First-Order Reasoning and Efficient Semi-Algebraic Proofs
Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms [3]: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization...
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| Vydáno v: | Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science s. 1 - 13 |
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IEEE
29.06.2021
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| Abstract | Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms [3]: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems (cf. [4]). Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's [7] dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule. |
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| AbstractList | Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms [3]: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems (cf. [4]). Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's [7] dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule. |
| Author | Part, Fedor Thapen, Neil Tzameret, Iddo |
| Author_xml | – sequence: 1 givenname: Fedor surname: Part fullname: Part, Fedor email: fedor.part@gmail.com organization: Czech Academy of Sciences,JetBrains Research,Czech Republic – sequence: 2 givenname: Neil surname: Thapen fullname: Thapen, Neil email: thapen@math.cas.cz organization: Czech Academy of Sciences,Institute of Mathematics,Czech Republic – sequence: 3 givenname: Iddo surname: Tzameret fullname: Tzameret, Iddo email: iddo.tzameret@gmail.com organization: Imperial College London,Department of Computing,United Kingdom |
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| Snippet | Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms [3]:... |
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| SubjectTerms | Approximation algorithms Calculus Cognition Computational modeling Computer science Lead Systematics |
| Title | First-Order Reasoning and Efficient Semi-Algebraic Proofs |
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