Higher-order constructs provide the necessary level of abstraction for concise and natural formulations in many areas of computer science. We present constructive methods for higher-order equational reasoning with applications ranging from theorem proving to novel programming concepts. A major problem of higher-order programming is the undecidability of higher-order unification. In the first part, we develop several classes with decidable second-order unification. As the main result, we show that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitely solvable. This is the unification needed for second-order functional-logic programming. The second main contribution is a framework for solving higher-order equational problems by narrowing. In the first-order case, narrowing is the underlying computation rule for the integration of logic programming and functional programming. We argue that there are...

We introduce several approaches for solving higher-order equational problems by higher-order narrowing and give first completeness results. The results apply to higher-order functional-logic programming languages and to higher-order unification modulo a higher-order equational theory. We lift the general notion of first-order narrowing to so-called higher-order patterns and argue that the full higher-order case is problematic. Integrating narrowing into unification, called lazy narrowing, can avoid these problems and can be adapted to the full higher-order case. For the second-order case, we develop a version where the needed second-order unification remains decidable. Finally we discuss a method that combines both approaches by using narrowing on higher-order patterns with full higher-order constraints.

Second-order unification is undecidable in general. Miller showed that unification of so-called higher-order patterns is decidable and unitary. Weshow that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitary. A few extensions of this unification problem are still decidable: unifying two second-order terms, where one term is linear, is undecidable if the terms contain bound variables but decidable if they don't.

We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n ! 1.

We introduce Context-sensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed -calculi possibly enriched with pattern-matching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbert-style proof systems, Gentzen-style sequent-calculi, rewrite systems with rule priorities, and the ß-calculus into CERSs. This last encoding is an (important) example of real context-sensitive rewriting. 1 Introduction A term rewriting system is a pair consisting of an alphabet and a set of rewrite rules. The alphabet is used freely to gene...

Abstract not found

. We study reductions in orthogonal (left-linear and non-ambiguous) Expression Reduction Systems, a formalism for Term Rewriting Systems with bound variables and substitutions. To generalise the normalization theory of Huet and L'evy, we introduce the notion of neededness with respect to a set of reductions \Pi or a set of terms S so that each existing notion of neededness can be given by specifying \Pi or S. We imposed natural conditions on S, called stability, that are sufficient and necessary for each term not in S-normal form (i.e., not in S) to have at least one S-needed redex, and repeated contraction of S-needed redexes in a term t to lead to an S-normal form of t whenever there is one. Our relative neededness notion is based on tracing (open) components, which are occurrences of contexts not containing any bound variable, rather than tracing redexes or subterms. 1 Introduction Since a normalizable term, in a rewriting system, may have an infinite reduction, it is important to...

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...

We study normalization relative to desirable sets S of `normal forms' by generalizing Huet&L'evy theory of `normalization by neededness'. We impose natural conditions on S, called stability, that are sufficient and necessary for each term not in S-normal form (i.e., not in S) to have at least one S-needed redex, and repeated contraction of S-needed redexes in a term t to lead to an S-normal form of t whenever there is one. Further, we prove existence of minimal normalizing reductions for regular stable sets of normal forms. For example, the sets of normal forms, head-normal-forms, and weak head-normal-forms, in the -calculus, are all stable and regular. Finally, we generalize L'evy's Optimality theorem to the case of all stable sets of normal forms, and establish a relationship between relative minimal and optimal reductions, revealing a conflict between minimality and optimality of a reduction -- for regular stable sets of normal forms, a term need not posses a reduction that is minim...

. 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...