96 research outputs found
Extending the Extensional Lambda Calculus with Surjective Pairing is Conservative
We answer Klop and de Vrijer's question whether adding surjective-pairing
axioms to the extensional lambda calculus yields a conservative extension. The
answer is positive. As a byproduct we obtain a "syntactic" proof that the
extensional lambda calculus with surjective pairing is consistent.Comment: To appear in Logical Methods in Computer Scienc
Unique normal forms for lambda calculus with surjective pairing
AbstractWe consider the equational theory λπ of λ-calculus extended with constants π, π0, π1 and axioms for surjective pairing: π0(πXY) = X, π1(πXY) = Y, π(π0X)(π1X) = X. Two reduction systems yielding the equality of λπ are introduced; the first is not confluent and, for the second, confluence is an open problem. It is shown, however, that in both systems each term possessing a normal form has a unique normal form. Some additional properties and problems in the syntactical analysis of λπ and the corresponding reduction systems are discussed
On the confluence of lambda-calculus with conditional rewriting
The confluence of untyped \lambda-calculus with unconditional rewriting is
now well un- derstood. In this paper, we investigate the confluence of
\lambda-calculus with conditional rewriting and provide general results in two
directions. First, when conditional rules are algebraic. This extends results
of M\"uller and Dougherty for unconditional rewriting. Two cases are
considered, whether \beta-reduction is allowed or not in the evaluation of
conditions. Moreover, Dougherty's result is improved from the assumption of
strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We
also provide examples showing that outside these conditions, modularity of
confluence is difficult to achieve. Second, we go beyond the algebraic
framework and get new confluence results using a restricted notion of
orthogonality that takes advantage of the conditional part of rewrite rules
An interpretation of the Sigma-2 fragment of classical Analysis in System T
We show that it is possible to define a realizability interpretation for the
-fragment of classical Analysis using G\"odel's System T only. This
supplements a previous result of Schwichtenberg regarding bar recursion at
types 0 and 1 by showing how to avoid using bar recursion altogether. Our
result is proved via a conservative extension of System T with an operator for
composable continuations from the theory of programming languages due to Danvy
and Filinski. The fragment of Analysis is therefore essentially constructive,
even in presence of the full Axiom of Choice schema: Weak Church's Rule holds
of it in spite of the fact that it is strong enough to refute the formal
arithmetical version of Church's Thesis
Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form
Higher-order matching is a special case of unification of simply-typed lambda-terms: in a matching equation, one of the two sides contains no unification variables. Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem. We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is that solutions must be in long beta-eta normal form
Extensional proofs in a propositional logic modulo isomorphisms
System I is a proof language for a fragment of propositional logic where
isomorphic propositions, such as and , or
and are made
equal. System I enjoys the strong normalisation property. This is sufficient to
prove the existence of empty types, but not to prove the introduction property
(every closed term in normal form is an introduction). Moreover, a severe
restriction had to be made on the types of the variables in order to obtain the
existence of empty types. We show here that adding -expansion rules to
System I permits to drop this restriction, and yields a strongly normalising
calculus with enjoying the full introduction property.Comment: 15 pages plus references and appendi
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
We give a semantics for the lambda-calculus based on a topological duality
theorem in nominal sets. A novel interpretation of lambda is given in terms of
adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is
necessary)
A general approach to define binders using matching logic
We propose a novel shallow embedding of binders using matching logic, where the binding behavior of object-level binders is obtained for free from the behavior of the built-in existential binder of matching logic. We show that binders in various logical systems such as lambda-calculus, System F, pi-calculus, pure type systems, etc., can be defined in matching logic. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic. An appealing aspect of our embedding of binders in matching logic is that it yields models to all binders, also for free. We show that models yielded by matching logic are deductively complete to the formal reasoning in the original systems. For lambda-calculus, we further show that the yielded models are representationally complete---a desired property that is not enjoyed by many existing lambda-calculus semantics.Ope
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