No abstract

Catalog is the language of logic programs without function symbols. It is used as a database query language. If it is possible to eliminate recursion from a Datalog program II, then II is said to be bounded. It is known that the problem of deciding whether a given Datalog program is bounded is undecidable, even for binary programs. We show here that boundedness is decidable for monadic programs, i.e., programs where the recursive predicates are monadic (the non-recursive predicates can have arbitrary arity). UnderIying our results are new tools for the optimization of Datalog programs based on automata theory and logic. In particular, one of the tools we develop is a theory of two-way alternating tree automata. We also use our techl Addre8n:

~0. Introduction. The idea of treating probability as a real valued function defined on sentences is an old one (see ['6] and [7], where other references can be found). Carnap's attempt to set up a theory of probability which will have a logical status analogous to that of two valued logic, is closely connected with it, ef. [1]. So far the sentences were used mainly from a "Boolean algebraic" point of view, that is, the operations that were involved were those of the sentential calculus. (The work of Carnap and his collaborators does, however, touch on probabilities which are defined for special cases of first order monadic sentences.)A measure on a sentential calculus which assigns real values to sentences is essentially the same as a measure on the Lindenbaum-Tarski algebra of that calculus, thus its investigation falls under the study of measures on Boolean algebras. These were studied quite a lot; see [3,5] were other references are given.In this work the notions of a measure on a first order calculus, and of a measuremodel, are introduced and investigated. This is done not from a point of view concerning the foundations of probability but with an eye to mathematical logic and measure theory; the concepts with which we shall deal form a natural generalization of the concepts of a theory and a model in the usual sense.In §1 the notion of a measure on a first order calculus is introduced. In §2 the notion of a measure-model is defined and a theorem analogous to the completeness theorem is proved. In §3 the case of a calculus with an equality is treated.§4 is concerned with measure-models in which the measure is invariant under permutations of the individuals, and §5 contains a specific example of such a model. Whereas the propositions of § §1, 2 are analogous to similar ones concerning theories and models (in the usual sense), § §4, 5 deal with situations which are typical to measures and measure-models and have no analogous counterpart.Received June 3, 1964. * The basic definitions and concepts of this paper were first presented by the author in a contributed paper to the 1960 Congress of Logic and Methodology of Science which took place at Stanford [2]. The paper contained part of the results appearing here. Other results, unpublished yet, were obtained since then by Ryll-Nardzewski, and presented by him in a t~k given at the International Symposium of Model Theory, 1963, which took place at Be~kel~,.

The basic concept underlying probability theory and statistics is a function assigning numerical values (probabilities) to events. An “event” in this context is any conceivable state of affairs including the so-called “empty event”—an a priori impossible state. Informally, events are described in everyday language (e.g. “by playing this strategy I shall win $1000 before going broke”). But in the current mathematical framework (first proposed by Kolmogoroff [Ko 1]) they are identified with subsets of some all-inclusive set Q. The family of all events constitutes a field, or σ-field, and the logical connectives ‘and’, ‘or’ and ‘not’ are translated into the set-theoretical operations of intersection, union and complementation. The points of Q can be regarded as possible worlds and an event as the set of all worlds in which it takes place. The concept of a field of sets is wide enough to accommodate all cases and to allow for a general abstract foundation of the theory. On the other hand it does not reflect distinctions that arise out of the linguistic structure which goes into the description of our events. Since events are always described in some language they can be indentified with the sentences that describe them and the probability function can be regarded as an assignment of values to sentences. The extensive accumulated knowledge concerning formal languages makes such a project feasible. The study of probability functions defined over the sentences of a rich enough formal language yields interesting insights in more than one direction.Our present approach is not an alternative to the accepted Kolmogoroff axiomatics. In fact, given some formal language L, we can consider a rich enough set, say Q, of models for L (called also in this work “worlds”) and we can associate with every sentence the set of all worlds in Q in which the sentence is true. Thus our probabilities can be considered also as measures over some field of sets. But the introduction of the language adds mathematical structure and makes for distinctions expressing basic intuitions that cannot be otherwise expressed. As an example we mention here the concept of a random sequence or, more generally, a random world, or a world which is typical to a certain probability distribution.