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Primitive Recursive Functions: Decidability Problems

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Primitive Recursive Functions: Decidability Problems Primitive recursive functions: decidability problems Armando B. Matos October 24, 2014 Abstract Although every primitive recursive (PR) function is total, many prob- lems related to PR functions are undecidable. In this work we study several questions associated with PR functions and characterize their de- gree of undecidability. For instance, we show that the injectivity problem 0 0 belongs to the class Π1 n ∆0 of the arithmetic hierarchy, while the surjec- 0 0 0 tivity and the bijectivity problems belong to Π2 n (Σ1 [ Π1). It follows that these subclasses of PR functions can not be enumerated { and thus cannot be characterized by an effective property. Several other PR deci- sion problems are studied such as, problems related to the existence and number of zeros, and problems related to the size of the codomain. We consider a specific but important decision problem, \does a given PR function has at least one zero?", and establish the necessary and suf- ficient input and output restrictions for decidability { the \frontiers of decidability" of this problem. We also study undecidability results for a more general class of problems, namely existential first order logic with equality and composition of PR functions. A PR function as a part of an \acyclic PR function graph" is also studied. This setting may be useful to evaluate the relevance of a specific PR function as a component of a larger PR structure. 1 Contents 1 Introduction3 2 Preliminaries5 2.1 Languages, reductions and the arithmetic hierarchy........6 2.2 Decision problems..........................8 2.3 Models of computation........................9 2.4 Primitive recursive functions.................... 10 2.5 Context free grammars........................ 10 3 Decision problems associated with PR functions 12 3.1 Some undecidable problems..................... 12 3.2 Size of the codomain......................... 18 3.3 Injectivity, surjectivity, and bijectivity............... 19 3.4 Primitive recursive versus partial recursive problems....... 22 4 Frontiers of decidability: the HAS-ZEROS problem 29 4.1 The meaning of restricted HAS-ZEROS problems......... 31 4.2 The HAS-ZEROS-f·g class of problems.............. 32 4.3 The HAS-ZEROS-h·f class of problems.............. 34 4.4 The HAS-ZEROS-h·f·g class of problems............. 34 4.5 A generalization: the output of g is a tuple............ 35 4.6 Using HAS-ZEROS-h·f·g to analyze other decision problems.. 35 4.7 Classifying a problem { testing the properties of g and h ..... 35 5 Generalizations of the PR decision problems 36 5.1 Existential first order logic with PR functions........... 37 5.2 A PR function f in an acyclic PR structure: normal form.... 38 5.3 Composition of f with itself..................... 42 6 Comment: on a paper by Stefan Kahrs 45 7 Conclusions and open problems 47 2 1 Introduction The primitive recursive (PR) functions form a vast and important enumerable sub-class of the class of recursive (total) functions. Its vastness is obvious from the fact that every total recursive function whose time complexity is bounded by a primitive recursive function is itself primitive recursive, while its importance is well expressed by the Kleene normal form Theorem [23,3, 32] and by the fact that every recursively enumerable set can be enumerated by a primitive recursive function1. The meta-mathematical and algebraic properties of PR functions have been widely studied, see for instance [23, 36, 34, 41, 37]. Special subclasses (such as unary PR functions, see for instance [35, 39]) and hierarchies (see for instance [18,1, 28, 30,2, 14] and [44, Ch. VI]) of PR functions were also analyzed. Efficiency lower bounds of primitive recursive algorithms have also been studied, see [4,5,6,8, 30, 31, 12, 43, 42]. In this work we concentrate on decidability questions related to PR functions. There are several questions related to PR functions that are known to be un- decidable. For instance, the question \given a PR function f(x; y) and an integer x, is there some y such that f(x; y) = 0?" is undecidable ([3]). In this paper we show that many other questions related to PR functions are unde- cidable, and classify their degree of undecidability according to the arithmetic hierarchy. Let us mention two examples of the problems studied in this work: \does a given PR function have exactly one zero?", and \is a given PR function surjective (onto)?". The undecidability of some of these problems has consequences. For instance, when studying reversible computations, it would be useful to have a method for enumerating the bijective PR functions. As we will see, that is not possible: although the set of PR functions is enumerable, the set of bijective PR functions 0 0 0 is not: it belongs Π2 n (Σ1 [ Π1). We also study classes of PR problems in which a PR function f is a part of a larger PR system S. We obtained a simple decidability condition for the case in which S(x) = h(f(g(x))), where g and h are fixed PR functions, but not for the general case illustrated in Figure1 (page4). Organization and contents of the paper. The main concepts and results needed for the rest of the paper are presented in Section2 (page5). The rest of the paper can be divided in two parts. 1In the sense that, if L is recursively enumerable, there is a primitive recursive function f such that L [ f0g = ff(i): i 2 Ng. 3 x f y S Figure 1: A PR function f is considered to be an important part of a complex structure S if the following problem is undecidable: instance: f, question: \9x : S(x) = 0?". We will show that an arbitrary acyclic system S containing one occurrence of f can always be reduced to the normal form illustrated in Figure9 (page 43). Part I, specific problems In Section3 (page 12) several decision problems associated with PR functions are studied. These problems include the existence of zeros, the equivalence of functions, \domain problems", and problems related with the injectivity, surjectivity, and bijectivity of PR functions. For each of these problems the degree of undecidability is established in terms of their exact location in the in the arithmetic hierarchy; see Figures2 (page 23) and3 (page 23). Part II, classes of problems Suppose that the PR function f, the instance of a decision problem, is \placed" somewhere inside a larger PR system S, see Figure1. In algebraic terms this means that S is obtained by composing fixed PR functions with the instance f. A more general form of composition, based on acyclic graphs is also studied, see Figure8 (page 40). Then we ask an important question about y = S(x), namely, \9x : S(x) = 0"? This is in fact a fundamental question: { In terms of the Kleene normal form, the existence of a zero is the basic undecidable question, and it is intimately related { in fact, equivalent { to the halting problem. { Many other decision problems can be reduced to this problem. The class of problems, parametrized by S, is: Instance: A PR function f. Question: Is there some x such that S(x) = 0? In some sense we can say that f can be considered an important component of S if (and only if) this problem is undecidable. 4 Part II of this work includes { Section4 (page 29), where we study the \frontiers of decidability" of the problem \does a given PR function has at least one zero?". More precisely, we find the input and output restrictions that are necessary and sufficient for decidability; these restrictions are obtained by pre- or post-applying fixed PR functions to the PR function which is the instance of the problem. This is a particular form of the arbitrary composition described above. { A more general setting in considered in Section5 (page 36), where we study existential first order logic with equality and composition of PR functions and also \acyclic PR" structures. The problem of composing the instance PR function f with itself is also studied. { Section6 includes some comments on Kahrs enumeration method. The initial motivation for this work was to generalize Rice Theorem to PR functions. Rice Theorem is simple: for any non trivial mathematical property of a partial recursive function, it is undecidable whether the function, defined by a given description (or index), has that property. A version of Rice Theorem applicable to to PR functions is necessarily more complex because, unlike the case of partial recursive functions, only quantified questions can be undecidable. Questions like \given a description of f, is f(2) = 5?" are clearly decidable. A result of this paper that can be seen as a first step to such a \Rice-like PR Theorem" is Theorem 24 (page 34). 2 Preliminaries The notation used in this paper is straightforward. “Iff" denotes \if and only if" and \def=" denotes \is defined as". The logical negation is denoted by \:". The complement of the set A is denoted by A, while the difference between the sets A and B is denoted by A n B. A function f : A ! B is injective (or one- to-one) if x 6= y implies f(x) 6= f(y); it is surjective (or onto) if for every y 2 B there is at least one x 2 A such that f(x) = y; and it is bijective if it is both injective and surjective. By f −1(y) we denote the set fx : f(x) = yg. The letters of the alphabet are denoted by 0, 1, a, b. ; the words by x, y. The length of the word x is denoted by jxj. For n ≥ 0, the sequence of arguments \x1; x2; : : : ; xn" is denoted by x.
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