TY - GEN

T1 - Complexity of semi-algebraic proofs

AU - Grigoriev, Dima

AU - Hirsch, Edward A.

AU - Pasechnik, Dmitrii V.

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2002.

PY - 2002

Y1 - 2002

N2 - Proof systems for polynomial inequalities in 0-1 variables include the well-studied Cutting Planes proof system (CP) and the Lov´asz-Schrijver calculi (LS) utilizing linear, respectively, quadratic, inequalities. We introduce generalizations LSd of LSinvolving polynomial inequalities of degree at most d.Surprisingly, the systems LSd turn out to be very strong. We construct polynomial-size bounded degree LSd proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded degree Positivstellensatz Calculus (PC) proofs), and Tseitin’s tautologies (hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coefficients, while other extra rules further reduce the proof degrees for the aforementioned examples. Finally, we prove lower bounds on Lov´asz-Schrijver ranks, demonstrating, in particular, their rather limited applicability for proof complexity.

AB - Proof systems for polynomial inequalities in 0-1 variables include the well-studied Cutting Planes proof system (CP) and the Lov´asz-Schrijver calculi (LS) utilizing linear, respectively, quadratic, inequalities. We introduce generalizations LSd of LSinvolving polynomial inequalities of degree at most d.Surprisingly, the systems LSd turn out to be very strong. We construct polynomial-size bounded degree LSd proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded degree Positivstellensatz Calculus (PC) proofs), and Tseitin’s tautologies (hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coefficients, while other extra rules further reduce the proof degrees for the aforementioned examples. Finally, we prove lower bounds on Lov´asz-Schrijver ranks, demonstrating, in particular, their rather limited applicability for proof complexity.

UR - http://www.scopus.com/inward/record.url?scp=84937409644&partnerID=8YFLogxK

U2 - 10.1007/3-540-45841-7_34

DO - 10.1007/3-540-45841-7_34

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AN - SCOPUS:84937409644

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 419

EP - 430

BT - STACS 2002 - 19th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings

A2 - Alt, Helmut

A2 - Ferreira, Afonso

T2 - 19th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2002

Y2 - 14 March 2002 through 16 March 2002

ER -