1,147 research outputs found
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is
accessible to a broad mathematical audience. Gentzen's cut elimination theorem
is not as well known as it deserves to be, and it is tied to a lot of
interesting mathematical structure. In particular we try to indicate some
dynamical and combinatorial aspects of cut elimination, as well as its
connections to complexity theory. We discuss two concrete examples where one
can see the structure of short proofs with cuts, one concerning feasible
numbers and the other concerning "bounded mean oscillation" from real analysis
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
Classes of representable disjoint NP-pairs
For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist
Introduction to Cirquent Calculus and Abstract Resource Semantics
This paper introduces a refinement of the sequent calculus approach called
cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.)
sequents are disjoint sequences of formulas, in cirquent calculus they are
permitted to share elements. Explicitly allowing or disallowing shared
resources and thus taking to a more subtle level the resource-awareness
intuitions underlying substructural logics, cirquent calculus offers much
greater flexibility and power than sequent calculus does. A need for
substantially new deductive tools came with the birth of computability logic
(see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed
formal theory of computational resources, which has stubbornly resisted any
axiomatization attempts within the framework of traditional syntactic
approaches. Cirquent calculus breaks the ice. Removing contraction from the
full collection of its rules yields a sound and complete system for the basic
fragment CL5 of computability logic. Doing the same in sequent calculus, on the
other hand, throws out the baby with the bath water, resulting in the strictly
weaker affine logic. An implied claim of computability logic is that it is CL5
rather than affine logic that adequately materializes the resource philosophy
traditionally associated with the latter. To strengthen this claim, the paper
further introduces an abstract resource semantics and shows the soundness and
completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio
Logical closure properties of propositional proof systems - (Extended abstract)
In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic
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