12,668 research outputs found
What is a logic, and what is a proof ?
International audienceI will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then preorder equivalence becomes equivalence of categories, whose arrows are the proofs. For identifying these, the concept of proof nets is discussed
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
An Abstract Approach to Stratification in Linear Logic
We study the notion of stratification, as used in subsystems of linear logic
with low complexity bounds on the cut-elimination procedure (the so-called
light logics), from an abstract point of view, introducing a logical system in
which stratification is handled by a separate modality. This modality, which is
a generalization of the paragraph modality of Girard's light linear logic,
arises from a general categorical construction applicable to all models of
linear logic. We thus learn that stratification may be formulated independently
of exponential modalities; when it is forced to be connected to exponential
modalities, it yields interesting complexity properties. In particular, from
our analysis stem three alternative reformulations of Baillot and Mazza's
linear logic by levels: one geometric, one interactive, and one semantic
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
Proof equivalence in MLL is PSPACE-complete
MLL proof equivalence is the problem of deciding whether two proofs in
multiplicative linear logic are related by a series of inference permutations.
It is also known as the word problem for star-autonomous categories. Previous
work has shown the problem to be equivalent to a rewiring problem on proof
nets, which are not canonical for full MLL due to the presence of the two
units. Drawing from recent work on reconfiguration problems, in this paper it
is shown that MLL proof equivalence is PSPACE-complete, using a reduction from
Nondeterministic Constraint Logic. An important consequence of the result is
that the existence of a satisfactory notion of proof nets for MLL with units is
ruled out (under current complexity assumptions). The PSPACE-hardness result
extends to equivalence of normal forms in MELL without units, where the
weakening rule for the exponentials induces a similar rewiring problem.Comment: Journal version of: Willem Heijltjes and Robin Houston. No proof nets
for MLL with units: Proof equivalence in MLL is PSPACE-complete. In Proc.
Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic
and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science, 201
Bipolar Proof Nets for MALL
In this work we present a computation paradigm based on a concurrent and
incremental construction of proof nets (de-sequentialized or graphical proofs)
of the pure multiplicative and additive fragment of Linear Logic, a resources
conscious refinement of Classical Logic. Moreover, we set a correspon- dence
between this paradigm and those more pragmatic ones inspired to transactional
or distributed systems. In particular we show that the construction of additive
proof nets can be interpreted as a model for super-ACID (or co-operative)
transactions over distributed transactional systems (typi- cally,
multi-databases).Comment: Proceedings of the "Proof, Computation, Complexity" International
Workshop, 17-18 August 2012, University of Copenhagen, Denmar
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