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Labelled tableaux for interpretability logics [article]

Tuomas A. Hakoniemi, Joost J. Joosten
2016 arXiv   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

Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs [article]

Albert Atserias, Tuomas Hakoniemi
2019 arXiv   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
more » ... 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.
arXiv:1811.01351v2 fatcat:w73ejvsl3fepdpgqebajs6cldu

Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

Albert Atserias, Tuomas Hakoniemi, Michael Wagner
2019 Computational Complexity Conference  
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 .  ... 
doi:10.4230/lipics.ccc.2019.24 dblp:conf/coco/AtseriasH19 fatcat:ega7hny7tngcbbl342szz7f3ye

Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

Tuomas Hakoniemi, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
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
more » ... Boolean axioms, but for Polynomial Calculus we do not assume their presence.
doi:10.4230/lipics.icalp.2020.63 dblp:conf/icalp/Hakoniemi20 fatcat:uyt7va4gwbdhxfk2krivqpnddy

Front Matter, Table of Contents, Preface, Conference Organization

Artur Czumaj, Anuj Dawar, Emanuela Merelli, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
and Sai Sandeep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62:1-62:12 Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares Tuomas Hakoniemi . . . .  ... 
doi:10.4230/lipics.icalp.2020.0 dblp:conf/icalp/X20 fatcat:xmcw4f32njfybpomfmdoe2isr4