4,311 research outputs found
From Proof Nets to the Free *-Autonomous Category
In the first part of this paper we present a theory of proof nets for full
multiplicative linear logic, including the two units. It naturally extends the
well-known theory of unit-free multiplicative proof nets. A linking is no
longer a set of axiom links but a tree in which the axiom links are subtrees.
These trees will be identified according to an equivalence relation based on a
simple form of graph rewriting. We show the standard results of
sequentialization and strong normalization of cut elimination. In the second
part of the paper we show that the identifications enforced on proofs are such
that the class of two-conclusion proof nets defines the free *-autonomous
category.Comment: LaTeX, 44 pages, final version for LMCS; v2: updated bibliograph
The Geometry of Synchronization (Long Version)
We graft synchronization onto Girard's Geometry of Interaction in its most
concrete form, namely token machines. This is realized by introducing
proof-nets for SMLL, an extension of multiplicative linear logic with a
specific construct modeling synchronization points, and of a multi-token
abstract machine model for it. Interestingly, the correctness criterion ensures
the absence of deadlocks along reduction and in the underlying machine, this
way linking logical and operational properties.Comment: 26 page
Canonical Proof nets for Classical Logic
Proof nets provide abstract counterparts to sequent proofs modulo rule
permutations; the idea being that if two proofs have the same underlying
proof-net, they are in essence the same proof. Providing a convincing proof-net
counterpart to proofs in the classical sequent calculus is thus an important
step in understanding classical sequent calculus proofs. By convincing, we mean
that (a) there should be a canonical function from sequent proofs to proof
nets, (b) it should be possible to check the correctness of a net in polynomial
time, (c) every correct net should be obtainable from a sequent calculus proof,
and (d) there should be a cut-elimination procedure which preserves
correctness. Previous attempts to give proof-net-like objects for propositional
classical logic have failed at least one of the above conditions. In [23], the
author presented a calculus of proof nets (expansion nets) satisfying (a) and
(b); the paper defined a sequent calculus corresponding to expansion nets but
gave no explicit demonstration of (c). That sequent calculus, called LK\ast in
this paper, is a novel one-sided sequent calculus with both additively and
multiplicatively formulated disjunction rules. In this paper (a self-contained
extended version of [23]), we give a full proof of (c) for expansion nets with
respect to LK\ast, and in addition give a cut-elimination procedure internal to
expansion nets - this makes expansion nets the first notion of proof-net for
classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and
Computation
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
The relational model is injective for Multiplicative Exponential Linear Logic (without weakenings)
We show that for Multiplicative Exponential Linear Logic (without weakenings)
the syntactical equivalence relation on proofs induced by cut-elimination
coincides with the semantic equivalence relation on proofs induced by the
multiset based relational model: one says that the interpretation in the model
(or the semantics) is injective. We actually prove a stronger result: two
cut-free proofs of the full multiplicative and exponential fragment of linear
logic whose interpretations coincide in the multiset based relational model are
the same "up to the connections between the doors of exponential boxes".Comment: 36 page
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