Homomorphism Preservation Theorems for Many-Valued Structures

The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a first-order formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existential-positive formula. Answering a long-standing question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the Łoś--Tarski theorem and Lyndon's positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential-positive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula of equal quantifier-rank .

Unions of conjunctive queries, also known as select-project-join-union queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive first-order formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that existential positive formulas are the only first-order formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. It is long-standing open problem in finite model theory, however, to determine whether the same homomorphism-preservation result holds in the finite, that is, whether every first-order formula preserved under homomorphisms on finite structures is logically equivalent to an existential positive formula on finite structures. In this paper, we show that the homomorphism-preservation theorem holds for several large classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.

We present some model theoretic results for Lukasiewicz predicate logic by using the methods of continuous model theory developed by Chang and Keisler. We prove compactness theorem with respect to the class of all structures taking values in the Lukasiewicz BL-algebra. We also prove some appropriate preservation theorems concerning universal and inductive theories. Finally, Skolemization and Morleyization in this framework are discussed and some natural examples of fuzzy theories are presented. ajek and Cintula wrote: \In the last few decades many formal systems of fuzzy logics have been developed. Since the main dierences between fuzzy and classical logics lie at propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this paper we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate In that paper the authors proved a generalized completeness result for a wide class of fuzzy predicate logics, namely core fuzzy logics. To do this they used a generalized Henkin method for a through survey of the present status of fuzzy predicate logic, see (3). In the present paper we continue the work on fuzzy predicate logics by developing further model theory of these logics. We choose Lukasiewicz predicate logic. The reason is that the Lukasiewicz BL-algebra is equipped with continuous operations and this is essential in proving the compactness theorem which is a fundamental result in model theory. Here, the compactness theorem says that if each nite subset of a set of sentences has a Lukasiewicz model, then the set itself has a Lukasiewicz model. As consequences of compactness, we prove the most basic model theoretic results including the upward and downward Lowenheim-Skolem theorems. We also prove suitable versions of the most well known preservation theorems, including preservation theorems for universal and inductive theories. At the end, we introduce some fuzzy theories and study their model theoretic properties. To prove the compactness theorem, we use ultraproduct and its basic properties which we rst investigate. We follow the old methods developed by Chang and

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

Unions of conjunctive queries, also known as select-project-join-union queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive first-order formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only first-order formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. The question of whether the homomorphism-preservation theorem holds for the class of all finite structures resisted solution for a long time. It was eventually shown that, unlike other classical preservation theorems, the homomorphism-preservation theorem does hold in the finite. In this article, we show that the homomorphism-preservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.

Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to model theory has proved to be so useful that it has become rare to see more advanced topology in model-theoretic articles. Prof. Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.The first part of the thesis will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces.In a similar spirit, the second part will show a new connection between the notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Dueñez, J. Iovino in a much simplified manner.The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead.The only part that did not originate from model-theoretic questions will be the fourth one. We show that $\operatorname {ZF} + \operatorname {DC} +$ “all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing determinacy implies the axiom of determinacy.Abstract prepared by Clovis HamelE-mail: chamel@math.toronto.edu

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

A model checking computation checks whether a given logical sentence is true in a given finite structure. Provenance analysis abstracts from such a computation mathematical information on how the result depends on the atomic data that describe the structure. In database theory, provenance analysis by interpretations in commutative semirings has been rather succesful for positive query languages (such a unions of conjunctive queries, positive relational algebra, or datalog). However, it did not really offer an adequate treatment of negation or missing information. Here we propose a new approach for the provenance analysis of logics with negation, such as first-order logic and fixed-point logics. It is closely related to a provenance analysis of the associated model-checking games, and based on new semirings of dual-indeterminate polynomials or dual-indeterminate formal power series. These are obtained by taking quotients of traditional provenance semirings by congruences that are generated by products of positive and negative provenance tokens. Beyond the use for model-checking problems in logics, provenance analysis of games is of independent interest. Provenance values in games provide detailed information about the number and properties of the strategies of the players, far beyond the question whether or not a player has a winning strategy from a given position.

A famous result by Jeavons, Cohen, and Gyssens shows that every Constraint Satisfaction Problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the so-called universal algebraic approach to a systematic theory of tractability and hardness in finite domain constraint satisfaction. Not surprisingly, the theorem of Jeavons et al. fails for arbitrary infinite domain CSPs. Many CSPs of practical interest, though, and in particular those CSPs that are motivated by qualitative reasoning calculi from artificial intelligence, can be formulated with constraint languages that are rather well-behaved from a model-theoretic point of view. In particular, the automorphism group of these constraint languages tends to be large in the sense that the number of orbits of n -subsets of the automorphism group is bounded by some function in n . In this article we present a generalization of the theorem by Jeavons et al. to infinite domain CSPs where the number of orbits of n -subsets grows subexponentially in n , and prove that preservation under a semi-lattice operation for such CSPs implies polynomial-time tractability. Unlike the result of Jeavons et al., this includes CSPs that cannot be solved by Datalog.

One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence.In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including L(Q) and L(aa), respectively the languages with the quantifiers “there exist un-countably many” and “for almost all countable subsets”.In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for L(Q). We also consider whether the natural class of “generalized Σ-sentences” in L(Q) characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open.In the third section we show that the analogous class of L(aa)-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3].In the fourth section we show that under rather general conditions, if L is a countably compact extension of first-order logic and T is an ℵ1-categorical first-order theory, then L is trivial relative to T.

This final chapter unites ideas from both model theory and complexity theory. Finite model theory is the part of model theory that disregards infinite structures. Examples of finite structures naturally arise in computer science in the form of databases, models of computations, and graphs. Instead of satisfiability and validity, finite model theory considers the following finite versions of these properties. • A first-order sentence is finitely satisfiable if it has a finite model. • A first-order sentence is finitely valid if every finite structure is a model. Finite model theory developed separately from the “classical” model theory of previous chapters. Distinct methods and logics are used to analyze finite structures. In Section 10.1, we consider various finite-variable logics that serve as useful languages for finite model theory. We define variations of the pebble games introduced in Section 9.2 to analyze the expressive power of these logics. Pebble games are one of the few tools from classical model theory that is useful for investigating finite structures. In Section 10.2, it is shown that many of the theorems from Chapter 4 are no longer true when restricted to finite models. There is no analog for the Completeness and Compactness theorems in finite model theory. Moreover, we prove Trakhtenbrot’s theorem which states that the set of finitely valid first-order sentences is not recursively enumerable. Descriptive complexity is the subject of 10.3. This subject describes the complexity classes discussed in Chapter 7 in terms of the logics introduced in Chapter 9. We prove Fagin’s theorem relating the class NP to existentional second-order logic. We prove the Cook–Levin theorem as a consequence of Fagin’s Theorem. This theorem states that the Satisfiability Problem for Propositional Logic is NP-complete. We conclude this chapter (and this book) with a section describing the close connection between logic and the P = NP problem. In this section, we discuss appropriate logics for the study of finite models. First-order logic, since it describes each finite model up to isomorphism, is too strong. For this reason, we must weaken the logic. It may seem counter-intuitive that we should gain knowledge by weakening our language.

Institutional semantics for many-valued logics

In this paper we mainly study preservation theorems for two fragments of the infinitary languagesLκκ, withκregular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, whenκisω, we obtain the corresponding theorems for the first-order case.The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only oneκ-ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic.

One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the self-embedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Such functions can be particularly well understood when the relational structure is countably infinite and has a first-order definition in another relational structure which has a finite language, is totally ordered and homogeneous, and has the Ramsey property. This is because in this situation, Ramsey theory provides the combinatorial tool for analyzing these functions -- in a certain sense, it allows to represent such functions by functions on finite sets. This is a survey of results in model theory and theoretical computer science obtained recently by the authors in this context. In model theory, we approach the problem of classifying the reducts of countably infinite ordered homogeneous Ramsey structures in a finite language, and certain decidability questions connected with such reducts. In theoretical computer science, we use the same combinatorial methods in order to classify the computational complexity for various classes of infinite-domain constraint satisfaction problems. While the first set of applications is obviously of an infinitary character, the second set concerns genuinely finitary problems -- their unifying feature is that the same tools from Ramsey theory are used in their solution.