We provide an explicit characterization of the properties of primitive
recursive functions that are decidable or semi-decidable, given a primitive
recursive index for the function. The result is much more general as it applies
to any c.e. class of total computable functions. This is an analog of Rice and
Rice-Shapiro theorem, for restricted classes of total computable functions

Hoyrup, Mathieu

English

arXiv

We provide an explicit characterization of the properties of primitive recursive functions that are decidable or semi-decidable, given a primitive recursive index for the function. The result is much more general as it applies to any c.e. class of total computable functions. This is an analog of Rice and Rice-Shapiro theorem, for restricted classes of total computable functions

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A Rice-like theorem for primitive recursive functions

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HAL Id: hal-01130868https://hal.inria.fr/hal-01130868Submitted on 12 Mar 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.A Rice-like theorem for primitive recursive functionsMathieu HoyrupTo cite this version:Mathieu Hoyrup. A Rice-like theorem for primitive recursive functions. [Research Report] Inria Nancy- Grand Est (Villers-lès-Nancy, France); Loria. 2015. ￿hal-01130868￿A Rice-like theorem for primitive recursive functionsMathieu HoyrupMarch 12, 2015AbstractWe provide an explicit characterization of the properties of primitive recursive functionsthat are decidable or semi-decidable, given a primitive recursive index for the function.The result is much more general as it applies to any c.e. class of total computable functions.This is an analog of Rice and Rice-Shapiro theorem, for restricted classes of total computablefunctions.1 IntroductionRice theorem [Ric53] states that no non-trivial property of partial computable functions canbe decided when the function is presented by one of its indices, or equivalently by a programcomputing it. Rice-Shapiro theorem [Sha56] is a refinement that identifies the properties thatare semi-decidable when the input function is given by one of its indices, and states that theyare exactly the properties that are semi-decidable if the input function is presented by an oracle.When restricting to the class of total computable functions, Kreisel-Lacombe-Schœnfield[KLS57] and Ceitin [Cei62] theorem implies that the decidable properties are the same whenpresenting the input function via an index or an oracle. However Friedberg [Fri58] constructeda set that is semi-decidable when the input function is given by an index, but not when it isgiven by an oracle.In this note we restrict our attention to subclasses of total computable functions, such as theclass of primitive recursive functions, the class FP of polynomial-time computable functions,the class of provably total functions in some fixed recursive system (Peano arithmetic, say), etc.The general problem is to understand what can be decided/semi-decided about a function fin such a class if f is given by one of its indices: here an index is essentially a program in arestricted language that only allows to compute functions in the class. The function that decidesor semi-decides the property is a usual computable function and is not assumed to belong tothe class.We give a characterization of the decidable and semi-decidable properties, i.e. an analogof Rice, Rice-Shapiro and Kreisel-Lacombe-Schœnfield/Ceitin theorems. This characterizationuses a version of Kolmogorov complexity restricted to the class of functions under consideration.Notations. Let N∗ be the set of finite sequences of natural numbers and NN the Baire space offunctions from N to N. Given f ∈ NN and n ∈ N, fn denotes the finite sequence (f(0), . . . , f(n−1)) ∈ N∗. We say that f ∈ NN extends v = (v0, . . . , vn−1) ∈ N∗ if f(0) = v0, . . . , f(n−1) = vn−1,i.e. if fn = v. We denote by [v] ⊆ NN the set of all extensions of v and call it a cylinder. Anopen set of the Baire space is a union of cylinders. An effective open set is the union of acomputable sequence of cylinders.12 The resultLet C be a class of total computable functions that is computably enumerable: there is anumbering C = {fi : i ∈ N} such that fi is computable uniformly in i, i.e. the mapping(i, n) 7→ fi(n) is computable. If f ∈ C then a C -index of f is any i such that f = fi (a functionmay have several indices).Examples of such classes are: the primitive recursive functions, the class FP of polynomial-time computable functions, the class of provably total computable functions in Peano arithmetic.First observe that if f ∈ C is given by an oracle then a set A ⊆ C is semi-decidable exactlywhen A is the intersection of an effective open subset of the Baire space with C ; hence A isdecidable exactly when both A and C \ A are the intersections of effective open sets with C .Indeed, the machine semi-deciding f ∈ A accepts f in finite time so it has only read a finitesegment of f hence it will accept all functions in some cylinder [fn].In order to identify the properties that are decidable or semi-decidable, given a C -index ofthe input function, we introduce a notion of Kolmogorov complexity adapted to the class C .Definition 2.1. The C -complexity of f : N→ N isKC (f) ={min{i : fi = f} if f ∈ C ,+∞ otherwise.If v = (v0, . . . , vn) is a finite sequence of natural numbers then its C -complexity isKC (v) = min{i : fi extends v} = min{KC (f) : f ∈ [v]}.Observe that for f : N → N, KC (fn) is nondecreasing and converges to KC (f). By theassumptions on C , the quantity KC (v) is computable from v, contrary to usual Kolmogorovcomplexity which is upper semi-computable only. However KC usually does not belong to theclass C (modulo encoding of N∗ in N).It could seem more consistent with usual notions of Kolmogorov complexity (see e.g. [LV93])to take for instance log(i) instead of i in the definition, or to use a machine that is universalfor the class C and define KC in terms of the size of its inputs. All these choices are equallyacceptable and lead exactly to the same result. The important point is that for each such notionof complexity K ′, an upper bound on KC (f) can be uniformly computed from any upper boundon K ′(f) and vice-versa. Here we take the simplest definition of complexity following directlyfrom the enumeration of C , to avoid technicality.First we give a class of decidable properties. Let h : N → N be a computable order, i.e. acomputable non-decreasing unbounded function.Definition 2.2. Let C be an effective class of functions and h a computable order. We definethe set AC ,h of (C , h)-anticomplex functions asAC ,h = {f : N→ N : ∀n,KC (fn) ≤ h(n)}.Proposition 2.1. For f ∈ C , the property f ∈ AC ,h is decidable given any C -index of f .Proof. Given an index i for f , one has KC (fn) ≤ KC (f) ≤ i for all n, so f belongs to AC ,hif and only if KC (fn) ≤ h(n) for all n such that h(n) < i, which is decidable as KC (fn) iscomputable from i and n and only a finite number of values of n has to be checked.In general AC ,h is no more decidable if instead of giving an index of f one is only given fas oracle. It contrasts with what happens on the class of partial or total computable functions.2Proposition 2.2. If C is dense in NN then AC ,h has empty interior in C (i.e. does not containany intersection of a cylinder with C ), therefore AC ,h is not semi-decidable when the inputfunction is given as oracle.Proof. For each u = (u0, . . . , un−1) ∈ N∗, there exist only finitely many i ∈ N such thatK(u0, . . . , un−1, i) ≤ h(n+ 1). Take any i outside this finite set: the cylinder [u0, . . . , un−1, i] isdisjoint from AC ,h but intersects C , so [u] ∩ C is not contained in AC ,h.All the usual classes of total computable functions are dense in NN. More generally,Proposition 2.3. If C contains a non-isolated function f then there is a computable order hsuch that f belongs to AC ,h but not to its interior, therefore AC ,h is not semi-decidable whenthe input function is given as oracle.Proof. We build h ≥ KC (f) so that f ∈ AC ,h. As f is not isolated in C , for each n there existinfinitely many functions in [fn]∩C . In particular there exist n+21 pairwise distinct functionsin that set and a p such that all these functions differ on their prefixes of length p. As C is ac.e. class one can compute such a p. In this way we can construct an increasing computablesequence pn. For each n there exists u ∈ Npn such that [u] intersects C and is contained in[fn] and KC (u) > n, as there are at most n + 1 sequences of a given length (here pn) whosecomplexity is bounded by n.Let h(p) = min{n ≥ KC (f) : p ≤ pn}. For each n ≥ KC (f), [fn] ∩ C is not contained inAC ,h as there exists u of length pn such that [u] intersects C but not AC ,h: indeed, KC (u) >n = h(|u|).Now observe that the proof of Proposition 2.1 actually shows that AC ,h remains decidableif one is given f via an oracle together with an upper bound on KC (f). This is actually the casefor every decidable, and even semi-decidable property.Proposition 2.4. Every semi-decidable set A ⊆ C is semi-decidable given an access to f asoracle and any upper bound on KC (f).Proof. Given f and k ≥ KC (f), one can progressively reject all numbers i ≤ k such that fi 6= f .In parallel one can progressively accept all numbers i ≤ k that are accepted by the semi-decisionprocedure. Wait for a stage when every number i ≤ k is accepted or rejected. If this happensthen accept f .It was proved in [HR15] that such a result also holds for the class of total computablefunctions and much more general classes of computable objects. The proof given here in thecase of an effective class C is much easier because we only deal with total programs.We can now state our main result: the cylinders and the sets of anticomplex functions arethe basic decidable properties, from which all decidable and semi-decidable properties can beobtained.Theorem 2.1. Let A ⊆ C . The following conditions are equivalent:• A is semi-decidable,• A is an effective union of sets of the form [v] ∩AC ,h, i.e.A = C ∩⋃n([vn] ∩AC ,hn)for some computable sequences of finite words vn ∈ N∗ and orders hn : N→ N.1Here we use the fact that there are at most n+ 1 functions of C -complexity at most n. For other notions ofC -complexity, this should be adapted, replacing n+ 2 with 2n+1 for instance.3Proof. For k, p ∈ N letCk = {f : KC (f) ≤ k},C pk = {u ∈ Np : KC (u) ≤ k} =⋃f∈Ck[fp].By Proposition 2.4 there exist uniformly effective open sets Uk ⊆ NN such that A ∩ Ck =Uk ∩ Ck. One can take Uk+1 ⊆ Uk, replacing Uk with Uk ∪ Uk+1 ∪ . . . if necessary. It followsthat A = C ∩⋂k Uk.Given k ∈ N and v ∈ N∗ such that [v] ⊆ Uk+1, we now build a computable order h suchthat [v] ∩ Ck+1 ⊆ [v] ∩ AC ,h ∩ C ⊆ A. In order to obtain the announced families vn and hn tocover the whole set A, we will simply start from all possible k ∈ N and all v in the effectiveenumeration of Uk+1.We now define h, by first constructing a kind of inverse of h. More precisely we define acomputable increasing sequence pi such that for all i,[v] ∩ C p1k+1 ∩ . . . ∩ Cpik+i ⊆ Uk+i+1.The base case i = 0 is satisfied as [v] ⊆ Uk+1. Once p1, . . . , pi have been defined,[v] ∩ C p1k+1 ∩ . . . ∩ Cpik+i ∩ Ck+i+1 ⊆ Uk+i+1 ∩ Ck+i+1⊆ A⊆ Uk+i+2.The left-hand side is a finite set. For each f in that set, there is p ∈ N such that [fp] ⊆Uk+i+2. As the set is finite there is a single p that works for each f in the finite set. As thisfinite set is computable, such a p can be computed. We then define pi+1 > pi such that[v] ∩ C p1k+1 ∩ . . . ∩ Cpik+i ∩ Cpi+1k+i+1 ⊆ Uk+i+2.We then have[v] ∩ Ck+1 ⊆ [v] ∩⋂i≥1C pik+i ∩ C ⊆⋂nUn ∩ C = A.Now, g ∈⋂i≥1 Cpik+i if and only if for all i, KC (gpi) ≤ k+ i. Let h be the computable orderdefined by h(p) = k + min{i ≥ 1 : p ≤ pi}. If g ∈ AC ,h then KC (gpi) ≤ h(pi) = k + i for all i.As h(p) ≥ k + 1 for all p, Ck+1 ⊆ AC ,h.A set A ⊆ C is then decidable from C -indices if and only if both A and C \ A can beexpressed as effective unions of sets [v] ∩AC ,h.References[Cei62] G. S. Ceitin. Algorithmic operators in constructive metric spaces. Trudy Matem-atiki Instituta Steklov, 67:295–361, 1962. English translation: American MathematicalSociety Translations, series 2, 64:1-80, 1967. 1[Fri58] Richard M. Friedberg. Un contre-exemple relatif aux fonctionnelles récursives.Comptes Rendus de l’Académie des Sciences, 247:852–854, 1958. 14[HR15] Mathieu Hoyrup and Cristóbal Rojas. On the information carried by programs aboutthe objects they compute. In Ernst W. Mayr and Nicolas Ollinger, editors, 32ndInternational Symposium on Theoretical Aspects of Computer Science, STACS 2015,March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 447–459. SchlossDagstuhl - Leibniz-Zentrum fuer Informatik, 2015. 3[KLS57] G. Kreisel, D. Lacombe, and J.R. Schœnfield. Fonctionnelles récursivementdéfinissables et fonctionnelles récursives. Comptes Rendus de l’Académie des Sciences,245:399–402, 1957. 1[LV93] Ming Li and Paul M. B. Vitanyi. An Introduction to Kolmogorov Complexity and ItsApplications. Springer-Verlag, Berlin, 1993. 2[Ric53] H. G. Rice. Classes of recursively enumerable sets and their decision problems. Trans-actions of the American Mathematical Society, 74(2):pp. 358–366, 1953. 1[Sha56] N. Shapiro. Degrees of computability. Transactions of the American MathematicalSociety, 82:281–299, 1956. 15

A Rice-like theorem for primitive recursive functions

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