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An informal tutorial is presented for program synthesis, with an emphasis on deductive methods. According to this approach, to construct a program meeting a given specification, we prove the existence of an object meeting the specified conditions. The proof is restricted to be sufficiently constructive, in the sense that, in establishing the existence of the desired output, the proof is forced to indicate a computational method for finding it. That method becomes the basis for a program that can be extracted from the proof. The exposition is based on the deductive-tableau system, a theorem-proving framework particularly suitable for program synthesis. The system includes a nonclausal resolution rule, facilities for reasoning about equality, and a well-founded induction rule. INTRODUCTION This is an introduction to program synthesis, the derivation of a program to meet a given specification. It focuses on the deductive approach, in which the derivation task is regarded as a problem of ...

We introduce novel, sound, complete, and locally optimal evaluation strategies for functional logic programming languages. Our strategies combine, in a non-trivial way, two landmark techniques in this area: the computation of unifiers performed by needed narrowing in inductively sequential rewrite systems and the simultaneous reduction of a necessary set of redexes performed by rewriting in weakly orthogonal, constructor-based rewrite systems. First, we define a sequential strategy similar in scope to other narrowing strategies used in modern lazy functional logic languages. Then, based on the sequential strategy, we define a parallel narrowing strategy that has several noteworthy characteristics: it is the first complete narrowing strategy which evaluates ground expressions in a fully deterministic, optimal way; it computes shortest derivations and minimal sets of solutions on inductively sequential rewrite systems; and when combined with term simplification, it subsumes and improves all r...

This paper proposes rewriting logic as an executable specification formalism for security protocols that offers some novel advantages. A messagepassing object-oriented approach seems particularly natural for communication protocols and can be naturally formalized in rewriting logic. This is illustrated by using the Needham-Schroeder Public-Key protocol as a running example. The rewriting logic-based Maude interpreter [CELM96] offers also some useful advantages. Efficient executability allows prototyping and debugging of protocol specifications. But since a concurrent system can have many different behaviors, to properly analyze the system it becomes important to explore not just the single execution provided by some default strategy, but many other executions. Maude supports user-defined execution strategies, including strategies such as breadth-first-search that can exhaustively explore all the executions of a system. This is very helpful in uncovering security flaws under unforeseen ...

In this paper we analyze completeness results for basic narrowing. We show that basic narrowing is not complete with respect to normalizable solutions for equational theories defined by confluent term rewriting systems, contrary to what has been conjectured. By imposing syntactic restrictions on the rewrite rules we recover completeness. We refute a result of Holldobler which states the completeness of basic conditional narrowing for complete (i.e. confluent and terminating) conditional term rewriting systems without extra variables in the conditions of the rewrite rules. In the last part of the paper we extend the completeness result of Giovannetti and Moiso for level-confluent and terminating conditional systems with extra variables in the conditions to systems that may also have extra variables in the right-hand sides of the rules. 1985 Mathematics Subject Classification: 68Q50 1987 CR Categories: F.4.1, F.4.2 Key Words and Phrases: narrowing, basic narrowing, conditional narrowin...

Functional and logic programming are the most important declarative programming paradigms, and interest in combining them has grown over the last decade. Early research concentrated on the definition and improvement of execution principles for such integrated languages, while more recently efficient implementations of these execution principles have been developed so that these languages became relevant for practical applications. In this paper we survey the development of the operational semantics as well as

We present the problem of integrating functional languages and logic languages. We explain why the narrowing-based techniques have so far prevailed as operational mechanisms for the functional logic interpreters. We then discuss various strategies of narrowing. Finally we explain how to simulate these strategies of narrowing using the leftmost SLD-resolution rule of Prolog, and compare some experimental results with those obtained with direct narrowing implementations. 1. Introduction There has been a flurry of research on the integration of functional programming (FP) and logic programming (LP). A natural framework would be to consider the union of a set H of Horn clauses with a set E of conditional equations as a program. The declarative semantics of a program is then given by first-order logic with equality [26], that is, first-order logic extended with an equality symbol and the standard equality axioms. The operational semantics of a program is usually given by a system of infere...

Solving equations in equational Horn-clause theories is a programming paradigm that combines logic programming and functional programming in a clean manner. Languages like EQLOG, SLOG and RITE, express programs as conditional rewrite rules and goals as equations to be solved. Procedures for completion of conditional equational theories, in a manner akin to that of Knuth and Bendix for unconditional theories, also require methods for solving equations appearing in conditions. Rewrite-based logic-programming uses (conditional) narrowing to solve equational goals. Recently a different, topdown equation solving procedure was proposed for unconditional rewrite systems. In this paper, we express equational goal solving using conditional rules. Some refinements are described: the notion of operator derivability is used to prune useless paths in the search tree and our use of oriented goals eliminates some redundant paths leading to non-normalized solutions. Our goal-directed method can also be extended to handle conditional systems. I. Equational Programming Several proposed programming languages use conditional equations as a means of combining the main features of logic programming and functional programming; such languages include RITE [Dershowitz-Plaisted-85], SLOG [Fribourg-85], and EQLOG [Goguen-Meseguer-86]. In this paradigm, a program is a set of rules, that is, directed (conditional) equations, and a goal is the question whether an equation s = t has a solution in the equational theory presented by the program. Computing consists of finding values (substitutions) for the variables in s and t for which the equality holds. Efficient methods of solving equations are therefore very important. So, too, is the ability to detect that equations are unsatisfiable.

. Languages that integrate functional and logic programming styles with a complete operational semantics are based on narrowing. In order to avoid useless computations and to deal with infinite data structures, lazy narrowing strategies have been proposed in the past. This paper presents an important improvement of lazy narrowing by incorporating deterministic simplification steps into lazy narrowing derivations. These simplification steps reduce the search space so that in some cases infinite search spaces are reduced to finite ones. We show that the completeness of lazy narrowing is not destroyed by the simplification process and demonstrate the improved operational behavior by means of several examples. 1 Introduction In recent years, a lot of proposals have been made to amalgamate functional and logic programming languages [19]. Functional logic languages with a sound and complete operational semantics are based on narrowing, a combination of the reduction principle of ...