Degree lower bounds of tower-type for approximating formulas with parity quantifiers.

Kolaitis and Kopparty have shown that for any first-order formula with parity quantifiers over the language of graphs there is a family of multi-variate polynomials of constant-degree that agree with the formula on all but a 2 −Ω(n)-fraction of the graphs with n vertices. The proof bounds the degree of the polynomials by a tower of exponentials whose height is the nesting depth of parity quantifiers in the formula. We show that this tower-type dependence is necessary. We build a family of formulas of depth q whose approximating polynomials must have degree bounded from below by a tower of exponentials of height proportional to q. Our proof has two main parts. First, we adapt and extend the results by Kolaitis and Kopparty that describe the joint distribution of the parities of the numbers of copies of small subgraphs in a random graph to the setting of graphs of growing size. Second, we analyse a variant of Karp’s graph canonical labeling algorithm and exploit its massive parallelism to get a formula of low depth that defines an almost canonical pre-order on a random graph.

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications