24,337 research outputs found
Context Semantics, Linear Logic and Computational Complexity
We show that context semantics can be fruitfully applied to the quantitative
analysis of proof normalization in linear logic. In particular, context
semantics lets us define the weight of a proof-net as a measure of its inherent
complexity: it is both an upper bound to normalization time (modulo a
polynomial overhead, independently on the reduction strategy) and a lower bound
to the number of steps to normal form (for certain reduction strategies).
Weights are then exploited in proving strong soundness theorems for various
subsystems of linear logic, namely elementary linear logic, soft linear logic
and light linear logic.Comment: 22 page
Normalization in Supernatural deduction and in Deduction modulo
Deduction modulo and Supernatural deduction are two extentions of predicate logic with computation rules. Whereas the application of computation rules in deduction modulo is transparent, these rules are used to build non-logical deduction rules in Supernatural deduction. In both cases, adding computation rules may jeopardize proof normalization, but various conditions have been given in both cases, so that normalization is preserved. We prove in this paper that normalization in Supernatural deduction and in Deduction modulo are equivalent, i.e. the set of computation rules for which one system strongly normalizes is the same as the set of computation rules for which the other is
On completeness of reducibility candidates as a semantics of strong normalization
This paper defines a sound and complete semantic criterion, based on
reducibility candidates, for strong normalization of theories expressed in
minimal deduction modulo \`a la Curry. The use of Curry-style proof-terms
allows to build this criterion on the classic notion of pre-Heyting algebras
and makes that criterion concern all theories expressed in minimal deduction
modulo. Compared to using Church-style proof-terms, this method provides both a
simpler definition of the criterion and a simpler proof of its completeness.Comment: 24 page
A simple proof that super consistency implies cut elimination
International audienceWe give a simple and direct proof that super-consistency implies the cut elimination property in deduction modulo. This proof can be seen as a simpli cation of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut elimination theorems in higher-order logic that involve V-complexes
A Simple Proof That Super-Consistency Implies Cut Elimination
We give a simple and direct proof that super-consistency implies the cut
elimination property in deduction modulo. This proof can be seen as a
simplification of the proof that super-consistency implies proof normalization.
It also takes ideas from the semantic proofs of cut elimination that proceed by
proving the completeness of the cut-free calculus. As an application, we
compare our work with the cut elimination theorems in higher-order logic that
involve V-complexes
Definitions by Rewriting in the Calculus of Constructions
The main novelty of this paper is to consider an extension of the Calculus of
Constructions where predicates can be defined with a general form of rewrite
rules. We prove the strong normalization of the reduction relation generated by
the beta-rule and the user-defined rules under some general syntactic
conditions including confluence. As examples, we show that two important
systems satisfy these conditions: a sub-system of the Calculus of Inductive
Constructions which is the basis of the proof assistant Coq, and the Natural
Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award
Cut elimination for Zermelo set theory
We show how to express intuitionistic Zermelo set theory in deduction modulo
(i.e. by replacing its axioms by rewrite rules) in such a way that the
corresponding notion of proof enjoys the normalization property. To do so, we
first rephrase set theory as a theory of pointed graphs (following a paradigm
due to P. Aczel) by interpreting set-theoretic equality as bisimilarity, and
show that in this setting, Zermelo's axioms can be decomposed into
graph-theoretic primitives that can be turned into rewrite rules. We then show
that the theory we obtain in deduction modulo is a conservative extension of (a
minor extension of) Zermelo set theory. Finally, we prove the normalization of
the intuitionistic fragment of the theory
The Stratified Foundations as a theory modulo
The Stratified Foundations are a restriction of naive set theory where the
comprehension scheme is restricted to stratifiable propositions. It is known
that this theory is consistent and that proofs strongly normalize in this
theory. Deduction modulo is a formulation of first-order logic with a general
notion of cut. It is known that proofs normalize in a theory modulo if it has
some kind of many-valued model called a pre-model. We show in this paper that
the Stratified Foundations can be presented in deduction modulo and that the
method used in the original normalization proof can be adapted to construct a
pre-model for this theory
Semantic A-translation and Super-consistency entail Classical Cut Elimination
We show that if a theory R defined by a rewrite system is super-consistent,
the classical sequent calculus modulo R enjoys the cut elimination property,
which was an open question. For such theories it was already known that proofs
strongly normalize in natural deduction modulo R, and that cut elimination
holds in the intuitionistic sequent calculus modulo R. We first define a
syntactic and a semantic version of Friedman's A-translation, showing that it
preserves the structure of pseudo-Heyting algebra, our semantic framework. Then
we relate the interpretation of a theory in the A-translated algebra and its
A-translation in the original algebra. This allows to show the stability of the
super-consistency criterion and the cut elimination theorem
Conservativity of embeddings in the lambda Pi calculus modulo rewriting (long version)
The lambda Pi calculus can be extended with rewrite rules to embed any
functional pure type system. In this paper, we show that the embedding is
conservative by proving a relative form of normalization, thus justifying the
use of the lambda Pi calculus modulo rewriting as a logical framework for
logics based on pure type systems. This result was previously only proved under
the condition that the target system is normalizing. Our approach does not
depend on this condition and therefore also works when the source system is not
normalizing.Comment: Long version of TLCA 2015 pape
- âŠ