A de nability theorem for rst order logic (original) (raw)
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A definability theorem for first order logic
1997
In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a | M |= ϕ(a)} defined by some formula ϕ) is invariant under all automorphisms of M. The same is of course true for subsets of M n defined by formulas with n free variables. Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M , in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L. Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models. The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models. * Both authors acknowledge support from the Netherlands Science Organisation (NWO).
On the Finite Model Property in Order-Sorted Logic
cs.wpi.edu
The Schoenfinkel-Bernays-Ramsey class is a fragment of first-order logic with the Finite Model Property: a sentence in this class is satisfiable if and only if it is satisfied in a finite model. Since an upper bound on the size of such a model is computable from the sentence, the satisfiability problem for this family is decidable. Sentences in this form arise naturally in a variety of application areas, and several popular reasoning tools explicitly target this class. Others have observed that the class of sentences for which such a finite model theorem holds is richer in a many-sorted framework than in the one-sorted case. This paper makes a systematic study of this phenomenon in the general setting of order-sorted logic supporting overloading and empty sorts. We establish a syntactic condition generalizing the Schoenfinkel-Bernays-Ramsey form that ensures the Finite Model Property. We give a linear-time algorithm for deciding this condition and a polynomial-time algorithm for computing the bound on model sizes. As a consequence, model-finding is a complete decision procedure for sentences in this class. Our algorithms have been incorporated into Margrave, a tool for analysis of access-control and firewall policies, and are available in a standalone application suitable for analyzing input to the Alloy model finder.
On orderings of the family of all logics
Archive for Mathematical Logic, 1980
The purpose of this paper is to examine the structural complexity of the sublogic relation between abstract logics. Let ~(") denote n 'h order logic. Then cp(,) is a proper subtogic of A °("÷ 1~ for each n < co, and we have the chain ~(1)<~(2)< ... <~cp(,)< .... The question naturally arises, what other kinds of chains or partial orderings we can have among sufficiently regular abstract logics. Can one have a family of logics ordered in the order type of the rationals (or perhaps the reals)? In Chapter 2 we prove that any distributive lattice, and therefore any partial ordering, can be embedded into the sublogic relation among what we call normal logics. Chapter 3 is devoted to a sublogic relation defined using PC-classes rather than elementary classes. In the last chapter we restrict ourselves to the very special logics of the form ~(Q1,-, Q,)-The situation becomes more problematic but we can still prove that any countable partial ordering is embeddable into the sublogic relation of these logics. We confine ourselves to single-sorted structures, but many of the results are equally true of many-sorted structures.
Lecture Notes in Computer Science, 1996
This paper proposes a simple, set-theoretic framework providing expressive typing, higher-order functions and initial models at the same time. Building upon Russell's ramified theory of types, we develop the theory of R n-logics, which are axiomatisable by an order-sorted equational Horn logic with a membership predicate, and of G n-logics, that provide in addition partial functions. The latter are therefore more adapted to the use in the program specification domain, while sharing interesting properties, like existence of an initial model, with R n-logics. Operational semantics of R n-/G n-logics presentations is obtained through order-sorted conditional rewriting. 1 Motivations The general goal of this work is to give a simple, set-theoretic framework providing expressive typing, higher-order functions and initial models at the same time. The decision to use set-theoretic interpretations is taken mainly because of their simplicity, and their intuitive appeal for formal software specification. Higherorder functions and highly expressive types including polymorphism, dependent and higher-order types are frequently used concepts that we want to handle in a uniform way. We also want to provide a concise and sufficiently simple deduction system, easily implemented by rewriting. Algebraic specification techniques model types as sets and subtypes as subsets, called sorts and subsorts, respectively. However, in conventional algebraic frameworks, sort expressions are generally restricted to constants, functions are first-order and sort assertions are static and unconditional. From the algebraic approach, we want to keep the initial semantics which provides a unique model up to isomorphism for classes of models, and rewrite techniques for operational semantics. Defining function graphs as sets of argument/value pairs for each function is a classical set-theoretic technique to give semantics to functions. This is the case in Russell's ramified theory of types from which we started our work. However, self-applicable functions, as in the untyped λ-calculus, are not possible in that theory; our approach will include references to sets (i.e. names) in order to cope with this problem.
Normalizable linear orders and generic computations in finite models
Archive for Mathematical Logic, 1999
Numerous results about capturing complexity classes of queries by means of logical languages work for ordered structures only, and deal with non-generic, or order-dependent, queries. Recent attempts to improve the situation by characterizing wide classes of finite models where linear order is definable by certain simple means have not been very promising, as certain commonly believed conjectures were recently refuted (Dawar's Conjecture). We take on another approach that has to do with normalization of a given order (rather than with defining a linear order from scratch). To this end, we show that normalizability of linear order is a strictly weaker condition than definability (say, in the least fixpoint logic), and still allows for extending Immerman-Vardi-style results to generic queries. It seems to be the weakest such condition. We then conjecture that linear order is normalizable in the least fixpoint logic for any finitely axiomatizable class of rigid structures. Truth of this conjecture, which is a strengthened version of Stolboushkin's conjecture, would have the same practical implications as Dawar's Conjecture. Finally, we suggest a series of reductions of the two conjectures to specialized classes of graphs, which we believe should simplify further work.
Chapter XVII: Set-Theoretic Definability of Logics
1985
model theory is the attempt to systematize the study of logics by studying the relationships between them and between various of their properties. The perspective taken in abstract model theory is discussed in Section 2 of Chapter I. The basic definitions and results of the subject were presented in Part A. Other results are scattered throughout the book. This final part of the book is devoted to more advanced topics in abstract model theory.
On Provability Logics with Linearly Ordered Modalities
Studia Logica, 2014
We introduce the logics GLP Λ , a generalization of Japaridze's polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variablefree fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP Λ .