Program Schemes, Arrays, Lindström Quantifiers and Zero-One Laws

Abstract

We characterize the class of problems accepted by a class of program schemes with arrays, NPSA, as the class of problems defined by the sentences of a logic formed by extending first-order logic with a particular uniform sequence of Lindström quantifiers. We prove that our logic, and consequently our class of program schemes, has a zero-one law. However, we show that there are problems definable in a basic fragment of our logic, and so also accepted by basic program schemes, which are not definable in bounded-variable infinitary logic. Hence, the class of problems NPSA is not contained in the class of problems defined by the sentences of partial fixed-point logic even though in the presence of a built-in successor relation, both NPSA and partial fixed-point logic capture the complexity class PSPACE.KeywordsFree VariableProgram SchemeRelation SymbolConstant SymbolProblem TokenThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.