6,743 research outputs found
The lambda-mu-T-calculus
Calculi with control operators have been studied as extensions of simple type
theory. Real programming languages contain datatypes, so to really understand
control operators, one should also include these in the calculus. As a first
step in that direction, we introduce lambda-mu-T, a combination of Parigot's
lambda-mu-calculus and G\"odel's T, to extend a calculus with control operators
with a datatype of natural numbers with a primitive recursor.
We consider the problem of confluence on raw terms, and that of strong
normalization for the well-typed terms. Observing some problems with extending
the proofs of Baba at al. and Parigot's original confluence proof, we provide
new, and improved, proofs of confluence (by complete developments) and strong
normalization (by reducibility and a postponement argument) for our system.
We conclude with some remarks about extensions, choices, and prospects for an
improved presentation
Lambda Dependency-Based Compositional Semantics
This short note presents a new formal language, lambda dependency-based
compositional semantics (lambda DCS) for representing logical forms in semantic
parsing. By eliminating variables and making existential quantification
implicit, lambda DCS logical forms are generally more compact than those in
lambda calculus
Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory
Classical (or Boolean) type theory is the type theory that allows the type
inference (the type counterpart of
double-negation elimination), where is any type and is
absurdity type. This paper first presents a denotational semantics for a
simplified version of Parigot's lambda-mu calculus, a premier example of
classical type theory. In this semantics the domain of each type is divided
into infinitely many ranks and contains not only the usual members of the type
at rank 0 but also their negative, conjunctive, and disjunctive shadows in the
higher ranks, which form an infinitely nested Boolean structure. Absurdity type
is identified as the type of truth values. The paper then presents a new
deduction system of classical type theory, a sequent calculus called the
classical type system (CTS), which involves the standard logical operators such
as negation, conjunction, and disjunction and thus reflects the discussed
semantic structure in a more straightforward fashion.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
Models and termination of proof reduction in the -calculus modulo theory
We define a notion of model for the -calculus modulo theory and
prove a soundness theorem. We then define a notion of super-consistency and
prove that proof reduction terminates in the -calculus modulo any
super-consistent theory. We prove this way the termination of proof reduction
in several theories including Simple type theory and the Calculus of
constructions
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