equation (9) makes sense for all\nt ∈dom h. The clause (c), however, could be slightly relaxed, since a small set of singularities\nin the integrand does not affect the integration. Define drC by replacing (c) with\n(c′) (v, τ, hτ) ∈dom g for any τ ∈dom h \\ S, where S is a countable set of isolated points.\nThis is due to Campagnolo [4, Definition 2.4.2], though he does not present a precise spec-\nification of the “unique” solution as we noted in the Section 3.2. The choice between (c)\nand (c′) is somewhat similar to the discussion regarding Principle 7. The issue is whether\ng(v, τ, hτ), where τ ∈[0, t], is a “subexpression” of the right-hand side of the equation (9).\nWithout going into the philosophical discussion to ask which is “natural,” we point out some\ndifferences this choice incurs.\nTheorem 12 (b) fails if we replace dr by drC, as the following example shows (Figure 2).\nThe function g :⊆R →R defined by dom g = R\\{1} and gt = inv+\n sqrt+\n sqrt+(t−1)2 \n=\n1/\np\n|t −1| is real primitive recursive by Lemma 11. But k = drC (−20→1, g ◦id2→1\n0\n) :⊆R →\nR, where −20→1 is the constant function with value −2, is the total function given by\nkt =\n(\n+2 · √t −1\nif t ⩾1,\n−2 · √−t + 1\nif t < 1,\n(16)\nwhich is not differentiable at 1. Note that dr (−20→1, g ◦id2→1\n0\n) is its restriction to (−∞, 1)\nand thus analytic. For a subtler example, recall the equation (13) for sqrt+; with drC, the\nsame equation produces the square root function defined on [0, ∞), rather than on (0, ∞).\nThis breaks the assumption of Theorem 12 (a) and thus gives rise to incomparable functions\nin Hv when, say, f = 10→1 and g = k ◦id2→1\n1\n, with k from (16); that is, the equation\nht = 1 +\nZ t\n0\nk(hτ) dτ\n(17)\nhas two solutions that take different values at a point.\nKeeping the class analytic also conforms to Moore’s intention [13, Definition 9] to make\nthe equation (9) equivalent to\nh0 = f v,\nD1ht = g(v, t, ht),\n(18)\nwhich would not make sense for non-differentiable h.\n8"},{"paragraph_id":"p23","order":23,"text":"3.4. A primitive recursive but not differentially algebraic function\nClaim 1 would not make sense if we adopted drC in defining real primitive recursive functions,\nbecause there would then arise non-analytic functions, as we noted above. We now show that,\neven under our restrictive definition with the analyticity-preserving dr, the claim fails.\nDefine ˇΓ :⊆R2 →R by dom Γ = (0, ∞)2 and\nˇΓ (R, x) =\nZ R\n1/R\nexp\n (x −1) · ln t −t"},{"paragraph_id":"p24","order":24,"text":"dt.\n(19)\nDefine Euler’s gamma function Γ :⊆R →R by dom Γ = (0, ∞) and\nΓ x = lim\nR→∞\nˇΓ (R, x).\n(20)\nIt can be verified that this value converges and satisfies\nDnΓ x = lim\nR→∞D(0,n) ˇΓ (R, x)\n(21)\nfor each n ∈N and x ∈(0, ∞). H ̈older showed that Γ is not differentially algebraic [9].\nWe do not know if Γ is real primitive recursive, but ˇΓ is easily shown real primitive\nrecursive, using Lemma 11. However, contrary to Claim 1, it is not differentially algebraic.\nFor assume that it were. We would then have a nonzero polynomial P such that\nP\n ˇΓ (R, x), D(0,1) ˇΓ (R, x), . . . , D(0,arity P −1) ˇΓ (R, x)"},{"paragraph_id":"p25","order":25,"text":"= 0\n(22)\nfor each (R, x) ∈(0, ∞)2. Note that we used the characterization (i) of Theorem 3 in order\nto take P independent of R. We take the limit of (22) as R →∞, which by (21) yields\nP (Γ x, D1Γ x, . . . , Darity P −1Γ x) = 0,\n(23)\ncontradicting H ̈older.\n4. Other classes and related works\nThis section discusses some other operators introduced by Moore and other authors.\n4.1. Minimization and Moore’s real recursive functions\nFor a function f :⊆Rm+1 →R, Moore defines mn f :⊆Rm →R by\nmn f v =\n(\nt+ = inf { t ≥0 | f (v, t) = 0 }\nif t+ < −t−,\nt−= sup { t ≤0 | f (v, t) = 0 }\notherwise.\n(24)\nThe class of real recursive functions5 is the smallest class containing all real primitive recursive\nfunctions and closed under jx, cm, dr and mn.\n5This “recursiveness” of Moore’s should not be confused with the same word also used in the context of\nComputable Analysis. As we see in Appendix C, Moore’s real recursive functions can even be discontinuous.\n9"},{"paragraph_id":"p26","order":26,"text":"Moore states the definition of mn in a way that leaves ambiguous whether (24) has a value\nwhen, say, dom f = Rm × [1, ∞) and f (v, t) = 2 −t for all t ≥1. Should it have the value 2,\nor be left undefined because “the zero-searching program gets stuck”?\nIt turns out that, whichever definition we choose, Moore’s claim about iteration [13, Propo-\nsition 11] remains true, in the following modified form. Since the original proof again forgets\npartial functions, we present a new proof in Appendix C.\nLemma 13. If f :⊆Rm →Rm is real recursive, there is a real recursive function g :⊆\nRm+1 →Rm that extends the function g′ defined by dom g′ ="},{"paragraph_id":"p27","order":27,"text":"(v, k) ∈R × (N \\ {0})\n v ∈\ndom f k \nand g′(v, k) = f kv for all (v, k) ∈dom g′, where f k = f ◦· · · ◦f\n|\n{z\n}\nk\n.\nWe have to note, however, that the class of real recursive functions is probably not well-\nbehaved, since, with mn producing non-smooth functions, the class no longer enjoys Theo-\nrem 12. We therefore doubt the significance of Claim 2, although it could be justified by\nusing Lemma 13 to simulate Turing machines as Moore did.\n4.2. Linear differential recursion\nWe have seen that many of the problems in Moore’s original work were caused by failure to\ndeal with partial functions properly. Some authors avoid this trouble by studying only oper-\nators preserving totality, so that partial functions never come into discussion. Campagnolo\nand Moore [5] take this path by considering linear differential recursion in place of dr. For\nclasses defined by this operator, some relationships with digital computation are known [2, 4].\n4.3. Open problems\nClaim 1 has been the main rationale for calling variants of Moore’s classes a model of analog\ncomputation. Now that we have lost it, an important challenge is the following.\nOpen Problem. Find a subclass of our real primitive recursive functions, preferably with an\nequally simple definition, that has a close relationship to the differentially algebraic functions.\nAnother direction would be to reformulate further the rest of Moore’s work, as well as other\nauthors’ works that also suffer from the same kind of ambiguity. For example, it may be\ninteresting to work out Mycka and Costa’s class arising from the operator of taking limits [14].\nAcknowledgement\nThe author thanks Mariko Yasugi at Kyoto Sangyo University for the discussion that led\nto this work. Comments by J. F. Costa at Instituto Superior T ́ecnico helped improve the\npresentation of Section 3.2.\n10"},{"paragraph_id":"p28","order":28,"text":"References\n[1] Blum, L., Cucker, F., Shub, M. and Smale, S. Complexity and Real Computation,\nSpringer-Verlag (1997).\n[2] Bournez, O. and Hainry, E. An Analog Characterization of Elementarily Computable Func-\ntions over the Real Numbers, Proceedings of the Thirty-First International Colloquium on\nAutomata, Languages and Programming, Springer (2004), Lecture Notes in Computer Science\n3142.\n[3] Bush, V. The Differential Analyzer: A New Machine for Solving Differential Equations, Jour-\nnal of the Franklin Institute, 212 (1931), 447–488.\n[4] Campagnolo, M. L. Computational Complexity of Real Valued Recursive Functions and Ana-\nlog Circuits, PhD thesis, Instituto Superior T ́ecnico (July 2001).\n[5] Campagnolo, M. L. and Moore, C. Upper and Lower Bounds on Continuous-Time Com-\nputation, Second International Conference on Unconventional Models of Computation (2001).\n[6] Campagnolo, M. L., Moore, C. and Costa, J. F. Iteration, Inequalities, and Differentia-\nbility in Analog Computers, Journal of Complexity, 16, 4 (December 2000), 642–660.\n[7] Grac ̧a, D. The General Purpose Analog Computer and Recursive Functions Over the Reals,\nMaster’s thesis, Instituto Superior T ́ecnico (July 2002).\n[8] Grac ̧a, D. Some recent developments on Shannon’s General Purpose Analog Computer, Math-\nematical Logic Quarterly, 50 (April 2004), 473–485.\n[9] H ̈older, O. L. Ueber die Eigenschaft der Gammafunction keiner algebraischen Differential-\ngleichung zu gen ̈ugen, Mathematische Annalen, 28 (1886), 1–13.\n[10] Krantz, S. G. and Parks, H. R. A Primer of Real Analytic Functions, Birkh ̈auser Advanced\nTexts, Birkh ̈auser Boston, second edition (June 2002).\n[11] Lang, S. Real and Functional Analysis, Vol. 142 of Graduate Texts in Mathematics, Springer-\nVerlag, third edition (1993).\n[12] Lipshitz, L. and Rubel, L. A. A Differentially Algebraic Replacement Theorem, and Analog\nComputability, Proceedings of the American Mathematical Society, 99, 2 (February 1987),\n367–372.\n[13] Moore, C. Recursion Theory on the Reals and Continuous-Time Computation, Theoretical\nComputer Science, 162 (1996), 23–44.\n[14] Mycka, J. and Costa, J. F. Real Recursive Functions and Their Hierarchy, Journal of\nComplexity, 20, 6 (December 2004), 835–857.\n[15] Orponen, P. A Survey of Continuous-Time Computation Theory, Advances in Algorithms,\nLanguages, and Complexity, Kluwer Academic Publishers (1997), 209–224.\n[16] Pour-El, M. B. Abstract Computability and its Relation to the General Purpose Analog\nComputer (Some Connections Between Logic, Differential Equations and Analog Computers),\nTransactions of the American Mathematical Society, 199 (1974), 1–28.\n[17] Ritt, J. F. and Gourin, E. An Assemblage-Theoretic Proof of the Existence of Transcen-\ndentally Transcendental Functions, Bulletin of the American Mathematical Society, 33 (1927),\n182–184.\n[18] Shannon, C. E. Mathematical Theory of the Differential Analyzer, Journal of Mathematics\nand Physics, 20, 4 (1941), 337–354.\n[19] Weihrauch, K. Computable Analysis: An Introduction, Texts in Theoretical Computer Sci-\nence, Springer-Verlag (2000).\n11"},{"paragraph_id":"p29","order":29,"text":"A. Old results\nWe list some known theorems that we used in Section 2.\nThe following Baire Category Theorem is used in the proof Theorem 3.\nTheorem 14. Let J be a subset of Rm. The union of countably many closed subsets of J with\nempty interior has empty interior.\nProof. Let J0, J1, . . . be closed subsets of J with empty interior, and U be any nonempty open\nsubset of J. We will show that U \\ S\nP ∈N JP is nonempty. For each P ∈N, we take xP ∈Rm\nand εP ∈R as follows. Write B(x, ε) for the open set of points in J whose distance from x is less\nthan ε. Let x0 ∈U and ε0 ∈(0, 1) be such that B(x0, ε0) ⊆U. For each P ∈N, let xP +1 ∈U\nand εP +1 ∈(0, 2−P −1) be such that B(xP +1, εP +1) ⊆B(xP , εP ) \\ JP . This is possible because\nB(xP , εP ) \\ JP is open and nonempty, since JP is closed and has empty interior. As P tends to\ninfinity, xP converges to a point in U \\ S\nP ∈N JP .\nThe proof of Theorem 3 also uses the following Identity Theorem (for real analytic functions\nof several variables), also known as the Principle of Analytic Continuation. It can be proved by\nstraightforwardly generalizing the same assertion for unary functions [10, Section 1.2].\nTheorem 15. An analytic function with open connected domain that vanishes on an open set\nvanishes everywhere.\nLet J ⊆R be an open interval. It is well known that functions u0, . . . , uk−1 ∈Cω[J] are linearly\ndependent if and only if the determinant\n (Diuj)i,j=0,...,k−1\n , called their Wronskian, is zero. Using\nthis fact, Ritt and Gourin [17] showed (iiiR) ⇒(iii) of Theorem 3.\nTheorem 16. Let J ⊆R be an open interval and let f ∈Cω[J]. If we have\nP (f, D1f, D2f, . . . , Darity P −1f) = 0\n(25)\nfor some R-coefficient nonzero polynomial P, then we have (25) for some Z-coefficient nonzero\npolynomial P.\nProof. By the assumption, there is a finite set B ⊆Narity P such that the functions\nf ν0 · (Df)ν1 · · · (Darity P −1f)νarity P −1,\nfor (ν0, . . . , νarity P −1) ∈B,\n(26)\nare linearly dependent. The Wronskian of (26) thus vanishes, which is a Z-coefficient polynomial\nin f, D1f, . . . , Darity P +|B|−1f. This polynomial is nonzero, since otherwise (26) would be linearly\ndependent for arbitrary f, which is absurd.\nOne direction of Lemma 5 uses the following Transcendence Degree Theorem.\nTheorem 17. Let F be a subfield of a field E and D be a subset of E. If D ⊆F(B) for some finite\nset B ⊆E, then D ⊆F(C) for some finite set C ⊆D.\nProof. For each d ∈D, the assumption gives\ndl =\nl−1\nP\nj=0\nβj · dj\n(27)\n12"},{"paragraph_id":"p30","order":30,"text":"for some l ∈N \\ {0} and βj ∈F(B). Suppose that for some d = d0 ∈D, this equation contains\nsome b ∈B \\ D, since otherwise we are done. Then we can rewrite (27) as\nbk =\nk−1\nP\ni=0\nαi · bi\n(28)\nfor some k ∈N \\ {0} and αi ∈F(B′), where B′ = (B \\ {b}) ∪{d0}.\nFor each d ∈D and t ∈N, we can substitute (27) and (28) repeatedly in dt to write\ndt =\nk−1\nP\ni=0\nl−1\nP\nj=0\nγi,j · ci · dj\n(29)\nfor some γi,j ∈F(B′). The k · l + 1 elements 1, d, d2, . . . , dk·l are hence linearly dependent over\nF(B′). We have thus found a set B′ with D ⊆F(B′) such that B′ \\ D has strictly less elements\nthan B \\ D. Repeat.\nB. Maximal unique function\nThis section shows that, from a set K of functions with a certain property, we can choose a function\nof which all functions in K is a restriction. This was used to justify Definition 9 in the presence of\nnon-analytic functions where Theorem 12 (a) does not apply.\nWe say that a set I ⊆R is 0-convex if it is either the empty set or a possibly unbounded interval\ncontaining 0. Note that the union of 0-convex sets is 0-convex.\nWe say that a set K of functions from R is consistent if for any t ∈R, the set { gt | g ∈K }\nhas at most one element. In this case, the union of K means the unique function k such that\ndom h = S\ng∈K dom g and for each t ∈dom h, there is some g ∈K with ht = gt.\nLemma 18. Let H be a set of functions from R with 0-convex domain. Then the set K of functions\nunique in H is consistent. Moreover, if its union belongs to H, it belongs to K.\nProof. For the first claim, suppose otherwise.\nThen there are functions k0, k1 ∈K and t ∈\ndom k0 ∩dom k1 such that k0t ̸= k1t. This contradicts the fact that k0 is unique in H.\nFor the second claim, suppose that the union k of K is not unique in H. That is, there are a\nfunction g ∈H and t ∈dom k ∩dom g such that gt ̸= kt. There is k0 ∈K for which t ∈dom k0.\nWe have k0t = kt ̸= gt, contradicting the fact that k0 is unique in H.\nThis lemma can be applied to H = Hv in the situation of Definition 9, because there the union\nof any consistent subset of Hv belongs to Hv.\nC. Iteration\nAs we noted, the definition (24) of the operator mn is ambiguous, as it contains a subexpression\nf (v, t) that may be undefined for some (v, t). So when is mn f v defined? Possible answers include:\n(a) When t+ and t−are defined.\n(b) When at least either t+ or t−is defined; the condition t+ < −t−will be used only when both\nare defined.\nAnd when is t+ (resp. t−) defined? Possible answers include:\n13"},{"paragraph_id":"p31","order":31,"text":"x = clk t\n1\n2\n1\nt\nx\nx = zigzag t\n1\n2\n1\nt\nx\nfv\nf 2v\nf 3v\nx = g(v, t)\nx = h(v, t)\n1\n2\nv\nt\nx\nFigure 3: Simulating iteration f kv by real recursive functions.\n(i) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈R.\n(ii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[−t, t]\n(resp. [t, −t]).\n(iii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[0, t]\n(resp. [t, 0]).\n(iv) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0.\nFor (i), (ii) and (iii), we may also consider adding the phrase “except for some countably many\nisolated τ” (compare (c′) in Section 3.3).\nMoore’s informal explanation by a programming language [13, Section 7] seems to suggest (b)\nand (ii). However, without discussing which is the “right” definition of mn, we show that, whichever\nwe choose, Lemma 13 holds. The following proof is consistent with any of the above 2 × 7 possible\ndefinitions.\nProof of Lemma 13. Denote mn f v by μt. f (v, t). Let\nzero?x = μy. (x2 + y2) · (1 −y),\n(30)\ninteger?x = zero?\n sin (π · x)"},{"paragraph_id":"p32","order":32,"text":",\n(31)\nroundx = x −μr. integer?(x −r),\n(32)\nso that (32) is the unique integer in (x −1/2, x + 1/2]. We get inv of (7) by\ninvx = μt. x · (x · t −1).\n(33)\nThe above four functions are total. Let\ndigit(x, b, i) = round\n x\nbi −1\n2"},{"paragraph_id":"p33","order":33,"text":"−b · round\n x\nbi+1 −1\n2"},{"paragraph_id":"p34","order":34,"text":"(34)\nfor b > 0, where bi = exp(i · ln b). When b > 1 and i are integers, digit(x, b, i) is the digit in bi’s\nplace when x is written in base-b notation. Define\nclkt = digit(t, 2, −1),\nzigzagt = 0 +\nZ t\n0\n(2 −4 · clkτ) dτ,\n(35)\n g(v, t)\nh(v, t)"},{"paragraph_id":"p35","order":35,"text":"=\n v\nv"},{"paragraph_id":"p36","order":36,"text":"+\nZ t\n0\n 2 · (1 −clkτ) ·\n f\n h(v, τ) −clkτ · (h(v, τ) −v)"},{"paragraph_id":"p37","order":37,"text":"−h(v, τ)"},{"paragraph_id":"p38","order":38,"text":"2 · clkτ ·\n h(v, τ) −g(v, τ)"},{"paragraph_id":"p39","order":39,"text":"· inv(zigzagτ)"},{"paragraph_id":"p40","order":40,"text":"dτ,\n(36)\nas in Figure 3. We have f kv = g(v, k −1/2) for k ∈N \\ {0}.\nNote that clkτ · (h(v, τ) −v) in (36) cannot be dropped, because of Principle 7.\n14"}],"pages":[{"page":1,"text":"arXiv:0704.0301v1 [cs.CC] 3 Apr 2007\nDifferential recursion and\ndifferentially algebraic functions∗\nAkitoshi Kawamura\nDepartment of Computer Science\nUniversity of Toronto\nMoore introduced a class of real-valued “recursive” functions by analogy with\nKleene’s formulation of the standard recursive functions. While his concise def-\ninition inspired a new line of research on analog computation, it contains some\ntechnical inaccuracies. Focusing on his “primitive recursive” functions, we pin\ndown what is problematic and discuss possible attempts to remove the ambiguity\nregarding the behavior of the differential recursion operator on partial functions.\nIt turns out that in any case the purported relation to differentially algebraic\nfunctions, and hence to Shannon’s model of analog computation, fails.\n1. Introduction\nThere are several different kinds of theoretical models that talk about “computability” and\n“complexity” of real functions. Computable Analysis [19] and some other equivalent models\nuse approximation in one way or another to bring real numbers into the framework of the\nstandard Computability Theory that deals with discrete data in discrete time. Another well-\nknown model is the Blum–Shub–Smale model [1] in which continuous quantities are treated\nas an entity in themselves but the machine still works with discrete clock ticks.\nA third approach is analog computation in which not only are the data real-valued, but\nalso the transition takes place in continuous time [15].\nOne of the oldest and the best-\nstudied models of such computation is Shannon’s General Purpose Analog Computer [18]\nthat models the Differential Analyzer [3], a computing device built and put to use during\nthe thirties through the fifties. The GPAC, after some refinements [8, 12, 16], was shown\ncapable of generating (in a sense) all and only the differentially algebraic functions. We will\nexplore this class in Section 2 and show that it can be characterized in many different ways.\nLittle is known about how such analog models relate to the standard (digital) computabil-\nity. Moore [13] addressed this question for his new function classes that also try to express\n∗Presented at the Second Conference on Computability in Europe (CiE 2006), Swansea, Wales, UK,\nJuly 2006. Supported in part by Research Fellowship (DC1, 18-11700) of the Japan Society for Promotion\nof Science while the author was at Tokyo Institute of Technology.\n1"},{"page":2,"text":"the power of GPAC-like computation. In imitation of Kleene’s characterization of the usual\nrecursive functions, these classes are defined as the closures under certain operators that\nare supposedly real-number versions of primitive recursion and minimization. He makes the\nfollowing claims, among others, that relate his classes of real primitive recursive and real re-\ncursive functions to analog and digital computation, respectively [13, Propositions 9 and 13].\nClaim 1. Real primitive recursive functions are differentially algebraic.1\nClaim 2. Each (partial) recursive function on the nonnegative integers (in the standard\nsense) is a restriction of some real recursive function.\nDespite its impact on the subsequent study on the classes and their variants [2, 4, 5, 8, 14],\nhis work lacked formality in some ways, as already pointed out [6, 7]. In fact, the definition of\nthe classes suffers from ambiguity. In Section 3 of this paper, we reformulate Moore’s theory\nup to the real primitive recursive functions in a mathematically sound way. With the aid of\nthe preparation in Section 2, we show that, even though our formulation of the class seems\nthe most restrictive possible, Claim 1 fails. In section 4, we discuss some issues about classes\nother than the real primitive recursive functions, including Claim 2.\nThroughout the paper, we write N, Z, R for the sets of nonnegative integers (including\n0), integers and real numbers, respectively.\nPartial functions\nIn this paper, a function f :⊆Rm →Rn may be partial, as opposed to\ntotal; that is, the set dom f of x ∈Rm for which the value f x ∈Rn is defined is allowed to\nbe a proper subset of Rm. By the restriction of f to a set J ⊆Rm we mean the function g\nwith dom g = J ∩dom f such that gx = f x for every x ∈dom g. When dom f is open, f\nis said to be (real) analytic if for every a = (a0, . . . , am−1) ∈dom f there are an open set\nJ ⊆dom f containing a and a family (cp)p∈Nm of n-tuples of real numbers such that the sum\nP\np=(p0,...,pm−1)∈Nm cp · (x0 −a0)p0 · · · (xm−1 −am−1)pm−1\n(1)\nconverges to f x for each x = (x0, . . . , xm−1) ∈J (regardless of the ordering of summation).\nSee Krantz and Parks [10, Chapters 1 and 2] for well-known properties of analytic functions.\nWhen f is analytic, we write D(a0,...,am−1)f (and not ∂a0+···+am−1f/∂ta0\n0 · · ·∂tam−1\nm−1 ) for the\nmixed partial derivative of f of order ai along the ith place (which is known to exist).\nMoore [13] does not explicitly deal with partial functions. We believe that this is responsible\nfor ambiguous and erroneous statements made in his seminal work as well as in some of the\nsubsequent works by other authors. Although there are some situations in mathematical\nanalysis where we can pretend that there are no partial functions (namely, when we are only\ndiscussing properties defined locally, such as continuity or analyticity), this is not the case\nwith the notions we want to discuss here. If, say, the above Claim 2 is to make any nontrivial\nsense, it is clearly inappropriate to talk about “real recursiveness at x,” as there is a real\nfunction which is simple locally but the restriction of which to N is highly complicated in the\nrecursion-theoretic sense. We therefore emphasize that partial functions must be dealt with\nseriously, and devote this paper to accordingly reformulating the theory wherever possible.\n1Moore writes M0 for the class of real primitive recursion functions. Claim 1 was later replaced by a similar\nclaim [6, Proposition 2] for a more “restricted” class G than M0, but its definition is again unclear.\n2"},{"page":3,"text":"2. Differentially algebraic functions\nWe show some facts about single- and multi-place differentially algebraic functions.\nTheorem 3. Let m, n and i < m be nonnegative integers and f :⊆Rm →Rn be an analytic\nfunction with open domain. Let (i), (ii), (iii) and (iv) be the following statements:\n(i) for any open connected set J ⊆dom f, there is a Zn-coefficient nonzero polynomial P\nsuch that\nP\n f x, Deif x, D2·eif x, . . . , D(arity P −1)·eif x\n \n= 0\n(2)\nfor all x ∈J, where ei ∈Nm is the vector whose ith component is 1 and others are 0;\n(ii) for each x0 ∈dom f, there are a Zn-coefficient nonzero polynomial P and an open set J\ncontaining x0 such that we have (2) for all x ∈J;\n(iii) for each x0 ∈dom f, there are a Zn-coefficient nonzero polynomial P and an open\ninterval J containing the ith component of x0 such that we have (2) for all x whose ith\ncomponent is in J and whose other components equal those of x0;\n(iv) for each x ∈dom f, there is a Zn-coefficient nonzero polynomial P satisfying (2).\nLet (iR), (iiR) and (iiiR) be the statements obtained by replacing Z by R in (i), (ii) and (iii),\nrespectively. Then (i), (ii), (iii), (iv), (iR), (iiR) and (iiiR) are equivalent.\nProof. The implications (i) ⇒(ii) ⇒(iii) ⇒(iv) and (i) ⇒(iR) ⇒(iiR) ⇒(iiiR) are obvious.\nIt has been known that (iiiR) ⇒(iii), see Theorem 16. To see (iv) ⇒(i), consider, for each Zn-\ncoefficient polynomial P, the set JP of all x ∈J satisfying (2). Since by (iv) these countably\nmany closed sets JP cover the open set J, one of them must have nonempty interior by Baire\nCategory Theorem 14. This JP must then equal J by the Identity Theorem 15.\nDefinition 4. Let m and n be nonnegative integers. An analytic function f :⊆Rm →Rn is\ndifferentially algebraic2 if for each i < m it satisfies one (or all) of the clauses in Theorem 3.\nNote that f need not be the unique solution of (2). For example, every function (with\nopen domain) that is constant on each connected component of its domain is differentially\nalgebraic because of the single set of equations Deif x = 0.\nThe clauses (ii)–(iv) show that being differentially algebraic is a “local” property.\nWhen dom f is connected, (i) reduces to the following statement:\n(i′) there is a Zn-coefficient nonzero polynomial P such that we have (2) for all x ∈dom f.\nHence, the clause (iii) shows that, as long as dom f is connected, our definition is equivalent\nto that of many authors, including Moore [13], who first state the definition for m = 1 by (i′)\nand then extend it to m > 1 by saying that a function is differentially algebraic when it is so\nas a unary function of each argument when all other arguments are held fixed.3 We proved\nTheorem 3 in order to use (i) to present a counterexample to Claim 1 later.\n2Also termed algebraic transcendental or hypotranscendental. Functions without this property is said to be\ntranscendentally transcendental or hypertranscendental.\n3Their definition for the case m = 1 is slightly different from (i′) in that it replaces (2) by P (x, f x, Deif x, . . . ,\nD(arity P −2)·eif x) = 0. But the proof of (a) ⇒(b) of Lemma 5 shows that this difference is superficial.\n3"},{"page":4,"text":"Let us characterize differentially algebraic functions in yet another way for the case n = 1.\nFor a field E, its subfield F and a set B ⊆E, we write F(B), agreeing tacitly on E, for the\nsmallest subfield of E that includes F and B. We write F for the algebraic closure of F, that\nis, the set of those elements of E that annuls some F-coefficient unary nonzero polynomial.\nLet J ⊆Rm be an open set and consider the ring Cω[J] of analytic functions g :⊆Rm →R\nwith dom g = J. Note that R is embedded into this ring by regarding each x ∈R as the\nconstant function taking the value x. To assert (2) for all x ∈J is to say that\nP\n f, Deif, D2·eif, . . . , D(arity P −1)·eif\n \n= 0\n(3)\nin Cω[J]. If J is connected, Cω[J] has a quotient field by the Identity Theorem 15, so the\nnotation R(D f) in the following lemma makes sense. We write D f = { Daf | a ∈Nm }.\nLemma 5. Let J ⊆Rm be open and connected. For f ∈Cω[J], the following are equivalent:\n(a) f is differentially algebraic;\n(b) D f ⊆R(B) for some finite set B ⊆D f;\n(c) D f ⊆R(B) for some finite set B ⊆R(D f).\nProof. The implication (b) ⇒(c) is trivial. The Transcendence Degree Theorem 17 shows\n(c) ⇒(a). For (a) ⇒(b), assume that for each i we have an R-coefficient polynomial Pi\nwith\nPi(f, Deif, D2·eif, . . . , DNi·eif) = 0,\n(4)\nwhere Ni = arity Pi −1. By choosing Ni to be smallest and then the degree of Pi in the last\nplace to be smallest, we may assume that\nΞ = (D(0,...,0,1)Pi)(f, Deif, D2·eif, . . . , DNi·eif)\n(5)\nis nonzero. Consider the order ≤on Nm defined by setting a ≤b when a + c = b for some\nc ∈Nm. We will show by induction on a ∈Nm that Daf ∈R({ Dbf | b ≤(N0, . . . , Nm−1) }).\nThe case a ≤(N0, . . . , Nm−1) being trivial, assume that a ≥(Ni + 1) · ei for some i. Apply\nDa−Ni·ei to both sides of (4) and calculate using chain rules to obtain\nΨ + Ξ · Daf = 0,\n(6)\nwhere Ψ can be written as a sum of products of several derivatives of f of order ≤a and\n̸= a, which hence enjoy the induction hypothesis.\nApart from purely theoretical interest, the significance of differentially algebraic functions\nlies in their relation to the General Purpose Analog Computer, an analog computation model\nintroduced by Shannon [18] and later refined by Pour-El [16]. More precisely, if a function\nf :⊆R →R with nonempty domain is differentially algebraic, then the restriction of f to\nsome nonempty subset of dom f is GPAC generable [16, Theorem 4]; conversely, if a function\nf :⊆R →R with nonempty domain is GPAC generable, then the restriction of f to some\nnonempty subset of dom f is differentially algebraic [12, Theorem 2]. Gra ̧ca later considered\nthe Polynomial GPAC, a simpler refinement than Pour-El’s, and proved analogous results [8].\n4"},{"page":5,"text":"3. Real primitive recursive functions\nThe class of real primitive recursive functions is defined [13] as the smallest class containing\nsome basic functions and closed under the operators specified below.\nUnfortunately, the\noriginal definition contains ambiguity, resulting in some inconsistent claims about the class.\nTo remedy this, we shall revisit the definitions carefully in Sections 3.1 and 3.2. Section 3.3\ndiscusses an alternative approach by Campagnolo. Section 3.4 disproves Claim 1.\n3.1. Two basic operators\nThe real primitive recursive functions are defined through three operators: juxtaposition,\ncomposition and differential recursion. The first two are very simple.\nDefinition 6. Given g0, . . . , gn−1 :⊆Rm →R, define their juxtaposition jx (g0, . . . , gn−1) :⊆\nRm →Rn by setting jx (g0, . . . , gn−1)x = (g0x, . . . , gn−1x) whenever x ∈dom g0 ∩· · · ∩\ndom gn−1. Given f :⊆Rm →Rn and g :⊆Rl →Rm, define their composition cm (f, g) :⊆\nRl →Rn by setting cm (f, g)x = f (gx) whenever x ∈dom g and gx ∈dom f.\nWe write f ◦g for cm (f, g). As remarked in Section 1, it is important to define precisely\nwhat the operators do on partial functions. Note how Definition 6 specifies the domain of\nthe functions constructed. If gx is not defined, neither is (f ◦g)x, even if, say, f is a constant\nfunction defined everywhere. We thus work in the following (informal) general principle.\nPrinciple 7. For the value of an expression to be defined, the value of each of its subexpres-\nsions has to be defined.\nWe remark that this was not explicitly intended by Moore. In fact, he presents an example\nto the contrary when he claims [13, Section 6] that the total function inv :⊆R →R given\nby\ninvx =\n(\n0\nif x = 0\n1/x\nif x ̸= 0\n(7)\ncan be obtained by composing the binary multiplication with jx (zero?, g), where zero? is\nfrom (30) and g is the restriction of inv to R\\{0}. Some authors point this out [4, p. 22] and\ncriticize it [7, p. 47]. Without discussing which definition is more “natural,” we adopt our\nrestrictive Definition 6, simply because it is not clear how to formulate a general definition\nthat would admit this construction of inv.\nThe operators jx and cm preserve analyticity [10, Proposition 2.2.8].\nTheorem 8. The property of being differentially algebraic is preserved by jx and cm.\nProof. This is trivial for jx. For cm, it suffices to show that if f :⊆Rm →R and g0, . . . ,\ngm−1 :⊆Rl →R are differentially algebraic, so is f ◦g, where g = jx (g0, . . . , gm−1). We\nmay assume that dom g and J = dom (f ◦g) are connected. We use the characterization (b)\nof Lemma 5. Calculate each element of D (f ◦g) by the chain rule to see that it belongs to\nR\n { d ◦g | d ∈D f } ∪\nm−1\nS\ni=0\n{ q↾J | q ∈D gi }\n \n,\n(8)\nwhere q↾J means the restriction of q to J. By the assumption, there are finite subsets A ⊆D f\nand Bi ⊆D gi with D f ⊆R(A) and D gi ⊆R(Bi) for each i = 0, . . . , m −1. This implies\nthat (8) stays unchanged by replacing D f by A and D gi by Bi.\n5"},{"page":6,"text":"initial value\nt\nx\nx = h0 t\nx = h1 t\nFigure 1: When the equation is satisfied by both h0 and h1 (as well as their restriction to\neach interval containing the origin), how do we say that the shaded interval is where\nthere is a “unique solution”?\n3.2. The differential recursion operator\nTo formulate the third operator, we need a notion of unique solution of an integral equation\nof the form (9) below, where h is the unknown. For example, it sounds natural to say that\nthe tangent function restricted to (−π/2, π/2) uniquely solves ht =\nR t\n0\n 1+(hτ)2 \ndτ. But as\nwe are talking about partial functions, the word “unique” should be used carefully, because\nthe restriction to any subinterval J ⊆(−π/2, π/2) containing 0 also satisfies the equation on\nJ. Thus, out of the set H of all solutions, we need to pick one function that deserves to be\ncalled the unique solution defined on the largest possible interval (Figure 1). Though Moore\ndid not discuss this, it is not hard to formulate this intuition: for a set H of functions of a\ntype, we say that a function h ∈H is unique in H if the restriction of any function in H to\ndom h is a restriction of h.\nDefinition 9. Let f :⊆Rm →Rn and g :⊆Rm+1+n →Rn. For each v ∈Rm, let Hv be the\nset of all functions h :⊆R →Rn such that\n(a) dom h is either the empty set or a possibly unbounded interval containing 0,\n(b) v ∈dom f if dom h is nonempty,\n(c) (v, τ, hτ) ∈dom g for each τ ∈dom h, and\n(d) every t ∈dom h satisfies\nht = f v +\nZ t\n0\ng(v, τ, hτ) dτ.\n(9)\nLet Kv be the set of functions unique in Hv. By Lemma 18 in the appendix, Kv has an\nelement hv of which all functions in Kv is a restriction. Define dr (f, g) :⊆Rm+1 →Rn by\ndom\n dr (f, g)\n \n= { (v, t) ∈Rm+1 | t ∈dom hv } and dr (f, g)(v, t) = hv t.\nDefinition 10. The class of real primitive recursive functions is the smallest class containing\nthe nullary functions 00→1, 10→1, −10→1 and closed under jx, cm and dr.\n6"},{"page":7,"text":"Lemma 11. The following functions are real primitive recursive: for each n ∈N, the n-ary\nconstants 0n→1, 1n→1, −1n→1; for n ∈N and i = 0, . . . , n −1, the n-ary projection idn→1\ni\nto\nthe ith component; binary add and mul; the functions inv+ (mapping x > 0 to 1/x), sqrt+\n(mapping x > 0 to √x) and ln (natural logarithm) defined on (0, ∞); the total functions sin,\ncos and exp; the circle ratio π as a nullary function.\nProof. The constant 0n→1 is built by 0n→1 = 00→1 ◦jx ( ); similarly for 1n→1 and −1n→1.\nThen inductively define idi+1→1\ni\n= dr (0i→1, 1i+2→1) and idn+1→1\ni\n= dr (idn→1\ni\n, 0n+2→1). Using\nthese, let add = dr (id1→1\n0\n, 13→1) and mul = dr (01→1, id3→1\n0\n). For inv+, define\nf = dr\n 10→1, mul ◦jx\n −11→1, mul ◦jx (id1→1\n0\n, id1→1\n0\n)\n \n◦id2→1\n1\n \n,\n(10)\ninv+ = f ◦\n add ◦jx (id1→1\n0\n, −11→1)\n \n,\n(11)\nor, more colloquially,\nf t = 1 −\nZ t\n0\n(f τ)2 dτ,\ninv+t = f (t −1).\n(12)\nSquare root is defined analogously by\nf t = 1 +\nZ t\n0\ninv+(2 · f τ) dτ,\nsqrt+t = f (t −1).\n(13)\nLogarithm and exponentiation are analogous, using suitable integral equations.\nFor the\ntrigonometric functions, let sin = id2→1\n0\n◦trig and cos = id2→1\n1\n◦trig, where\ntrig = dr\n jx (00→1, 10→1), jx\n id3→1\n2\n,\n mul ◦jx (−13→1, id3→1\n1\n)\n \n,\n(14)\nwhich is to say,\n \nsint\ncost\n \n=\n \n0\n1\n \n+\nZ t\n0\n \ncosτ\n−sinτ\n \ndτ.\n(15)\nThe circle ratio is π( ) = 4·Arctan1, with Arctan defined by a suitable integral equation.\nSome authors say “the function 1/x is real primitive recursive” to mean that inv+ is. It is\nnot clear how such assertions without specification of domain can be justified.\nThe reader may have felt uncomfortable with the unwieldy process of Definition 9 in picking\nthe right solution hv out of Hv.\nThis can be simplified if we discuss only real primitive\nrecursive functions, because of the following facts that result from the Uniqueness Theorem\nfor initial value problems [11] and the Cauchy–Kowalewsky Theorem [10, Section 2.4].\nTheorem 12. Let f, g, v, Hv and hv be as in Definition 9.\n(a) If g is an analytic4 function with open domain, Hv is the set of all restrictions of hv.\n(b) If f and g are analytic functions with open domain, so is dr (f, g).\nThe fact (a) says that a solution of (9) may diverge to infinity at some point but can never\n“branch” as in Figure 1, provided g is smooth enough. We therefore could have dispensed\nwith Lemma 18 and simply let hv be the (graph) union of Hv, so far as real primitive recursive\nfunctions are concerned, because they are analytic by (b).\n4This fact is often stated with a weaker assumption that g be Lipschitz continuous.\n7"},{"page":8,"text":"t\nx\n1\nx = gt\nIntegrate?\ny = kt\nt\ny\n1\nFigure 2: Integrand with a singularity.\n3.3. Campagnolo’s differential recursion\nThe clauses (a)–(c) of Definition 9 guarantee that the integral equation (9) makes sense for all\nt ∈dom h. The clause (c), however, could be slightly relaxed, since a small set of singularities\nin the integrand does not affect the integration. Define drC by replacing (c) with\n(c′) (v, τ, hτ) ∈dom g for any τ ∈dom h \\ S, where S is a countable set of isolated points.\nThis is due to Campagnolo [4, Definition 2.4.2], though he does not present a precise spec-\nification of the “unique” solution as we noted in the Section 3.2. The choice between (c)\nand (c′) is somewhat similar to the discussion regarding Principle 7. The issue is whether\ng(v, τ, hτ), where τ ∈[0, t], is a “subexpression” of the right-hand side of the equation (9).\nWithout going into the philosophical discussion to ask which is “natural,” we point out some\ndifferences this choice incurs.\nTheorem 12 (b) fails if we replace dr by drC, as the following example shows (Figure 2).\nThe function g :⊆R →R defined by dom g = R\\{1} and gt = inv+\n sqrt+\n sqrt+(t−1)2 \n=\n1/\np\n|t −1| is real primitive recursive by Lemma 11. But k = drC (−20→1, g ◦id2→1\n0\n) :⊆R →\nR, where −20→1 is the constant function with value −2, is the total function given by\nkt =\n(\n+2 · √t −1\nif t ⩾1,\n−2 · √−t + 1\nif t < 1,\n(16)\nwhich is not differentiable at 1. Note that dr (−20→1, g ◦id2→1\n0\n) is its restriction to (−∞, 1)\nand thus analytic. For a subtler example, recall the equation (13) for sqrt+; with drC, the\nsame equation produces the square root function defined on [0, ∞), rather than on (0, ∞).\nThis breaks the assumption of Theorem 12 (a) and thus gives rise to incomparable functions\nin Hv when, say, f = 10→1 and g = k ◦id2→1\n1\n, with k from (16); that is, the equation\nht = 1 +\nZ t\n0\nk(hτ) dτ\n(17)\nhas two solutions that take different values at a point.\nKeeping the class analytic also conforms to Moore’s intention [13, Definition 9] to make\nthe equation (9) equivalent to\nh0 = f v,\nD1ht = g(v, t, ht),\n(18)\nwhich would not make sense for non-differentiable h.\n8"},{"page":9,"text":"3.4. A primitive recursive but not differentially algebraic function\nClaim 1 would not make sense if we adopted drC in defining real primitive recursive functions,\nbecause there would then arise non-analytic functions, as we noted above. We now show that,\neven under our restrictive definition with the analyticity-preserving dr, the claim fails.\nDefine ˇΓ :⊆R2 →R by dom Γ = (0, ∞)2 and\nˇΓ (R, x) =\nZ R\n1/R\nexp\n (x −1) · ln t −t\n \ndt.\n(19)\nDefine Euler’s gamma function Γ :⊆R →R by dom Γ = (0, ∞) and\nΓ x = lim\nR→∞\nˇΓ (R, x).\n(20)\nIt can be verified that this value converges and satisfies\nDnΓ x = lim\nR→∞D(0,n) ˇΓ (R, x)\n(21)\nfor each n ∈N and x ∈(0, ∞). H ̈older showed that Γ is not differentially algebraic [9].\nWe do not know if Γ is real primitive recursive, but ˇΓ is easily shown real primitive\nrecursive, using Lemma 11. However, contrary to Claim 1, it is not differentially algebraic.\nFor assume that it were. We would then have a nonzero polynomial P such that\nP\n ˇΓ (R, x), D(0,1) ˇΓ (R, x), . . . , D(0,arity P −1) ˇΓ (R, x)\n \n= 0\n(22)\nfor each (R, x) ∈(0, ∞)2. Note that we used the characterization (i) of Theorem 3 in order\nto take P independent of R. We take the limit of (22) as R →∞, which by (21) yields\nP (Γ x, D1Γ x, . . . , Darity P −1Γ x) = 0,\n(23)\ncontradicting H ̈older.\n4. Other classes and related works\nThis section discusses some other operators introduced by Moore and other authors.\n4.1. Minimization and Moore’s real recursive functions\nFor a function f :⊆Rm+1 →R, Moore defines mn f :⊆Rm →R by\nmn f v =\n(\nt+ = inf { t ≥0 | f (v, t) = 0 }\nif t+ < −t−,\nt−= sup { t ≤0 | f (v, t) = 0 }\notherwise.\n(24)\nThe class of real recursive functions5 is the smallest class containing all real primitive recursive\nfunctions and closed under jx, cm, dr and mn.\n5This “recursiveness” of Moore’s should not be confused with the same word also used in the context of\nComputable Analysis. As we see in Appendix C, Moore’s real recursive functions can even be discontinuous.\n9"},{"page":10,"text":"Moore states the definition of mn in a way that leaves ambiguous whether (24) has a value\nwhen, say, dom f = Rm × [1, ∞) and f (v, t) = 2 −t for all t ≥1. Should it have the value 2,\nor be left undefined because “the zero-searching program gets stuck”?\nIt turns out that, whichever definition we choose, Moore’s claim about iteration [13, Propo-\nsition 11] remains true, in the following modified form. Since the original proof again forgets\npartial functions, we present a new proof in Appendix C.\nLemma 13. If f :⊆Rm →Rm is real recursive, there is a real recursive function g :⊆\nRm+1 →Rm that extends the function g′ defined by dom g′ =\n \n(v, k) ∈R × (N \\ {0})\n v ∈\ndom f k \nand g′(v, k) = f kv for all (v, k) ∈dom g′, where f k = f ◦· · · ◦f\n|\n{z\n}\nk\n.\nWe have to note, however, that the class of real recursive functions is probably not well-\nbehaved, since, with mn producing non-smooth functions, the class no longer enjoys Theo-\nrem 12. We therefore doubt the significance of Claim 2, although it could be justified by\nusing Lemma 13 to simulate Turing machines as Moore did.\n4.2. Linear differential recursion\nWe have seen that many of the problems in Moore’s original work were caused by failure to\ndeal with partial functions properly. Some authors avoid this trouble by studying only oper-\nators preserving totality, so that partial functions never come into discussion. Campagnolo\nand Moore [5] take this path by considering linear differential recursion in place of dr. For\nclasses defined by this operator, some relationships with digital computation are known [2, 4].\n4.3. Open problems\nClaim 1 has been the main rationale for calling variants of Moore’s classes a model of analog\ncomputation. Now that we have lost it, an important challenge is the following.\nOpen Problem. Find a subclass of our real primitive recursive functions, preferably with an\nequally simple definition, that has a close relationship to the differentially algebraic functions.\nAnother direction would be to reformulate further the rest of Moore’s work, as well as other\nauthors’ works that also suffer from the same kind of ambiguity. For example, it may be\ninteresting to work out Mycka and Costa’s class arising from the operator of taking limits [14].\nAcknowledgement\nThe author thanks Mariko Yasugi at Kyoto Sangyo University for the discussion that led\nto this work. Comments by J. F. Costa at Instituto Superior T ́ecnico helped improve the\npresentation of Section 3.2.\n10"},{"page":11,"text":"References\n[1] Blum, L., Cucker, F., Shub, M. and Smale, S. Complexity and Real Computation,\nSpringer-Verlag (1997).\n[2] Bournez, O. and Hainry, E. An Analog Characterization of Elementarily Computable Func-\ntions over the Real Numbers, Proceedings of the Thirty-First International Colloquium on\nAutomata, Languages and Programming, Springer (2004), Lecture Notes in Computer Science\n3142.\n[3] Bush, V. The Differential Analyzer: A New Machine for Solving Differential Equations, Jour-\nnal of the Franklin Institute, 212 (1931), 447–488.\n[4] Campagnolo, M. L. Computational Complexity of Real Valued Recursive Functions and Ana-\nlog Circuits, PhD thesis, Instituto Superior T ́ecnico (July 2001).\n[5] Campagnolo, M. L. and Moore, C. Upper and Lower Bounds on Continuous-Time Com-\nputation, Second International Conference on Unconventional Models of Computation (2001).\n[6] Campagnolo, M. L., Moore, C. and Costa, J. F. Iteration, Inequalities, and Differentia-\nbility in Analog Computers, Journal of Complexity, 16, 4 (December 2000), 642–660.\n[7] Grac ̧a, D. The General Purpose Analog Computer and Recursive Functions Over the Reals,\nMaster’s thesis, Instituto Superior T ́ecnico (July 2002).\n[8] Grac ̧a, D. Some recent developments on Shannon’s General Purpose Analog Computer, Math-\nematical Logic Quarterly, 50 (April 2004), 473–485.\n[9] H ̈older, O. L. Ueber die Eigenschaft der Gammafunction keiner algebraischen Differential-\ngleichung zu gen ̈ugen, Mathematische Annalen, 28 (1886), 1–13.\n[10] Krantz, S. G. and Parks, H. R. A Primer of Real Analytic Functions, Birkh ̈auser Advanced\nTexts, Birkh ̈auser Boston, second edition (June 2002).\n[11] Lang, S. Real and Functional Analysis, Vol. 142 of Graduate Texts in Mathematics, Springer-\nVerlag, third edition (1993).\n[12] Lipshitz, L. and Rubel, L. A. A Differentially Algebraic Replacement Theorem, and Analog\nComputability, Proceedings of the American Mathematical Society, 99, 2 (February 1987),\n367–372.\n[13] Moore, C. Recursion Theory on the Reals and Continuous-Time Computation, Theoretical\nComputer Science, 162 (1996), 23–44.\n[14] Mycka, J. and Costa, J. F. Real Recursive Functions and Their Hierarchy, Journal of\nComplexity, 20, 6 (December 2004), 835–857.\n[15] Orponen, P. A Survey of Continuous-Time Computation Theory, Advances in Algorithms,\nLanguages, and Complexity, Kluwer Academic Publishers (1997), 209–224.\n[16] Pour-El, M. B. Abstract Computability and its Relation to the General Purpose Analog\nComputer (Some Connections Between Logic, Differential Equations and Analog Computers),\nTransactions of the American Mathematical Society, 199 (1974), 1–28.\n[17] Ritt, J. F. and Gourin, E. An Assemblage-Theoretic Proof of the Existence of Transcen-\ndentally Transcendental Functions, Bulletin of the American Mathematical Society, 33 (1927),\n182–184.\n[18] Shannon, C. E. Mathematical Theory of the Differential Analyzer, Journal of Mathematics\nand Physics, 20, 4 (1941), 337–354.\n[19] Weihrauch, K. Computable Analysis: An Introduction, Texts in Theoretical Computer Sci-\nence, Springer-Verlag (2000).\n11"},{"page":12,"text":"A. Old results\nWe list some known theorems that we used in Section 2.\nThe following Baire Category Theorem is used in the proof Theorem 3.\nTheorem 14. Let J be a subset of Rm. The union of countably many closed subsets of J with\nempty interior has empty interior.\nProof. Let J0, J1, . . . be closed subsets of J with empty interior, and U be any nonempty open\nsubset of J. We will show that U \\ S\nP ∈N JP is nonempty. For each P ∈N, we take xP ∈Rm\nand εP ∈R as follows. Write B(x, ε) for the open set of points in J whose distance from x is less\nthan ε. Let x0 ∈U and ε0 ∈(0, 1) be such that B(x0, ε0) ⊆U. For each P ∈N, let xP +1 ∈U\nand εP +1 ∈(0, 2−P −1) be such that B(xP +1, εP +1) ⊆B(xP , εP ) \\ JP . This is possible because\nB(xP , εP ) \\ JP is open and nonempty, since JP is closed and has empty interior. As P tends to\ninfinity, xP converges to a point in U \\ S\nP ∈N JP .\nThe proof of Theorem 3 also uses the following Identity Theorem (for real analytic functions\nof several variables), also known as the Principle of Analytic Continuation. It can be proved by\nstraightforwardly generalizing the same assertion for unary functions [10, Section 1.2].\nTheorem 15. An analytic function with open connected domain that vanishes on an open set\nvanishes everywhere.\nLet J ⊆R be an open interval. It is well known that functions u0, . . . , uk−1 ∈Cω[J] are linearly\ndependent if and only if the determinant\n (Diuj)i,j=0,...,k−1\n , called their Wronskian, is zero. Using\nthis fact, Ritt and Gourin [17] showed (iiiR) ⇒(iii) of Theorem 3.\nTheorem 16. Let J ⊆R be an open interval and let f ∈Cω[J]. If we have\nP (f, D1f, D2f, . . . , Darity P −1f) = 0\n(25)\nfor some R-coefficient nonzero polynomial P, then we have (25) for some Z-coefficient nonzero\npolynomial P.\nProof. By the assumption, there is a finite set B ⊆Narity P such that the functions\nf ν0 · (Df)ν1 · · · (Darity P −1f)νarity P −1,\nfor (ν0, . . . , νarity P −1) ∈B,\n(26)\nare linearly dependent. The Wronskian of (26) thus vanishes, which is a Z-coefficient polynomial\nin f, D1f, . . . , Darity P +|B|−1f. This polynomial is nonzero, since otherwise (26) would be linearly\ndependent for arbitrary f, which is absurd.\nOne direction of Lemma 5 uses the following Transcendence Degree Theorem.\nTheorem 17. Let F be a subfield of a field E and D be a subset of E. If D ⊆F(B) for some finite\nset B ⊆E, then D ⊆F(C) for some finite set C ⊆D.\nProof. For each d ∈D, the assumption gives\ndl =\nl−1\nP\nj=0\nβj · dj\n(27)\n12"},{"page":13,"text":"for some l ∈N \\ {0} and βj ∈F(B). Suppose that for some d = d0 ∈D, this equation contains\nsome b ∈B \\ D, since otherwise we are done. Then we can rewrite (27) as\nbk =\nk−1\nP\ni=0\nαi · bi\n(28)\nfor some k ∈N \\ {0} and αi ∈F(B′), where B′ = (B \\ {b}) ∪{d0}.\nFor each d ∈D and t ∈N, we can substitute (27) and (28) repeatedly in dt to write\ndt =\nk−1\nP\ni=0\nl−1\nP\nj=0\nγi,j · ci · dj\n(29)\nfor some γi,j ∈F(B′). The k · l + 1 elements 1, d, d2, . . . , dk·l are hence linearly dependent over\nF(B′). We have thus found a set B′ with D ⊆F(B′) such that B′ \\ D has strictly less elements\nthan B \\ D. Repeat.\nB. Maximal unique function\nThis section shows that, from a set K of functions with a certain property, we can choose a function\nof which all functions in K is a restriction. This was used to justify Definition 9 in the presence of\nnon-analytic functions where Theorem 12 (a) does not apply.\nWe say that a set I ⊆R is 0-convex if it is either the empty set or a possibly unbounded interval\ncontaining 0. Note that the union of 0-convex sets is 0-convex.\nWe say that a set K of functions from R is consistent if for any t ∈R, the set { gt | g ∈K }\nhas at most one element. In this case, the union of K means the unique function k such that\ndom h = S\ng∈K dom g and for each t ∈dom h, there is some g ∈K with ht = gt.\nLemma 18. Let H be a set of functions from R with 0-convex domain. Then the set K of functions\nunique in H is consistent. Moreover, if its union belongs to H, it belongs to K.\nProof. For the first claim, suppose otherwise.\nThen there are functions k0, k1 ∈K and t ∈\ndom k0 ∩dom k1 such that k0t ̸= k1t. This contradicts the fact that k0 is unique in H.\nFor the second claim, suppose that the union k of K is not unique in H. That is, there are a\nfunction g ∈H and t ∈dom k ∩dom g such that gt ̸= kt. There is k0 ∈K for which t ∈dom k0.\nWe have k0t = kt ̸= gt, contradicting the fact that k0 is unique in H.\nThis lemma can be applied to H = Hv in the situation of Definition 9, because there the union\nof any consistent subset of Hv belongs to Hv.\nC. Iteration\nAs we noted, the definition (24) of the operator mn is ambiguous, as it contains a subexpression\nf (v, t) that may be undefined for some (v, t). So when is mn f v defined? Possible answers include:\n(a) When t+ and t−are defined.\n(b) When at least either t+ or t−is defined; the condition t+ < −t−will be used only when both\nare defined.\nAnd when is t+ (resp. t−) defined? Possible answers include:\n13"},{"page":14,"text":"x = clk t\n1\n2\n1\nt\nx\nx = zigzag t\n1\n2\n1\nt\nx\nfv\nf 2v\nf 3v\nx = g(v, t)\nx = h(v, t)\n1\n2\nv\nt\nx\nFigure 3: Simulating iteration f kv by real recursive functions.\n(i) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈R.\n(ii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[−t, t]\n(resp. [t, −t]).\n(iii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[0, t]\n(resp. [t, 0]).\n(iv) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0.\nFor (i), (ii) and (iii), we may also consider adding the phrase “except for some countably many\nisolated τ” (compare (c′) in Section 3.3).\nMoore’s informal explanation by a programming language [13, Section 7] seems to suggest (b)\nand (ii). However, without discussing which is the “right” definition of mn, we show that, whichever\nwe choose, Lemma 13 holds. The following proof is consistent with any of the above 2 × 7 possible\ndefinitions.\nProof of Lemma 13. Denote mn f v by μt. f (v, t). Let\nzero?x = μy. (x2 + y2) · (1 −y),\n(30)\ninteger?x = zero?\n sin (π · x)\n \n,\n(31)\nroundx = x −μr. integer?(x −r),\n(32)\nso that (32) is the unique integer in (x −1/2, x + 1/2]. We get inv of (7) by\ninvx = μt. x · (x · t −1).\n(33)\nThe above four functions are total. Let\ndigit(x, b, i) = round\n x\nbi −1\n2\n \n−b · round\n x\nbi+1 −1\n2\n \n(34)\nfor b > 0, where bi = exp(i · ln b). When b > 1 and i are integers, digit(x, b, i) is the digit in bi’s\nplace when x is written in base-b notation. Define\nclkt = digit(t, 2, −1),\nzigzagt = 0 +\nZ t\n0\n(2 −4 · clkτ) dτ,\n(35)\n g(v, t)\nh(v, t)\n \n=\n v\nv\n \n+\nZ t\n0\n 2 · (1 −clkτ) ·\n f\n h(v, τ) −clkτ · (h(v, τ) −v)\n \n−h(v, τ)\n \n2 · clkτ ·\n h(v, τ) −g(v, τ)\n \n· inv(zigzagτ)\n \ndτ,\n(36)\nas in Figure 3. We have f kv = g(v, k −1/2) for k ∈N \\ {0}.\nNote that clkτ · (h(v, τ) −v) in (36) cannot be dropped, because of Principle 7.\n14"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"with dom g = J ∩dom f such that gx = f x for every x ∈dom g. When dom f is open, f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"is said to be (real) analytic if for every a = (a0, . . . , am−1) ∈dom f there are an open set","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"p=(p0,...,pm−1)∈Nm cp · (x0 −a0)p0 · · · (xm−1 −am−1)pm−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"converges to f x for each x = (x0, . . . , xm−1) ∈J (regardless of the ordering of summation).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"algebraic because of the single set of equations Deif x = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"to that of many authors, including Moore [13], who first state the definition for m = 1 by (i′)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"3Their definition for the case m = 1 is slightly different from (i′) in that it replaces (2) by P (x, f x, Deif x, . . . ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"D(arity P −2)·eif x) = 0. But the proof of (a) ⇒(b) of Lemma 5 shows that this difference is superficial.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"Let us characterize differentially algebraic functions in yet another way for the case n = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"with dom g = J. Note that R is embedded into this ring by regarding each x ∈R as the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"notation R(D f) in the following lemma makes sense. We write D f = { Daf | a ∈Nm }.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"Pi(f, Deif, D2·eif, . . . , DNi·eif) = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"where Ni = arity Pi −1. By choosing Ni to be smallest and then the degree of Pi in the last","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"Ξ = (D(0,...,0,1)Pi)(f, Deif, D2·eif, . . . , DNi·eif)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"is nonzero. Consider the order ≤on Nm defined by setting a ≤b when a + c = b for some","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"Ψ + Ξ · Daf = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"̸= a, which hence enjoy the induction hypothesis.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"Rm →Rn by setting jx (g0, . . . , gn−1)x = (g0x, . . . , gn−1x) whenever x ∈dom g0 ∩· · · ∩","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"Rl →Rn by setting cm (f, g)x = f (gx) whenever x ∈dom g and gx ∈dom f.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"invx =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"if x = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"if x ̸= 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"gm−1 :⊆Rl →R are differentially algebraic, so is f ◦g, where g = jx (g0, . . . , gm−1). We","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"may assume that dom g and J = dom (f ◦g) are connected. We use the characterization (b)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"and Bi ⊆D gi with D f ⊆R(A) and D gi ⊆R(Bi) for each i = 0, . . . , m −1. This implies","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"x = h0 t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"x = h1 t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"the tangent function restricted to (−π/2, π/2) uniquely solves ht =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"ht = f v +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"= { (v, t) ∈Rm+1 | t ∈dom hv } and dr (f, g)(v, t) = hv t.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"constants 0n→1, 1n→1, −1n→1; for n ∈N and i = 0, . . . , n −1, the n-ary projection idn→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"Proof. The constant 0n→1 is built by 0n→1 = 00→1 ◦jx ( ); similarly for 1n→1 and −1n→1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"= dr (0i→1, 1i+2→1) and idn+1→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"= dr (idn→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"these, let add = dr (id1→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":", 13→1) and mul = dr (01→1, id3→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"f = dr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"inv+ = f ◦","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"f t = 1 −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"inv+t = f (t −1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"f t = 1 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"sqrt+t = f (t −1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"trigonometric functions, let sin = id2→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"◦trig and cos = id2→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"trig = dr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"The circle ratio is π( ) = 4·Arctan1, with Arctan defined by a suitable integral equation.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"x = gt","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"y = kt","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"The function g :⊆R →R defined by dom g = R\\{1} and gt = inv+","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"|t −1| is real primitive recursive by Lemma 11. But k = drC (−20→1, g ◦id2→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"kt =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"in Hv when, say, f = 10→1 and g = k ◦id2→1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"ht = 1 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"h0 = f v,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"D1ht = g(v, t, ht),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"Define ˇΓ :⊆R2 →R by dom Γ = (0, ∞)2 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"ˇΓ (R, x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"Define Euler’s gamma function Γ :⊆R →R by dom Γ = (0, ∞) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"Γ x = lim","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"DnΓ x = lim","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"P (Γ x, D1Γ x, . . . , Darity P −1Γ x) = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"mn f v =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"t+ = inf { t ≥0 | f (v, t) = 0 }","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"t−= sup { t ≤0 | f (v, t) = 0 }","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"when, say, dom f = Rm × [1, ∞) and f (v, t) = 2 −t for all t ≥1. Should it have the value 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"Rm+1 →Rm that extends the function g′ defined by dom g′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"and g′(v, k) = f kv for all (v, k) ∈dom g′, where f k = f ◦· · · ◦f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"(Diuj)i,j=0,...,k−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"P (f, D1f, D2f, . . . , Darity P −1f) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"dl =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"j=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"for some l ∈N \\ {0} and βj ∈F(B). Suppose that for some d = d0 ∈D, this equation contains","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"bk =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"for some k ∈N \\ {0} and αi ∈F(B′), where B′ = (B \\ {b}) ∪{d0}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"dt =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"j=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"dom h = S","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"g∈K dom g and for each t ∈dom h, there is some g ∈K with ht = gt.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"dom k0 ∩dom k1 such that k0t ̸= k1t. This contradicts the fact that k0 is unique in H.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"function g ∈H and t ∈dom k ∩dom g such that gt ̸= kt. There is k0 ∈K for which t ∈dom k0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"We have k0t = kt ̸= gt, contradicting the fact that k0 is unique in H.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"This lemma can be applied to H = Hv in the situation of Definition 9, because there the union","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"x = clk t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"x = zigzag t","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"x = g(v, t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"x = h(v, t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"(i) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈R.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"(ii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[−t, t]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"(iii) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0 and (v, τ) ∈dom f for all τ ∈[0, t]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"(iv) When there is t ≥0 (resp. ≤0) such that f (v, t) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"zero?x = μy. (x2 + y2) · (1 −y),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"integer?x = zero?","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"roundx = x −μr. integer?(x −r),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"invx = μt. x · (x · t −1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"digit(x, b, i) = round","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"for b > 0, where bi = exp(i · ln b). When b > 1 and i are integers, digit(x, b, i) is the digit in bi’s","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"clkt = digit(t, 2, −1),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"zigzagt = 0 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"as in Figure 3. We have f kv = g(v, k −1/2) for k ∈N \\ {0}.","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":35832,"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}}