A First Order Logic for Logic Programming

Pure Lisp style recursive function programs are represented in a new way by sentences and schemata of first order logic. This permits easy and natural proofs of extensional properties of such programs by methods that generalize structural induction. It also systematizes known methods such as recursion induction, subgoal induction, inductive assertions by interpreting them as first order axiom schemata. We discuss the metatheorems justifying the representation and techniques for proving facts about specific programs. We also give a simpler version of the Godel-Kleene way of representing computable functions by first order sentences.

On arithmetical first-order theories allowing encoding and decoding of lists

The problem of reasoning about recursive programs is considered. Utilizing a simple analogy between iterative and recursive programs viewed as unfinite unions of finite terms, we carry out an investigation analogous to that carried out recently for iterative programs. The main results are the arithmetical completeness of axiom systems for (1) context-free dynamic logic and (2) its extension for dealing with infinite computations. Having the power of expression of these logics in mind, these results can be seen to supply (as corollaries) complete proof methods for the various kinds of correctness of recursive programs.

Method of applying Lamb waves in ultrasonic testing

In his paper [12] Ross extends the well-founded semantics for normal logic programs [16] to disjunctive logic programs. His definition is top-down and it is closer to a procedural semantics than to the elegant fixpoint definition of the well-founded semantics for normal logic programs. In the present paper, we propose a declarative, bottom-up fixpoint definition of the well-founded semantics for disjunctive logic programs. Our construction of the greatest unfounded set of extended literals is similar with the construction of the greatest unfounded set for normal programs. As a consequence, the connection between the well-founded semantics for normal programs and the well-founded semantics for disjunctive programs is made clearer.KeywordsLogic ProgramDeductive DatabaseNormal ProgramDisjunctive ProgramExtended AtomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A meta-program, regardless of the nature of the programming language, is a program whose data denotes another (object) program. The importance of meta-programming can be gauged from its large number of applications. These include compilers, interpreters, program analysers, and program transformers. Furthermore, if the object program is a logic or functional program formalizing some knowledge, then the meta-program may be regarded as a meta-reasoner for reasoning about this knowledge. In this chapter, the meta-program is assumed to be a logic program. The object program does not have to be a logic program although much of the work in this chapter assumes this. We have identified three major topics for consideration. These are the theoretical foundations of meta-programming, the suitability of the alternative meta-programming techniques for different applications, and methods for improving the efficiency of meta-programs. As with logic programs generally, meta-programs have declarative and procedural semantics. The theoretical study of meta-programming shows that both aspects of the semantics depend crucially on the manner in which object programs are represented as data in a meta-program. The second theme of the paper is the problem of designing and choosing appropriate ways of specifying important meta-programming problems, including dynamic metaprogramming and problems involving self-application. The third theme concerns efficient implementation of meta-programs. Meta-programming systems require representations with facilities that minimize the overhead of interpreting the object program. In addition, efficiency can be gained by transforming the meta-program, specializing it for the particular object program it is reasoning about. This chapter, which concentrates on these aspects of meta-programming, is not intended to be a survey of the field. A more complete survey of meta-programming for logic programming can be found in [Barklund, 1995]. Many issues in meta-programming have their roots in problems in logic which have been studied for several decades. This chapter emphasizes meta-programming solutions. It is not intended to give a full treatment of the underlying logical problems, though we try to indicate some connections to wider topics in meta-logic. The meta-programs in this chapter are logic programs based on first order logic.

Languages that integrate functional and logic programming with a complete operational semantics are based on narrowing, a unification-based goal-solving mechanism which subsumes the reduction principle of functional languages and the resolution principle of logic languages. In this article, we present a partial evaluation scheme for functional logic languages based on an automatic unfolding algorithm which builds narrowing trees. The method is formalized within the theoretical framework established by Lloyd and Shepherdson for the partial deduction of logic programs, which we have generalized for dealing with functional computations. A generic specialization algorithm is proposed which does not depend on the eager or lazy nature of the narrower being used. To the best of our knowledge, this is the first generic algorithm for the specialization of functional logic programs. We also discuss the relation to work on partial evaluation in functional programming, term-rewriting systems, and logic programming. Finally, we present some experimental results with an implementation of the algorithm which show in practice that the narrowing-driven partial evaluator effectively combines the propagation of partial data structures (by means of logical variables and unification) with better opportunities for optimization (thanks to the functional dimension).

A game semantics for disjunctive logic programming

We introduce a new semantics for logic programming languages. It generalises the traditional Herbrand universe semantics, and specialises the semantics of logical relations, as used in analysing parametricity in functional and imperative programming languages. We outline a typed logic programming language, give it this semantics, and show how it supports structured development of logic programs as advocated by Sterling et al. In particular, it gives semantics for some dynamic aspects of logic programs.

A new declarative programming paradigm, constraint functional logic programming (CFLP) is presented, which unifies systematically all features of constraint, functional and logic programming. CFLP extends constraint functional programming (CFP), which supports equation solving over a functional program, to general constraint programming permitting the use of constraints for programming as well as for posing queries. It also generalises constraint logic programming (CLP), by admitting user-defined functions as a purely functional subsystem of a CLP language. This paper presents the motivation for designing CFLP and its semantics. Programming examples are given in the concrete CFLP language Falcon.KeywordsLogic ProgramLogic ProgrammingConstraint ProgrammingConstraint SystemFunctional ProgrammingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper typed and untyped functional programs are considered. Typed functional programs use variables of any order and constants of order $\le$1, where constants of order 1 are strong computable, $\lambda$-definable functions with indeterminate values of arguments. The basic semantics of a typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. The basic semantics of an untyped functional program is an untyped $\lambda$-term, which is defined by means of a fixed point combinator. An algorithm that translates typed functional program $P$ into untyped functional program $P'$ is suggested. It is proved that the basic semantics of the program $P'$ $\lambda$-defines the basic semantics of the program $P$.

We study the problem of automated hypersafety verification of infinite-state recursive programs . We propose an infinite class of product programs , specifically designed with recursion in mind, that reduce the hypersafety verification of a recursive program to standard safety verification. For this, we combine insights from language theory and concurrency theory to propose an algorithmic solution for constructing an infinite class of recursive product programs. One key insight is that, using the simple theory of visibly pushdown languages , one can maintain the recursive structure of syntactic program alignments which is vital to constructing a new product program that can be viewed as a classic recursive program — that is, one that can be executed on a single stack. Another key insight is that techniques from concurrency theory can be generalized to help define product programs based on the view that the parallel composition of individual recursive programs includes all possible alignments from which a sound set of alignments that faithfully preserve the satisfaction of the hypersafety property can be selected. On the practical side, we formulate a family of parametric canonical product constructions that are intuitive to programmers and can be used as building blocks to specify recursive product programs for the purpose of relational and hypersafety verification, with the idea that the right product program can be verified automatically using existing techniques. We demonstrate the effectiveness of these techniques through an implementation and highly promising experimental results.

We consider prescriptive type systems for logic programs (as in Godel or Mercury). In such systems, the typing is static, but it guarantees an operational property: if a program is well-typed, then all derivations starting in a well-typed query are again well-typed. This property has been called subject reduction. We show that this property can also be phrased as a property of the proof-theoretic semantics of logic programs, thus abstracting from the usual operational (top-down) semantics. This proof-theoretic view leads us to questioning a condition which is usually considered necessary for subject reduction, namely the head condition. It states that the head of each clause must have a type which is a variant (and not a proper instance) of the declared type. We provide a more general condition, thus reestablishing a certain symmetry between heads and body atoms. The condition ensures that in a derivation, the types of two unified terms are themselves unifiable. We discuss possible implications of this result. We also discuss the relationship between the head condition and polymorphic recursion, a concept known in functional programming.

In the article, we consider verification of programs with mutual recursion in the data driven functional parallel language Pifagor. In this language the program could be represented as a data flow graph, that has no control connections, and only has data relations. Under these conditions it is possible to simplify the process of formal verification, since there is no need to analyse resource conflicts, which are present in the systems with ordinary architectures. The proof of programs correctness is based on the elimination of mutual recursions by program transformation. The universal method of mutual recursion of an arbitrary number of functions elimination consists in constructing the universal recursive function that simulates all the functions in mutual recursion. A natural number is assigned to each function in mutual recursion. The universal recursive function takes as its argument the number of a function to be simulated and the arguments of this function. In some cases of the indirect recursion it is possible to use a simpler method of program transformation, namely, the merging of the functions code into a single function. To remove mutual recursion of an arbitrary number of functions, it is suggested to construct a graph of all connected functions and transform this graph by removing functions that are not connected with the target function, then by merging functions with indirect recursion and finally by constructing the universal recursive function. It is proved that in the Pifagor language such transformations of functions as code merging and universal recursive function construction do not change the correctness of the initial program. The example of partial correctness proof is given for the program that parses a simple arithmetic expression. We construct the graph of all connected functions and demonstrate two methods of proofs: by means of code merging and by means of the universal recursive function.

M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.