The extensions of first-order logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abiteboul and Vianu, 1991b] investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ! 1! (see, for instance, [Kolaitis and Vardi, 1990]). We investigate this logic of finite structures and provide a normal form for it. We also present a treatment of the results in [Abiteboul and Vianu, 1991b] from this point of view. In particular, we show that we can write a formula of FO + LFP that defines ...

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences ' and `, we consider the structures with domain f1; : : : ; Ng that satisfy `, and compute the fraction of them in which ' is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii[31] and Grove, Halpern, and Koller [22], in the general case, asymptotic conditional probabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on the assumption that ` can use a nonunary predicate symbol. Liogon'kii [31] shows that if we condition on formulas ` involving unary predicate symbols only (but no equality or constant symbols), then the asymptotic conditional probability does exist and can be effectively computed. This is the case even if we place no corresponding restrictions on '. We extend this result here to the case where ` involves equality and constants. We show that the complexity of computing the limit depends on various factors, such as the depth of quantifier nesting, or whether the vocabulary is finite or infinite. We completely characterize the complexity of the problem in the different cases, and show related results for the associated approximation problem.

We study 0-1 laws for extensions of first-order logic by Lindstrom quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q] -- the extension of first-order logic by means of the quantifier Q -- to have a 0-1 law. We use these conditions to show, in particular, that FO[Rig], where Rig is the quantifier expressing rigidity, has a 0-1 law. We also show that FO[Ham], where Ham is the quantifier expressing Hamiltonicity, does not have a 0-1 law. Blass and Harary pose the question whether there is a logic which is powerful enough to express Hamiltonicity or rigidity and which has a 0-1 law. It is a consequence of our results that there is no such regular logic (in the sense of abstract model theory) in the case of Hamiltonicity, but there is one in the case of rigidity. We also consider sequences of vectorized quantifiers, and show that the extensions of first-order logic obtained by adding such sequences generated by quantifiers that are...

In this survey article we discuss some aspects of finite variable logics. We translate some well-known fixed-point logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this connection we consider definable linear orderings of types realised in finite structures. We then show that the Craig interpolation and Beth definability properties fail for L ! 1! . Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout. 1 Some extensions of first-order logic Quisani: Hello. Who are you? I am Yuri's imaginary student, and I usually talk to him at this time. Author: I'm afraid he may be a bit late. I am a computer scientist from London, England. I have some imaginary students myself, so maybe I can help. I was reading your earlier conversation on 0--1 laws [Gu3]. Quisani: I remember it. We examined the...

: This paper continues our work on infinite, recursive structures. We investigate the descriptive complexity of several logics over recursive structures, including first-order, second-order, and fixpoint logic, exhibiting connections between expressibility of a property and its computational complexity. We then address 0--1 laws, proposing a version that applies to recursive structures, and using it to prove several non-expressibility results. 0 Introduction Infinite recursive structures, with recursive graphs as a special case, have been studied quite extensively in the past. Most interesting properties of recursive graphs have been shown to be undecidable, and many are actually outside the arithmetic hierarchy; see, e.g., [AMS, Be1, Be2, BG, H, HH1]. In [HH2] we considered recursive structures to be generalizations of finite relational data bases, and investigated the class of computable queries over them, the motivation being borrowed from [CH1]. A computable query is a (partial) r...

We consider L | first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relationship between the size of a finite structure and the number of distinct types it realizes, with respect to L . Some open questions, formulated as finitary Löwenheim-Skolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an Ehrenfeucht-Mostowski property.

We study 0-1 laws for extensions of first-order logic by Lindström quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q] – the extension of first-order logic by means of the quantifier Q – to have a 0-1 law. We use these conditions to show, in particular, that FO[Rig], where Rig is the quantifier expressing rigidity, has a 0-1 law. We also show that extensions of first-order logic with quantifiers for Hamiltonicity, regularity and self-complementarity of graphs do not have a 0-1 law. Blass and Harary pose the question whether there is a logic which is powerful enough to express Hamiltonicity or rigidity and which has a 0-1 model theory) in the case of Hamiltonicity, but there is one in the case of rigidity. We also consider sequences of vectorized quantifiers, and show that the extensions of first-order logic obtained by adding such sequences generated by quantifiers that are closed under substructures have 0-1 laws. The positive results also extend to the infinitary logic with finitely many variables.

Structures" [Mos74], where they are called inductive relations. It should also be pointed out that in Immerman's book on "Descriptive Complexity" LFP is denoted by FO(LFP) (the closure of FO under least fixed-points) and LFP 1 is denoted by LFP(FO) (least fixed-points of first-order formulas).

it cannot express a variety of properties. For example, it follows immediately that evenness is not expressible in FO. Indeed, n (even) is 1 if n is even and 0 if n is odd. Thus, n (even) does not converge, so evenness is not expressible in any language that has a 0-1 law. While 0-1 laws provide an elegant and powerful tool, they require the development of some non-trivial machinery. Interestingly, this is one of the rare occasions when it is needed to consider infinite structures while proving something about finite structures! For simplicity, we consider only the case vocabularies consisting of a binary relation G (representing edges in a directed graph with no edges of the form ha; ai). It is straightforward to generalize the development to arbitrary vocabularies. We will us