9 research outputs found
On the relative proof complexity of deep inference via atomic flows
We consider the proof complexity of the minimal complete fragment, KS, of
standard deep inference systems for propositional logic. To examine the size of
proofs we employ atomic flows, diagrams that trace structural changes through a
proof but ignore logical information. As results we obtain a polynomial
simulation of versions of Resolution, along with some extensions. We also show
that these systems, as well as bounded-depth Frege systems, cannot polynomially
simulate KS, by giving polynomial-size proofs of certain variants of the
propositional pigeonhole principle in KS.Comment: 27 pages, 2 figures, full version of conference pape
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
On linear rewriting systems for Boolean logic and some applications to proof theory
Linear rules have played an increasing role in structural proof theory in
recent years. It has been observed that the set of all sound linear inference
rules in Boolean logic is already coNP-complete, i.e. that every Boolean
tautology can be written as a (left- and right-)linear rewrite rule. In this
paper we study properties of systems consisting only of linear inferences. Our
main result is that the length of any 'nontrivial' derivation in such a system
is bound by a polynomial. As a consequence there is no polynomial-time
decidable sound and complete system of linear inferences, unless coNP=NP. We
draw tools and concepts from term rewriting, Boolean function theory and graph
theory in order to access some required intermediate results. At the same time
we make several connections between these areas that, to our knowledge, have
not yet been presented and constitute a rich theoretical framework for
reasoning about linear TRSs for Boolean logic.Comment: 27 pages, 3 figures, special issue of RTA 201
Enumerating Independent Linear Inferences
A linear inference is a valid inequality of Boolean algebra in which each
variable occurs at most once on each side. Equivalently, it is a linear rewrite
rule on Boolean terms that constitutes a valid implication. Linear inferences
have played a significant role in structural proof theory, in particular in
models of substructural logics and in normalisation arguments for deep
inference proof systems.
In this work we leverage recently developed graphical representations of
linear formulae to build an implementation that is capable of more efficiently
searching for switch-medial-independent inferences. We use it to find four
`minimal' 8-variable independent inferences and also prove that no smaller ones
exist; in contrast, a previous approach based directly on formulae reached
computational limits already at 7 variables. Two of these new inferences derive
some previously found independent linear inferences. The other two (which are
dual) exhibit structure seemingly beyond the scope of previous approaches we
are aware of; in particular, their existence contradicts a conjecture of Das
and Strassburger.
We were also able to identify 10 minimal 9-variable linear inferences
independent of all the aforementioned inferences, comprising 5 dual pairs, and
present applications of our implementation to recent `graph logics'.Comment: 33 pages, 3 figure
Enumerating Independent Linear Inferences
A linear inference is a valid inequality of Boolean algebra in which each
variable occurs at most once on each side.
In this work we leverage recently developed graphical representations of
linear formulae to build an implementation that is capable of more efficiently
searching for switch-medial-independent inferences. We use it to find four
`minimal' 8-variable independent inferences and also prove that no smaller ones
exist; in contrast, a previous approach based directly on formulae reached
computational limits already at 7 variables. Two of these new inferences derive
some previously found independent linear inferences. The other two (which are
dual) exhibit structure seemingly beyond the scope of previous approaches we
are aware of; in particular, their existence contradicts a conjecture of Das
and Strassburger.
We were also able to identify 10 minimal 9-variable linear inferences
independent of all the aforementioned inferences, comprising 5 dual pairs, and
present applications of our implementation to recent `graph logics'
On the relative proof complexity of deep inference via atomic flows
We consider the proof complexity of the minimal complete fragment, KS, of
standard deep inference systems for propositional logic. To examine the size of
proofs we employ atomic flows, diagrams that trace structural changes through a
proof but ignore logical information. As results we obtain a polynomial
simulation of versions of Resolution, along with some extensions. We also show
that these systems, as well as bounded-depth Frege systems, cannot polynomially
simulate KS, by giving polynomial-size proofs of certain variants of the
propositional pigeonhole principle in KS