Partial Fixed-Point Logic on Infinite Structures

Abstract

We consider an alternative semantics for partial fixed-point logic (PFP). To define the fixed point of a formula in this semantics, the sequence of stages induced by the formula is considered. As soon as this sequence becomes cyclic, the set of elements contained in every stage of the cycle is taken as the fixed point. It is shown that on finite structures, this fixed-point semantics and the standard semantics for PFP as considered in finite model theory are equivalent, although arguably the formalisation of properties might even become simpler and more intuitive. Contrary to the standard PFP semantics which is only defined on finite structures the new semantics generalises easily to infinite structures and transfinite inductions. In this generality we compare - in terms of expressive power - partial with other known fixed-point logics. The main result of the paper is that on arbitrary structures, PFP is strictly more expressive than inflationary fixed-point logic (IFP). A separation of these logics on finite structures would prove Ptime different from Pspace.