Maria Luisa Bonet, Juan Luis Esteban, Jan Johannsen

We prove an exponential lower bound for tree-like Cutting Planes

refutations of a set of clauses which has polynomial size resolution

refutations. This implies an exponential separation between tree-like

and dag-like proofs for both Cutting Planes and resolution; in both

cases only superpolynomial separations were known before.

In order to ...
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Dima Grigoriev, Edward Hirsch

We introduce two algebraic propositional proof systems F-NS

and F-PC. The main difference of our systems from (customary)

Nullstellensatz and Polynomial Calculus is that the polynomials

are represented as arbitrary formulas (rather than sums of

monomials). Short proofs of Tseitin's tautologies in the

constant-depth version of F-NS provide ...
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Dima Grigoriev, Edward Hirsch, Dmitrii V. Pasechnik

It is a known approach to translate propositional formulas into

systems of polynomial inequalities and to consider proof systems

for the latter ones. The well-studied proof systems of this kind

are the Cutting Planes proof system (CP) utilizing linear

inequalities and the Lovasz-Schrijver calculi (LS) utilizing

quadratic ...
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Jan Krajicek

We describe a general method how to construct from

a propositional proof system P a possibly much stronger

proof system iP. The system iP operates with

exponentially long P-proofs described ``implicitly''

by polynomial size circuits.

As an example we prove that proof system iEF, implicit EF,

corresponds to bounded ...
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Edward Hirsch, Sergey I. Nikolenko

Goerdt (1991) considered a weakened version of the Cutting Plane proof system with a restriction on the degree of falsity of intermediate inequalities. (The degree of falsity of an inequality written in the form $\sum a_ix_i+\sum b_i(1-x_i)\ge c,\ a_i,b_i\ge0$ is its constant term $c$.) He proved a superpolynomial lower bound ... more >>>

Edward Hirsch, Dmitry Itsykson, Ivan Monakhov, Alexander Smal

The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajicek and Pudlak (1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (2009) ... more >>>

Titus Dose, Christian Glaßer

The article investigates the relation between three well-known hypotheses.

1) Hunion: the union of disjoint complete sets for NP is complete for NP

2) Hopps: there exist optimal propositional proof systems

3) Hcpair: there exist complete disjoint NP-pairs

The following results are obtained:

a) The hypotheses are pairwise independent ...
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