The Average Case Complexity of Multilevel Syllogistic

An approach to the problem of developing provably correct programs has been to enrich a theorem prover for Hoare logic with decision procedures for a number of decidable sublanguages of set theory (EMLS, MLS, and extensions) and arithmetic (FPILP) Sch77]. Citing results of Goldberg Gol79] on average case behavior of algorithms for SAT, it was hoped that these decision procedures would perform well on average. So far, it has been fairly diicult to prove average case NP-hardness under the various deenitions ((Lev86], BDCGL89], BG91], Gur91], VR92], SY92], RS93]). We should note that the deenitions in the literature haven't yet been standardized. We survey some of the results of the average case analysis of NP-complete problems, and compare the results of Goldberg with more pessimistic results. We prove that FPILP, EMLS and related fragments of set theory are NP-average complete, and show that there are simple distributions that will frustrate any algorithm for these decision problems.