10 research outputs found
Combinatorial Flows and Their Normalisation
This paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic proofs are translated into combinatorial flows and vice versa
Towards a Combinatorial Proof Theory
International audienceThe main part of a classical combinatorial proof is a skew fi-bration, which precisely captures the behavior of weakening and contraction. Relaxing the presence of these two rules leads to certain substruc-tural logics and substructural proof theory. In this paper we investigate what happens if we replace the skew fibration by other kinds of graph homomorphism. This leads us to new logics and proof systems that we call combinatorial
From Syntactic Proofs to Combinatorial Proofs
International audienceIn this paper we investigate Hughes’ combinatorial proofs as a notion of proof identity for classical logic. We show for various syntactic formalisms including sequent calculus, analytic tableaux, and resolution, how they can be translated into combinatorial proofs, and which notion of identity they enforce. This allows the comparison of proofs that are given in different formalisms
Intuitionistic proofs without syntax
International audienceWe present Intuitionistic Combinatorial Proofs (ICPs), a concrete geometric semantics of intuitionistic logic based on the principles of the second author's classical Com-binatorial Proofs. An ICP naturally factorizes into a linear fragment, a graphical abstraction of an IMLL proof net (an arena net), and a parallel contraction-weakening fragment (a skew fibration). ICPs relate to game semantics, and can be seen as a strategy in a Hyland-Ong arena, generalized from a tree-like to a dag-like strategy. Our first main result, Polynomial Full Completeness, is that ICPs as a semantics are complexity-aware: the translations to and from sequent calculus are size-preserving (up to a polynomial). By contrast, lambda-calculus and game semantics incur an exponential blowup. Our second main result, Local Canonicity, is that ICPs abstract fully and faithfully over the non-duplicating permutations of the sequent calculus, analogously to the first and second authors' recent result for MALL
An Analytic Propositional Proof System on Graphs
In this paper we present a proof system that operates on graphs instead of
formulas. Starting from the well-known relationship between formulas and
cographs, we drop the cograph-conditions and look at arbitrary undirected)
graphs. This means that we lose the tree structure of the formulas
corresponding to the cographs, and we can no longer use standard proof
theoretical methods that depend on that tree structure. In order to overcome
this difficulty, we use a modular decomposition of graphs and some techniques
from deep inference where inference rules do not rely on the main connective of
a formula. For our proof system we show the admissibility of cut and a
generalization of the splitting property. Finally, we show that our system is a
conservative extension of multiplicative linear logic with mix, and we argue
that our graphs form a notion of generalized connective
An Analytic Propositional Proof System on Graphs
In this paper we present a proof system that operates on graphs instead of
formulas. Starting from the well-known relationship between formulas and
cographs, we drop the cograph-conditions and look at arbitrary undirected)
graphs. This means that we lose the tree structure of the formulas
corresponding to the cographs, and we can no longer use standard proof
theoretical methods that depend on that tree structure. In order to overcome
this difficulty, we use a modular decomposition of graphs and some techniques
from deep inference where inference rules do not rely on the main connective of
a formula. For our proof system we show the admissibility of cut and a
generalisation of the splitting property. Finally, we show that our system is a
conservative extension of multiplicative linear logic with mix, and we argue
that our graphs form a notion of generalised connective
Combinatorial Flows and their Normalisation
International audienceThis paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic proofs are translated into combinatorial flows and vice versa