Partial Functions in a Total Setting

Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic

Many-valued logic based interval reasoning plays an increasingly important role in fuzzy and other many-valued extensions of two-valued (crisp) logic. It is often overlooked that many-valued logic (MVL) reasoning has a richer inference rule base than the crisp (i.e. classical two-valued) logic. Indeed some rules that are not possible in the crisp logic (which is point-based) do come to existence when we accept intervals as the basic elements of the semantic space ([0, 1] or a more general lattice) into which the logic expressions are valuated [11, 8, 37].

We consider some advanced topics in logic including fuzzy logic, temporal logic, intuitionist logic, undefined values in logic, logic and AI and theorem provers. Fuzzy logic is an extension of classical logic that acts as a mathematical model for vagueness. Temporal logic is concerned with the expression of properties that have time dependencies. Brouwer and others developed intuitionist logic as the logical foundation for intuitionism, which was a controversial theory of the foundations of mathematics based on a rejection of the law of the excluded middle and an insistence on constructive existence. We discuss several approaches that have been applied to dealing with undefined values that arise with partial functions including the logic of partial functions; Dijkstra’s approach with his cand and cor operators; and Parnas’s approach which preserves a classical two-valued logic.

Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. We could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second evaluates them, and the last computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor. Every partial recursive function is associated with an inductive domain predicate; evaluation of the function requires a proof that the input values satisfy the predicate. We give a constructive justification for the new construct by interpreting it into the base type theory. This shows that the extended theory is consistent and constructive.

Minimization and NP multifunctions

4 - Recursive Enumerability

Chapter 20 - Automated Deduction for Many-Valued Logics

In what follows, by retracing function we will always mean special retracing function: a partial recursive function f whose range is contained in its domain and which is associated with a unique point b such that for each x in the domain of f, f(x) x and for some n, fn (x) = b. Suppose R were a recursively enumerable set which was the union of n immune retraceable sets with n as small as possible. Then R must be infinite by definition of immunity. Let R=S1U .. USn where each Si is retraced by partial recursive function fi to basepoint bi. We observe that if T is any infinite recursive subset of R, for each i, Tn'Si is immune retraceable. For let B= {bi, b2, * * , bn} and T* = TUB. Then T* = S* U ... US* wrhere St = T*flSi. Now St is retraceable for it is retraced by the function f* where domain ft = T n Dom f, and xEDom f =>ft (x) is the largest element y of T* such that for some m f7(x) =y. Hence St could only be immune or finite, but the minimality of n insures that it must be immune. But B is finite so our conclusion holds without adjunction of B. Next, we use the previous observation to show that the intersection of the domains of the fi is infinite. For if this were not the case we could choose a largest subcollection of the fi whose domains had infinite intersection and enumerate the intersection of their domains. Tlhis infinite recursively enumerable subset of R contains only finitely imany members of the domain of some fj not in the subcollection hence only finitely many members of the corresponding Sj. But this set contains an infinite recursive subset with finite intersection with Sj contradicting our observation. Since the intersection of the domains of thefi is infinite recursively enumerable we may choose an infinite recursive subset of this intersection which (by our observation) is the union of n immune retraceable sets; or equivalently, assume that R is recursive and each fi has R as domain. If, for some n, fn(x) =y we will write i: x->y. If this happens for no n, we write i: x-f-y. We notice that if yE Sj then {z|j:zy} is finite. For if it were infinite, since if it is disjoint from Sj by the defi-

With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying don't need to be explicitly included in Turing-complete computation models: in this paper we show that the class of partial recursive functions that are syntactically linear (that is, partial recursive functions where no argument is erased or copied) is Turing-complete

We introduce a new model of a learner learning an unknown concept from examples with a teacher's help. In such models, “outright coding” refers to a situation in which the teacher sends the learner a representation of the concept, either directly or encoded via the examples. Previous models have used adversarial learners or adversarial teachers to try to prevent outright coding. Our model is an attempt to reflect more directly some of the reasons that outright coding is not a common mode of human learning. We model the learner as a Turing machine with oracle access to another programming system, called its “function box.” The programming system in its function box is initially unknown to the learner. The target concept is a partial recursive function and the goal of the learner is to find in its function box a function that is equal to or extends the target concept. We exhibit a class of learner/teacher pairs in which the learner can learn any partial recursive function, provided that the learner's function box is “not too much slower” than the teacher's. This result is shown not to hold if the learner's function box can contain an arbitrary acceptable programming system.

Complexity classes of partial recursive functions

We provide motivation for composition-nominative approach and its main underlying principles. We give a brief introduction to the hierarchy of composition-nominative logics (CNL) and discuss possible applications of these logics in formal approaches for software development. CNL are algebra-based logics of partial predicates and functions. They are constructed in a semantic-syntactic style on the methodological basis, which is common with programming. Main attention in the paper is paid to the first-order CNL. The satisfiability problem for the first-order CNL is addressed. We show the reduction of the problem in hand to the satisfiability problem for classical first-order logic with function symbols. The reduction proposed requires extension of logic language and logic models with an infinite number of unessential variables. The method presented in the paper enables one to use existent satisfiability checking algorithms and tools for solving instances of the CNL satisfiability problem. Thus, many standard techniques developed for classical logics can be applied in CNL-based approaches for program reasoning.KeywordsComposition-nominative approachformal methodspartial predicatepartial logicfirst-order logicsatisfiabilityvalidity

Inspired by human’s remarkable capability to perform a wide variety of physical and mental tasks without any measurements and computations and dissatisfied with classical logic as a tool for modeli...

This document is intended to contribute to the area of validation and verification of communicating real time systems, with emphasis put on parametric reachability analysis of systems modeled using timed automata.Reachability analysis is a crucial aspect of validation and verification of software and hardware systems. The reachability analysis for real time systems is area that is studied by many researchers in academic and industrial communities. However, not much work has been done for systems, where time constraints are expressed using parameters. This is serious disproportion with real world, where specifications of most of the communication protocols or embedded software and hardware systems are indeed parameterized.This thesis presents a complete framework for forward and backward parametric reachability analysis. The solution presented here can be used as a base of algorithms for validation and verification of software and hardware real-time systems, modeled as timed automata with parameters. The results of the thesis can be easily applied to model checking or test generation tools and algorithms.The core idea of the thesis is a concept of Extended Difference Bound Matrix (EDBM). This is a data structure that stores relations between all system"s clocks and parameters. In contrast to Parametric DBM, that is the state-of-the-art data structure for parametric analysis, EDBM does not require storing constraints on clocks and constraints on parameters separately. This leads to significant benefits regarding memory consumption and time necessary to perform basic operations for symbolic analysis.The maturity of the solution was proven by implementation of a proof-of-concept tool and by experiments performed with modern communication protocol. The results show that even complex systems can be efficiently handled by the framework.

There are essentially two results described in this paper. First, it is shown that for any random access machine (RAM) time-constructable function $T(n) \geqq n$, there are languages L such that L can be recognized in time $O(T(n))$ by a RAM (using the unit cost measure), but L cannot be recognized by any deterministic multitape Turing machine in time $O(T(n))$. Secondly, a family of random access stored program machines (RASP’S) are considered. For these RASP’S it is shown that there is an arbitrarily complex (infinitely often) partial recursive function $f(n)$. which has only 0–1 values (whenever defined) such that $f(n)$ can be computed in time $F(n)$ by some RASP, but cannot be computed in time $(1 - \varepsilon )F(n)$, for any $\varepsilon > 0$, by any RASP in this family.