Dominoes and the complexity of subclasses of logical theories

Abstract

The complexity of subclasses of logical theories (mainly Presburger and Skolem arithmetic) is studied. The subclasses are defined by the structure of the quantifier prefix. For this purpose finite versions of dominoes (tiling problems) are used. Dominoes were introduced in the sixties as a tool to prove the undecidability of the ∀∃∀-case of the predicate calculus and have found in the meantime many other applications. Here it is shown that problems in complexity classes NTIME( T( n)) are reducible to domino problems where the space to be tiled is a square of size T( n). Because of their simple combinatorial structure these dominoes provide a convinient method for providing lower complexity bounds for simple formula classes in logical theories. Using this method it is shown that the class of ∃∀ ∗-formulas in Presburger arithmetic has exponential complexity. This seems to be the simplest class with this property because the set of ∃ ∗-sentences in Presburger arithmetic is NP-complete and the classes which is shown to be fixed prefixes (i.e. where also the number of variables is limited) are all contained in appropriate levels of the polynomial time-hierarchy. Skolem arithmetic is the theory of positive natural numbers with multiplication and 's thus (isomorphic to) the weak direct power of Presburger arithmetic. For the theory in general as well as for most subclasses the complexity is one exponential step higher than in the case of Presburger arithmetic. An exception is the class of ∃ ∗-formulas which is shown to be NP-complete. On the other hand there is a formula class with fixed dimension which already has an exponential lower complexity bound. The last section mentions some results on other logical theories and indicates some possible lines of future research.