This paper discusses the technique of structural induction for proving theorems about programs. This technique is closely related to recursion induction but makes use of the inductive definition of the data structures handled by the programs. It treats programs with recursion but without assignments or jumps. Some syntactic extensions to Landin's functional programming language ISWIM are suggested which make it easier to program the manipulation of data structures and to develop proofs about such programs. Two sample proofs are given to demonstrate the technique, one for a tree sorting algorithm and one for a simple compiler for expressions. (First received April 1968 and in revised form August 1968) Since the problem of proving that computer programs really do what their inventors allege them to do was discussed by McCarthy (1963), there has been considerable progress and proofs have been produced for non-trivial programs such as a simple compiler (Painter, 1967;

We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications.

The Maude LTL model checker supports on-the-y explicit-state model checking of concurrent systems expressed as rewrite theories with performance comparable to that of current tools of that kind, such as SPIN. This greatly expands the range of applications amenable to model checking analysis. Besides traditional areas well supported by current tools, such as hardware and communication protocols, many new applications in areas such as rewriting logic models of cell biology, or nextgeneration reective distributed systems can be easily speci ed and model checked with our tool.

this paper we present algorithm Hyper which can be used to find all hypergeometric solutions of (1.3). To give some motivation, we describe first an application of algorithm Hyper to definite hypergeometric summation. The problem of indefinite hypergeometric summation was solved by Gosper (1978) who discovered an algorithm for finding hypergeometric solutions of the non-homogeneous first-order recurrence

. The purpose of this paper is to make some practically relevant results in automated theorem proving available to many-valued logics with suitable modifications. We are working with a notion of many-valued first-order clauses which any finitely-valued logic formula can be translated into and that has been used several times in the literature, but in an ad hoc way. We give a many-valued version of polarity which in turn leads to natural many-valued counterparts of Horn formulas, hyperresolution, and a Davis-Putnam procedure. We show that the many-valued generalizations share many of the desirable properties of the classical versions. Our results justify and generalize several earlier results on theorem proving in many-valued logics. KEYWORDS: many-valued logic, polarity, Horn formula, direct products of structures, resolution, Davis-Putnam procedure Introduction The purpose of this paper is to make some practically relevant results in automated theorem proving available to many-value...

We introduce and investigate new methods to define parallel composition of words and languages. The operation of parallel composition leads to new shuffle-like operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of non-context-free languages in terms of context-free and even regular languages. This paper concetrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution. TUCS Research Group

We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a two-level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our two-level approach is illustrated by some examples developed in Lego.

Current methodology for compiler construction evolved from the need to release programmers form the burden of writing machine-language programs. This methodology does not assume a formal concept of a programming language and is not based on mathematical algorithms that model the behavior of a compiler. The side effect is that compiler implementation is a difficult task and the correctness of a compiler usually is not proven mathematically. Moreover, a compiler may be based on assumptions about its source and target languages that are not necessarily acceptable for another compiler that has the same source and target languages. The consequence is that programs are not portable between platforms of machines and between generations of languages. In addition, while a conventional compiler freezes the notation that programmers can use to develop their programs the problem domain evolves and requires extensions that are not supported by the compiler. These problems are addressed by two direc...

Introduction and notation Let F be a field of characteristic zero and ht n i 1 n=0 a sequence of elements from F which is eventually non-zero. Call t n : 1. hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (n)t n+1 = p 2 (n)t n for all n; 2. q-hypergeometric or basic hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (q n )t n+1 = p 2 (q n<

. We review Quillen's concept of a model category as the proper setting for defining derived functors in non-abelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas -- most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in non-abelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such non-abelian derived functors is the E 2 -term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 -term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...