We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the -calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...

Invariance of interpretation by #-conversion is one of the minimal requirements for any standard model for the #-calculus. With the intersection type systems being a general framework for the study of semantic domains for the #-calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.

This paper surveys a part of the theory of fi-reduction in -calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from -terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in -terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in -calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in type-free -calculus [4, 6, 7, 15, 38, 44, 81]---see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...

We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the -calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Context-sensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...

This paper surveys a part of the theory of fi-reduction in λ-calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λ-terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λ-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λ-calculus and type theory.

In this work we present a categorical approach for modeling the pure (i.e., without constants) call-by-value -calculus, defined by Plotkin as a restriction of the classical one. In particular, we study the properties a category must enjoy for give rise to a model of such a language. This definition is enough general for grasping models in different settings. 1 Introduction The call-by-value -calculus is a restriction of the classical -calculus (fi-calculus, for short), based on the notion of value. A value is a term which is either a variable or an abstraction. In particular, the call-by-value -calculus is obtained from the classical one by restricting the evaluation rule (the fi-rule) to redexes whose operand is a value. This leads to a call-by-value parameter passing mechanism, which is a feature present in many real programming languages, where an evaluation is callby -value if it evaluates parameters before they have been passed. This feature, together with the lazy evaluation, wh...

. We define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted -calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higher-order) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...

The paper explores different approaches for modeling the lazy -calculus, which is a paradigmatic language for studying the operational behaviour of programming languages, like Haskell, using a call-by-name and lazy evaluation mechanism. Two models for lazy -calculus in the coherence spaces setting are built. They give a new insight in the behaviour of the language since their local structures are different from the one of all existing models in the literature. In order to compare different models, a class of models for lazy -calculus is defined, namely, the lazy regular models class. All the models adequate for the lazy -calculus studied in the literature belong to this class. Moreover, all the lazy regular model share important properties, like the approximation property, which is a key tool for studying their local structure. 1

Many predominantly lazy languages now incorporate strictness enforcing primitives, for example a strict let or sequential composition seq. Reasons for doing this include gains in time or space efficiencies, or control of parallel evaluation. This thesis studies how to prove equivalences between programs in languages with selective strictness, specifically, we use a restricted core lazy functional language with a selective strictness operator seq whose operational semantics is a variant of one considered by van Eckelen and de Mol, which itself was derived from Launchbury’s natural semantics for lazy evaluation. The main research contributions are as follows: We establish some of the first ever equivalences between programs with selective strictness. We do this by manipulating operational semantics derivations, in

This paper presents a useful induction measure on -terms. Combining leftmost reduction with the proper subterm relation, we introduce a new notion called H-relation for untyped -calculus. We then prove the equivalence between strong normalisability and H-normalisability. Exploiting the new notion, we present some simplified proofs for several fundamental theorems such as finiteness of developments, the conservation theorem for K-calculus, and the strong normalisation theorem for simply typed -calculus. Also a simplified proof of the characterisation theorem on perpetual redexes in [BK82] is included. Compared with other proofs in the literature, all presented proofs are quite concise and perspicuous. Finally, we give a brief comparison between H-relation and other methods such as perpetual strategies. 1 1. Introduction In -calculus and some other rewriting systems, an induction measure on terms usually plays a pivotal role in the proofs of various theorems related to strong normalis...