Postponement of fi K -contractions and the conservation theorem do not hold for ordinary fi but have been established by de Groote for a mixture of fi with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation fi e which generalises fi. This then is used to solve an open problem of fi e : the Preservation of Strong Normalisation 1 . Keywords: Generalised fi-reduction, Postponement of K-contractions, Generalised Conservation, Preservation of Strong Normalisation. 1 Introduction 1.1 Background and Motivation In the term (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P . It is also the case that the function starting with y and the argument Q will result in another redex when the first redex is contracted. This idea has been exploited by many researchers and reduction has been extended so that the generalised redex based on the matching y and Q is given the same priority a...

We propose a new framework for representing logics, called LF + and based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions which capture how well a logic has been represented. These definitions are possible since we are able to distinguish in a generic way that part of the LF + entailment which corresponds to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction first-order logic can be well-represented in LF + , whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between type-theoretic and categorical approaches to frameworks. 1 Introduction Much effort has been devoted to building systems for supporting the construction of f...

In this article, we extend the Barendregt Cube with \Pi-conversion (which is the analogue of beta-conversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term.

. We present a technique to study relations between weak and strong fi-normalisations in various typed -calculi. We first introduce a translation which translates a -term into a I-term, and show that a -term is strongly fi-normalisable if and only if its translation is weakly fi-normalisable. We then prove that the translation preserves typability of -terms in various typed -calculi. This enables us to establish the equivalence between weak and strong fi-normalisations in these typed -calculi. This translation can deal with Curry typing as well as Church typing, strengthening some recent closely related results. This may bring some insights into answering whether weak and strong fi-normalisations in all pure type systems are equivalent. 1 Introduction In various typed -calculi, one of the most interesting and important properties on -terms is how they can be fi-reduced to fi-normal forms. A -term M is said to be weakly fi-normalisable (WN fi (M )) if it can be fi-reduced to a fi-n...

Abstract. Recursive path ordering (RPO) is a well-known reduction ordering introduced by Dershowitz [6], that is useful for proving termination of term rewriting systems (TRSs). Jouannaud and Rubio generalized this ordering to the higher-order case thus creating the higher-order recursive path ordering (HORPO) [8]. They proved that this ordering can be used for proving termination of higher-order TRSs which essentially comes down to proving well-foundedness of the union of HORPO and βreduction relation of simply typed lambda calculus (λ →), [1]. This result entails well-foundedness of RPO and termination of λ →. This paper describes author’s undertaking of providing a complete, axiomfree, fully constructive formalization of those results in the theorem prover Coq. Formalization is complete and hence it contains all the dependant results for λ → , multisets and multiset extension of the relation. Also decidability of HORPO has been proven and due to constructive nature of this proof a certified algorithm to verify whether two terms can be oriented with HORPO can be extracted from this proof. 1

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Self-representation – the ability to represent programs in their own language – has important applications in reflective languages and many other domains of programming language design. Although approaches to designing typed program representations for sublanguages of some base language have become quite popular recently, the question whether a fully metacircular typed selfrepresentation is possible is still open. This paper makes a big step towards this aim by defining the F ∗ ω calculus, an extension of the higher-order polymorphic lambda calculus Fω that allows typed self-representations. While the usability of these representations for metaprogramming is still limited, we believe that our approach makes a significant step towards a new generation of reflective languages that are both safe and efficient.

We will analyze some extensions of Martin-Löf’s constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency. 1

In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of fi-conversion. A similar observation holds for the generalized Cartesian product types, \Pi x:oe : . In fact, many versions of type theory assume that fi holds of both types and terms, yet only a few attempt to study the theory where terms and types are really treated equally and where fi-conversion is used for both. A unified treatment however, of types and terms is becoming indispensible especially in the approaches which try to generalise many systems under a unique one. For example, [Barendregt 91] provides the Barendregt cube and the Pure Type Systems (PTSs) which are a generalisation of many type theories. Yet even such a generalisation does not use fi-conversion for both types and terms. This is unattractive, in a calculus where types have the same syntax as terms (such as the calculi of the cube or the PTSs). For example, in those systems, even though compatibility holds for the typing of abstraction, it does not hold for the typing of application. That is, even though M : N ) y:P :M : \Pi y:P :N holds, the following does not hold: Based on this observation, we present a -calculus in which the conversion rules apply to types as well as terms. Abstraction and application, moreover, range over both types and terms. We extend the calculus with a canonical type operator in order to associate types to terms. The type of fa will then be Fa, where F is the type of f and the statement \Gamma ` t : oe from usual type theory is split in two statements in our system: \Gamma ` t and (\Gamma; t) = oe. Such a splitting enables us to discuss the two questio...

We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching. We obtain this way already known NP-hardness bounds for the order two and three and a new result that the fourth order matching is NEXPTIME-hard. 1 Introduction Consider two normalized simply typed lambda terms M and N, where N is closed (does not contain free variables). The higher order matching problem M ? = N (also known as pattern matching, 1 range problem or instantiation problem) is to decide whether there exists a substitution # for free variables in M, such that M# is ##-reducible to N. Matching is a special case of unification, where the restriction that N is closed is removed (and a solution of M ? = N is a substitution # such that M# and N# are equal modulo ##-conversion). The order of a problem M ? = N is the highest functionality order of free variables occurring in M. At the time of writin...