Matrix Rank Arguments for Proving Lower Bounds on the Size of Branching Programs.pdf

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Matrix Rank Arguments for Proving Lower Bounds on the Size of Branching Programs.pdf

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Matrix Rank Arguments for Proving Lower Bounds on the Size of Branching Programs

Matrix Rank Arguments for Proving LowerBounds on the Size of Branching Programs Christoph MeinelFB IV - Informatik, Universitat TrierD-54286 Triermeinel@uni-trier.deStephan WaackInst. fur Num. u. Angew. Mathematik,Georg-August-UniversitatD-37083 Gottingenwaack@namu01.gwdg.deIntroductionIn order to characterize and investigate dierent logarithmic space{bounded com-plexity classes like L; NL; coNL; L; or MODpL by means of combinatorial com-putational devices, branching programs [Lee59], -branching programs, IB2[Mei89] (this model includes the nondeterministic and co{nondeterministic branch-ing programs), and MODp-branching programs [DKMW94] were introduced andinvestigated. Unfortunately, up to now, it was impossible to prove a superpolyno-mial lower bound on the size of any of these devices which would be a prerequistefor separating the corresponding complexity classes. In order to get a better under-standing of the diculties in deriving such lower bounds, certain restricted branchingprogram models like ordered binary decision diagrams (OBDDs) or oblivious linearlength branching programs were considered.Based on results of [AM86, KMW89, Kra90, DKMW94], we have developed a the-ory of communication within deterministic, nondeterministic, co-nondeterministic,parity and MODp{branching programs. The restriction the considered branchingProc. of the Workshop on Algebraic Methods in Complexity Theory, Madras, 5{21, 19941 programs are supposed to fulll is the bounded alternating property: A branchingprogram P with this property is characterized by the fact that there are two disjointsubsets Y , Z X in the set X of variables tested in P , #Y = #Z = (#X),such that, on each path of P , the number of alternations between testing variablesfrom Y and testing variables from Z is bounded by (#X)o(1). With algebraical rankarguments concerning certain communication matrices, we are able to derive expo-nential lower bounds on the sizes of all the mentioned types of boun