Zeilschr. zyxwvutsrqponm f. zyxwvutsrqponmlk m&. zyxwvutsrqponm Logik und Cnrndlagen d. M&. Bd. 21, S. 89-96 (1975) zyxwvutsrqponmlkji MONADIC GENERALIZED SPECTRA by RONALD FAGIN in Yorktown Heights, New York (U.S.A.) l) 1. Introduction Let d be the class of finite models of a second-order existential sentence 3Pl.. .3P,o, where u is an arbitrary first-order sentence (with equality). Thus, d is a PC class zyxw in the sense of TARSKI zyxwvutsrq [8], where we restrict our attention to the class of finite structures. If PI, . . . , P,, are the only nonlogical symbols appearing in u, then A? can be identified with the set of cardinalities of finite models of 6. H. SCHOLZ [7] called this set the spectrum of (T. Hence, in the general case, we call d a generalized spectrum. If P,, .. . , P, are each unary predicate symbols, then we call d a monadic generalized spectrum. In this paper, we show, by using FRAIssE-type games, that the class of monadic gener- alized spectra is not closed under complement. If zyxwvuts Y is a similarity type, that is, a finite set of predicate and constant symbols, then by an Y-structure, we mean a relational structure appropriate for zyxwv 9'. We will show that the class of non-connected, finite {P}-structures (where P is a binary predicate symbol) is a monadic generalized spectrum, but that the class of connected, finite {P}-structures is not (although the latter class is a generalized spectrum with just one existentialized binary predicate symbol, as we will see). Assume throughout this paper that P is a binary predicate symbol and that U, , U,, . . . are unary predicate symbols. Define a cycle (of lengthn) to be a {P}-structure ' % = = (A; Q), where for some n distinct elements a,, . . . , zyxwv a,,, Q = {(ai, ai+J : 1 A = {al, . . ., a,}, i < n} zyxwvu u {(an, u,)}. Write card(%) = n. If % = (A; Q) and ' $3 = (B; R) are cycles, and An B = 0, then by the cardinal sum 9.i @ b, we mean the {P}-structure (A u B; Q u R). We dl show that if z is gu,. . . 3Uda, where d is a first-order {P, u,, . . ., ud}' sentence (that is, its nonlogical symbols are a subset of {P, U,, . . ., U,}), then there is a constant N such that for each cycle 9.i with zyxwvut 9l k z and card(%) 2 N, there is a cycle b such that B @ B k z. It easily follows that the class of connected, finite {P}-structures is not a monadic generalized spectrum. This result is related to monadic second-order decidability results in Bihxi~ [2] and RABIN [6}, but it does not seem to be directly derivable from them. In any case, this result can be derived very directly by the use of FaGssE-type games [5], which is the approach we will use. G. ASSER [l] posed the question of whether the complement of every spectrum is a spectrum. We remark that the author showed in [3] and [4] that there is a par- ticular monadic generalized spectrum d (namely, the class of all finite models of 3u v 23 ! y(Pxy A Uy), where U is unary, P is binary, and 3 ! y is read "There is exactly l) This paper is based on a part of the author's doctoral dissertation [3] in the Department of Mathematics at the University of California, Berkeley. Part of this work was carried out while the author was a National Science Foundation Graduate Fellow; also, part of this work wm supported by NSF Grant No. GP-24532. The author is grateful to ROBERT VAUGHT and WILLIAM CRAIQ for useful suggestions which improved readability.