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We present a simple extension of typed -calculus where functions can be overloaded by putting different "branches of code" together. When the function is applied, the branch to execute is chosen according to a particular selection rule which depends on the type of the argument. The crucial feature of the present approach is that the branch selection depends on the "run-time type" of the argument, which may differ from its compile-time type, because of the existence of a subtyping relation among types. Hence overloading cannot be eliminated by a static analysis of code, but is an essential feature to be dealt with during computation. We obtain in this way a type-dependent calculus, which differs from the various -calculi where types do not play any role during computation. We prove Confluence and a generalized Subject-Reduction theorem for this calculus. We prove Strong Normalization for a "stratified" subcalculus. The definition of this calculus is guided by the understand...

This paper is concerned with the model for security of cryptographic protocols suggested by Dolev and Yao. The Dolev and Yao model deals with a restricted class of protocols, known as Two-Party Ping-Pong Protocols. In such a protocol, messages are exchanged in a memoryless manner. That is, the message sent by each party results from applying a predetermined operator to the message he has received. The Dolev and Yao model is presented, generalized in various directions and the affect of these generalizations is extensively studied. First, the model is trivially generalized to deal with multi-party ping-pong protocols. However, the problems which arise from this generalization are very far from being trivial. In particular, it is no longer clear how many saboteurs (adversaries) should be considered when testing the security of p-party ping-pong protocols. We demonstrate an upper bound of 3(p \Gamma 2) + 2 and a lower bound of 3(p \Gamma 2) + 1 on this number. Thus, for every fixed p, th...

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We present a correctness proof of a graph-directed variational geometric constraint solver. First, we prove that the graph reduction that establishes the sequence in which to apply the construction steps defines a terminating confluent reduction system, in the case of well-constrained graphs. For overconstrained problems there may not be a unique normal form. Underconstrained problems, on the other hand, do have a unique normal form. Second, we prove that all geometric solutions found using simple root-selection rules must place certain triples of elements in the same topological order, no matter which graph reduction sequence they are based on. Moreover, we prove that this implies that the geometric solutions derived by different reduction sequences must be congruent. Again, this result does not apply to overconstrained problems. Keywords: geometric constraint solving, computer aided design 1. Introduction Geometric constraint solving has broad applications in a wide range of subje...

We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of term rewriting systems. These criteria provide interesting possibilities to infer completeness, i.e. termination plus confluence, from restricted termination and confluence properties. Using these basic results we are also able to prove some new results about modular termination of rewriting. In particular, we show that termination is modular for some classes of innermost terminating and locally confluent term rewriting systems, namely for non-overlapping and even for overlay systems. As an easy consequence this latter result also entails a simplified proof of the fact that completeness is a decomposable property of so-called constructor systems. Furthermore we show how to obtain similar re...

A property P of term rewriting systems (TRSs, for short) is said to be persistent if for any many-sorted TRS R, R has the property P if and only if its underlying unsorted TRS (R) has the property P. This notion was introduced by H. Zantema (1994). In this paper, it is shown that confluence is persistent.

In this paper we introduce M-LISP, a simple new dialect of LISP which is designed with an eye toward reconciling LISP's metalinguistic power with the structural style of operational semantics advocated by Plotkin [Plo75]. We begin by reviewing the original denition of LISP [McC61] in an attempt to clarify the source of its metalinguistic power. We nd that it arises from a problematic clause in this denition. We then dene the abstract syntax and operational semantics of M-LISP, essentially a hybrid of M-expression LISP and Scheme. Next, we tie the operational semantics to the corresponding equational logic. As usual, provable equality in the logic implies operational equality. Having established this framework we then extend M-LISP with the metalinguistic eval and reify operators (the latter is a non-strict operator which converts its argument to its metalanguage representation.) These operators encapsulate the metalinguistic representation conversions that occur globall...

This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.

In this paper two formats of higher-order rewriting are compared: Combinatory Reduction Systems introduced by Klop [Klo80] and Higher-order Rewrite Systems defined by Nipkow [Nipa]. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into Higher-Order Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very `neat' in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as rewrite theory is concerned, Combinatory Reduction Systems and Higher-Order Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more `lazy' evaluation strategy. Moreover, due to this result...