Provably Total Primitive Recursive Functions: Theories with Induction

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In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains. It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.

Minimization and NP multifunctions

In the first part of this paper we investigate the intuitionistic version $iI\!\Sigma_1$ of $I\!\Sigma_1$ (in the language of $PRA$ ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for $HA$ and establishes a number of nice properties for $iI\!\Sigma_1$ , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: $iI\!\Sigma_1$ together with induction over arbitrary prenex formulas is $\Pi_2$ -conservative over $iI\!\Pi_2$ . In the second part of the article we study the relation of $iI\!\Sigma_1$ to $iI\!\Pi_1$ (in the usual arithmetical language). The situation here is markedly different from the classical case in that $iI\!\Pi_1$ and $iI\!\Sigma_1$ are mutually incomparable, while $iI\!\Sigma_1$ is significantly stronger than $iI\!\Pi_1$ as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in $iI\!\Sigma_1$ whereas the provably recursive functions of $iI\!\Pi_1$ are all majorized by polynomials over ${\Bbb N}$ . 0 $iI\!\Pi_1$ is unusual also in that it lacks closure under Markov's Rule $\mbox{MR}_{PR}$ .

The built-in resources of L A – the first-order language of basic arithmetic which we first introduced in Section 5.2 – are minimal: there are no non-logical predicates, and just three primitive functions, successor, addition and multiplication. We have previously noted, though, a few examples of what else L A can express (‘express’ in the sense of Section 5.4). We now radically extend our list of examples by proving that L A can in fact express any primitive recursive function. Starting the proof Our target, then, is to prove Theorem 15.1 Every p.r. function can be expressed in L A . The proof strategy Suppose that the following three propositions are all true: L A can express the initial functions. If L A can express the functions g and h , then it can also express a function f defined by composition from g and h . If L A can express the functions g and h , then it can also express a function f defined by primitive recursion from g and h . Now, any p.r. function f must be specifiable by a chain of definitions by composition and/or primitive recursion, building up from initial functions. So as we follow through the full chain of definitions which specifies f, we start with initial functions which are expressible in L A , by (1). By (2) and (3), each successive definitional move takes us from expressible functions to expressible functions. So, given (1) to (3) are true, f must therefore also be expressible in L A .

4 - Recursive Enumerability

Continuous-time computation with restricted integration capabilities

Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be one-one, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursive functions, we thus define a sub-computability as an enumeration of the r.e. sets that are themselves one-one enumerable by total functions of the given collection. Notions similar to the classical computability ones are introduced and variants of the classical theorems are shown. We also introduce sub-reducibilities and study the related completeness notions. One of the striking results is the existence of natural (recursive) sets which play the role of low (non-recursive) solutions to Post's problem for these sub-reducibilities. The similarity between sub-computabilities and (complete) computability is surprising, since there are so many missing r.e. sets in sub-computabilities. They can be seen as toy models of computability.

The contribution of Mostowski to the foundations of second order arithmetic

We consider the computation of finite semigroups using unbounded fan-in circuits. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f, circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.

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Analysis of algorithms is a part of computing curriculums throughout the world in both undergraduate and graduate levels. However, students in general find this topic to be difficult subject due its theoretical and mathematical nature. This paper presents BiGO, a tool to support the students to understand and learn to analyze the time complexity of algorithms in rigorous manner. BiGO has four main features: i. it enables students to encode any given primitive recursive function with a specific scripting language in order to ii. automatically compute the upper bounds of the script and to, iii. visualize the behavior of the upper bounds in comparison to the most commonly occurring time complexity classes and, iv. view how long it takes for a computer to process the algorithm based on the different input values. This paper outlines the pedagogical goals of the tool and the system implementation architecture. Furthermore, we discuss future developmental plans for the tool and an evaluation within undergraduate and graduate level courses such as Data structures and algorithms and Design and analysis of algorithms.

The design of programs as the composition of smaller ones is a wide spread approach to programming. In functional programming, this approach raises the necessity of creating a good amount of intermediate data structures with the only aim of passing data from one function to another. Using program fusion techniques, it is possible to eliminate many of those intermediate data structures by an appropriate combination of the codes of the involved functions. In the standard case, no mention to the eliminated data structure remains in the code obtained from fusion. However, there are situations in which parts of that data structure becomes an internal value manipulated by the fused program. This happens, for example, when primitive recursive functions (so-called paramorphisms) are involved. We show, for example, that the result of fusing a primitive recursive function p with another function f may give as result a function that contains calls to f . Moreover, we show that in some cases the result of fusion may be less efficient than the original composition. We also investigate a general recursive version of paramorphism. This study is strongly motivated by the development of a fusion tool for Haskell programs called HFUSION.

Isabelle/HOL has recently acquired new versions of Definitional packages for inductive datatypes and primitive recursive functions. In contrast to its predecessors and most other implementations, Isabelle/HOL datatypes may be mutually and indirect recursive, even infinitely branching. We also support inverted datatype definitions for characterizing existing types as being inductive ones later. All our constructions are fully Definitional according to established HOL tradition. Stepping back from the logical details, we also see this work as a typical example of what could be called Formal-Logic Engineering. We observe that building realistic theorem proving environments involves further issues rather than pure logic only.

Eleven different definitions for functions to be computable by Petri nets are given and compared. For nine of these definitions it was possible to show that they are not powerful enough to describe all the primitive recursive functions.