The strength of multilinear proofs

Ran Raz, Iddo Tzameret

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show the following: 1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) proofs; 2. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for the functional pigeonhole principle; 3. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for Tseitin's graph tautologies. By known lower bounds, this demonstrates that algebraic proof systems manipulating depth 3 multilinear formulas are strictly stronger than Resolution, PC and PCR, and have an exponential gap over bounded-depth Frege for both the functional pigeonhole principle and Tseitin's graph tautologies. We also illustrate a connection between lower bounds on multilinear proofs and lower bounds on multilinear circuits. In particular, we show that (an explicit) super-polynomial size separation between proofs manipulating general arithmetic circuits and proofs manipulating multilinear circuits implies a super-polynomial size lower bound on multilinear circuits for an explicit family of polynomials.

Original languageEnglish (US)
Pages (from-to)407-457
Number of pages51
JournalComputational Complexity
Issue number3
StatePublished - Oct 2008
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Mathematics
  • Computational Theory and Mathematics
  • Computational Mathematics


  • Multilinear arithmetic formulas
  • Polynomial calculus
  • Proof complexity
  • Propositional pigeonhole principle
  • Tseitin tautologies


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