The existential fragment of S1S over element and successor is the co-Buchi languages

Büchi’s theorem, in establishing the equivalence between languages definable in S1S over pP,ăq and the ω-regular languages also demonstrated that S1S over pP,ăq is no more expressive than its existential fragment. It is also easy to see that S1S over pP,ăq is equi-expressive with S1S over pP, sq. However, it is not immediately obvious whether it is possible to adapt Büchi’s argument to establish equivalence between expressivity in S1S over pP, sq and its existential fragment. In this paper we show that it is not: the existential fragment of S1S over pP, sq is strictly less expressive, and is in fact equivalent to the co-Büchi languages. 1 Preliminaries 1.1 Second order theory of one successor Definition 1 (S1S syntax). We introduce the following components: • A set of first order variables, denoted by lower case letters, possibly with subscripts. • A set of second order variables, denoted by upper case letters, possibly with subscripts. S1SpP,ăq has the following set of well formed formulae: φ ::“ t ă t | t P X | φ^ φ | φ | Dxiφ | DXiφ. t is understood to range over terms, which in this case are just the first order variables, and X to range over second order variables.