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Labelled tableaux for interpretability logics
[article]

2016
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arXiv
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pre-print

In is paper we present a labelled tableau proof system that serves a wide class of interpretability logics. The system is proved sound and complete for any interpretability logic characterised by a frame condition given by a set of universal strict first order Horn sentences. As such, the current paper adds to a better proof-theoretical understanding of interpretability logics.

arXiv:1605.05612v1
fatcat:4vkfmnnyj5drrgxxqa3pnic7em
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Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs
[article]

2019
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arXiv
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pre-print

We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree is of the order of the square root of n s plus k. A similar statement holds for the more general Positivstellensatz (PS) proofs. This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof

arXiv:1811.01351v2
fatcat:w73ejvsl3fepdpgqebajs6cldu
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... for the LP and SDP hierarchies of Lovász and Schrijver. As a corollary to this, and to the known degree lower bounds, we get optimal integrality gaps for exponential size SOS proofs for sparse random instances of the standard NP-hard constraint optimization problems. We also get exponential size SOS lower bounds for Tseitin and Knapsack formulas. The proof of our main result relies on a zero-gap duality theorem for pre-ordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.##
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Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

2019
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Computational Complexity Conference
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*Hakoniemi*24:11 Lemma 10. ...

*Hakoniemi*24:9 Note that deg(r j ) ≤ deg(q j ) − 1 since α i ≥ 1 for α ∈ J j and α n+i ≥ 1 for α ∈ K j . ...

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Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

2020
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International Colloquium on Automata, Languages and Programming
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We prove that both Polynomial Calculus and Sums-of-Squares proof systems admit a strong form of feasible interpolation property for sets of polynomial equality constraints. Precisely, given two sets P(x,z) and Q(y,z) of equality constraints, a refutation Π of P(x,z) ∪ Q(y,z), and any assignment a to the variables z, one can find a refutation of P(x,a) or a refutation of Q(y,a) in time polynomial in the length of the bit-string encoding the refutation Π. For Sums-of-Squares we rely on the use of

doi:10.4230/lipics.icalp.2020.63
dblp:conf/icalp/Hakoniemi20
fatcat:uyt7va4gwbdhxfk2krivqpnddy
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... Boolean axioms, but for Polynomial Calculus we do not assume their presence.##
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Front Matter, Table of Contents, Preface, Conference Organization

2020
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International Colloquium on Automata, Languages and Programming
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and Sai Sandeep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62:1-62:12 Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

doi:10.4230/lipics.icalp.2020.0
dblp:conf/icalp/X20
fatcat:xmcw4f32njfybpomfmdoe2isr4
*Tuomas**Hakoniemi*. . . . ...