Finite Model Theory Research Articles

UNARY FA-PRESENTABLE SEMIGROUPS

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classification of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a difficult problem in general. A restricted problem, also of significant interest, is to ask this question for unary automatic presentations: automatic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally finite, but non-finitely generated unary FA-presentable semigroups may be infinite. Every unary FA-presentable semigroup satisfies some Burnside identity. We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classification is given of the unary FA-presentable completely simple semigroups.

Highly acyclic groups, hypergraph covers, and the guarded fragment

We construct finite groups whose Cayley graphs have large girth even with respect to a discounted distance measure that contracts arbitrarily long sequences of edges from the same color class (subgroup), and only counts transitions between color classes (cosets). These groups are shown to be useful in the construction of finite bisimilar hypergraph covers that avoid any small cyclic configurations. We present two applications to the finite model theory of the guarded fragment: a strengthening of the known finite model property for GF and the characterization of GF as the guarded bisimulation invariant fragment of first-order logic in the sense of finite model theory.

On vectorizations of unary generalized quantifiers

Vectorization of a class of structures is a natural notion in finite model theory. Roughly speaking, vectorizations allow tuples to be treated similarly to elements of structures. The importance of vectorizations is highlighted by the fact that if the complexity class PTIME corresponds to a logic with reasonable syntax, then it corresponds to a logic generated via vectorizations by a single generalized quantifier (Dawar in J Log Comput 5(2):213---226, 1995). It is somewhat surprising, then, that there have been few systematic studies of the expressive power of vectorizations of various quantifiers. In the present paper, we consider the simplest case: the cardinality quantifiers C S . We show that, in general, the expressive power of the vectorized quantifier logic $${{\rm FO}(\{{\mathsf C}_S^{(n)}\, | \, n \in \mathbb{Z}_+\})}$$ is much greater than the expressive power of the non-vectorized logic FO(C S ).

Randomisation and Derandomisation in Descriptive Complexity Theory

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures.

Safety Evaluation of Marine Derrick Steel Structures Based on Dynamic Measurement and Updated Finite Element Model

In order to forecast the safety loading capacity of marine derrick steel structures, a novel method is proposed, which bases on modal parameters identified from vibration measurement and updated finite element model with optimization, and takes the safety assessment theory as judgment criterion. Firstly, according to the design feature and the shock excitation mode of marine derrick steel structures, the formulas are deduced for identifying modal parameters. Secondly, taking the frequency and modal shape as the key indicators and the relevant design parameters as the updating objects, the dynamic model for the updating finite element model theory is established with optimization. Thirdly, safety judgment criterion is obtained based on formerly criterion and specification. And then, the modal test of marine derrick steel structures is carried out using ocean wave stationary random motivation. The first left to right and the first front to rear natural frequencies and mode shapes are identified. At last, safety evaluation is carried out on the marine derrick steel structures.

Stanley Peters and Dag Westerståhl: Quantifiers in language and logic

Quantifiers in Language and Logic (QLL) is a major contribution to natural language semantics, specifically to quantification. It integrates the extensive recent work on quantifiers in logic and linguistics. It also presents new observations and results. QLL should help linguists understand the mathematical generalizations we can make about natural language quantification, and it should interest logicians by presenting an extensive array of quantifiers that lie beyond the pale of classical logic. Here we focus on those aspects of QLL we judge to be of specific interest to linguists, and we contribute a few musings of our own, as one mark of a worthy publication is whether it stimulates the reader to seek out new observations, and QLL does. QLL is long and fairly dense, so we make no attempt to cover all the points it makes. But QLL has a topic index, a special symbols index and two tables of contents, a detailed one and an overview one, all of which help make it user friendly. QLL is presented in four parts: I, “The Logical Conception of Quantifiers and Quantification” with an introductory section “Quantification”. II, “Quantifiers of Natural Language”, the most extensive section in the book and of the most direct interest to linguists. III, “Beginnings of a Theory of Expressiveness, Translation, and Formalization” introduces notions of expressive power and definability, and IV, presents recent work and techniques concerning quantifier definability over finite domains, making accessible to linguists recent work in finite model theory.

E. Grädel, P.G. Kolaitis, L. Libkin, M. Marx, J. Spencer, M.Y. Vardi, Y. Venema and S. Weinstein. Finite model theory and its applications. Texts in Theoretical Computer Science. Springer, Berlin, 2007, xiii + 437 pp.

E. Grädel, P.G. Kolaitis, L. Libkin, M. Marx, J. Spencer, M.Y. Vardi, Y. Venema and S. Weinstein. Finite model theory and its applications. Texts in Theoretical Computer Science. Springer, Berlin, 2007, xiii + 437 pp. - Volume 16 Issue 3

First order properties on nowhere dense structures

AbstractA set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

Expressing versus Proving: Relating Forms of Complexity in Logic

Complexity in logic comes in many forms. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here, we consider several notions of complexity in logic, the connections among them and their relationship with computational complexity. In particular, we show how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic. For example, the transitive closure function (testing reachability between two given points in a directed graph) is definable using only NL-concepts (where NL is the non-deterministic logspace complexity class) and its totality (and, thus, the closure of NL under complementation) is provable within NL-reasoning. Lastly, we will touch upon the topic of formalizing complexity theory using logic, and the meta-question of complexity of logical reasoning about complexity-theoretic statements. This is intended to be a high-level overview, suitable for readers who are not familiar with complexity theory and complexity in logic.

Recursive Definitions and Fixed-Points

An expression such as @?x(P(x)@[email protected](P)), where P occurs in @f(P), does not always define P. When such expression implicitly defines P, in the sense of Beth [Beth, E. W., On Padoa's method in the theory of definitions., Indagationes Mathematicae 15 (1953), pp. 330-339] and Padoa [Padoa, A., Essai d'une theorie algebrique des nombres entiers, precede d'une introduction logique a une theorie deductive quelconque, in: Bibliotheque Du Congres International de Philosophie, 1900, pp. 118-123], we call it a recursive definition. In the Least Fixed-Point Logic (LFP), we have theories where interesting relations can be recursively defined [Ebbinghaus, H.-D. and J. Flum, ''Finite Model Theory,'' Springer-Verlag, 1995; Libkin, L., ''Elements of Finite Model Theory,'' Springer, 2004]. We will show that for some sorts of recursive definitions there are explicit definitions on sufficiently strong theories of LFP. It is known that LFP, restricted to finite models, does not have Beth's Definability Theorem [Gurevich, Y. and S. Shelah, On finite rigid structures, Journal of Symbolic Logic 61 (1996), pp. 549-562; Hodkinson, I.M., Finite variable logics, Bulletin of the EATCS 51 (1993), pp. 111-140; Dawar, A., L. Hella and P.G. Kolaitis, Implicit definability and infinitary logic in finite model theory, in: ICALP '95: Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (1995), pp. 624-635]. Beth's Definability Theorem states that, if a relation is implicitly defined, then there is an explicit definition for it. We will also give a proof that Beth's Definability Theorem fails for LFP without this finite model restriction. We intend to investigate fragments of LFP for which Beth's Definability Theorem holds.

Modal characterisation theorems over special classes of frames

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem’s theorem, which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes–for instance, rooted frames, finite rooted frames, finite transitive frames, well-founded transitive frames, finite equivalence frames–as these are not elementary. Instead we develop and extend the game-based analysis (first-order Ehrenfeucht–Fraïssé versus bisimulation games) over such classes and provide bisimulation preserving model constructions within these classes. Over most of the classes considered, we obtain finite model theory analogues of the classically expected characterisations, with new proofs also for the classical setting. The class of transitive frames is a notable exception, with a marked difference between the classical and the finite model theory of bisimulation invariant first-order properties. Over the class of all finite transitive frames in particular, we find that monadic second-order logic is no more expressive than first-order as far as bisimulation invariant properties are concerned — though both are more expressive here than basic modal logic. We obtain ramifications of the de Jongh–Sambin theorem and a new and specific analogue of the Janin–Walukiewicz characterisation of bisimulation invariant monadic second-order for finite transitive frames.

Homomorphism preservation theorems

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 .

An Infinitary System for the Least Fixed-Point Logic restricted to Finite Models

The notion of the least fixed-point of an operator is widely applied in computer science as, for instance, in the context of query languages for relational databases. Some extensions of first-order classical logic (FOL) with fixed-point operators, as the least fixed-point logic (LFP), were proposed to deal with problems related to the expressivity of FOL. LFP captures the complexity class PTIME over the class of finite ordered structures. The descriptive characterization of computational classes is a central issue within finite model theory (FMT). Trakhtenbrot's theorem states that validity over finite models is not recursively enumerable, that is, completeness fails over finite models. This result is based on an underlying assumption that any deductive system is of finite nature. However, we can relax such assumption as done in the scope of proof theory for arithmetic. Motivated by Gödel incompleteness theorems, proof theory for arithmetic offer an example of a true mathematically meaningful principle non derivable in first-order arithmetic. One way of presenting this proof is based on a definition of a proof system with an infinitary rule, the ω-rule, that establishes the consistency of first-order arithmetic through a proof-theoretical perspective. Inspired by this proof, here we will propose an infinitary natural deduction system for FOL and LFP restricted to finite models, FOLfin and LFPfin, respectively, and we will prove soundness and completeness for them, and also a normalization theorem for fragments of these systems. With this infinitary deductive system for LFPfin, we aim to present a proof theory for a logic traditionally investigated within the scope of FMT. It opens up an alternative way of proving results already obtained within FMT and also new results through a proof theoretical perspective.

A combinatorial characterization of resolution width

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the relationship between size and width cannot be made subpolynomial.

Definability of Languages by Generalized First-Order Formulas over $(\mathbb{N},+)$

We consider an extension of first-order logic by modular quantifiers of a fixed modulus $q$. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1's is divisible by a prime $p$ that does not divide $q$. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class $ACC$ from $NC^1$. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.

The metamathematics of random graphs

We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work connects developments in stability theory, finite model theory, abstract model theory, and probability. We conclude by linking this context with work on the Urysohn space.

A case study in pathway knowledgebase verification

BackgroundBiological databases and pathway knowledgebases are proliferating rapidly. We are developing software tools for computer-aided hypothesis design and evaluation, and we would like our tools to take advantage of the information stored in these repositories. But before we can reliably use a pathway knowledgebase as a data source, we need to proofread it to ensure that it can fully support computer-aided information integration and inference.ResultsWe design a series of logical tests to detect potential problems we might encounter using a particular knowledgebase, the Reactome database, with a particular computer-aided hypothesis evaluation tool, HyBrow. We develop an explicit formal language from the language implicit in the Reactome data format and specify a logic to evaluate models expressed using this language. We use the formalism of finite model theory in this work. We then use this logic to formulate tests for desirable properties (such as completeness, consistency, and well-formedness) for pathways stored in Reactome.We apply these tests to the publicly available Reactome releases (releases 10 through 14) and compare the results, which highlight Reactome's steady improvement in terms of decreasing inconsistencies. We also investigate and discuss Reactome's potential for supporting computer-aided inference tools.ConclusionThe case study described in this work demonstrates that it is possible to use our model theory based approach to identify problems one might encounter using a knowledgebase to support hypothesis evaluation tools. The methodology we use is general and is in no way restricted to the specific knowledgebase employed in this case study. Future application of this methodology will enable us to compare pathway resources with respect to the generic properties such resources will need to possess if they are to support automated reasoning.

Locality of Queries and Transformations

Locality is a standard notion of finite model theory. There are two well known flavors of it, based on Hanf's and Gaifman's theorems. Essentially they say that structures that locally look alike cannot be distinguished by first-order sentences. Very recently these standard notions have been generalized in two ways. The first extension makes the notion of “looking alike” depend on logical indistinguishability, rather than isomorphism, of local neighborhoods. The second extension considers transformations defined by FO formulae, and requires that small neighborhoods be preserved by those transformations. In this survey we explain these new notions – as well as the standard ones – and show how they behave with respect to Hanf's and Gaifman's conditions.

The Theory of Finite Models without Equal Sign

In this paper, it is the first time ever to suggest that we study the model theory of all finite structures and to put the equal sign in the same situtation as the other relations. Using formulas of infinite lengths we obtain new theorems for the preservation of model extensions, submodels, model homomorphisms and inverse homomorphisms. These kinds of theorems were discussed in Chang and Keisler's Model Theory, systematically for general models, but Gurevich obtained some different theorems in this direction for finite models. In our paper the old theorems manage to survive in the finite model theory. There are some differences between into homomorphisms and onto homomorphisms in preservation theorems too. We also study reduced models and minimum models. The characterization sentence of a model is given, which derives a general result for any theory T to be equivalent to a set of existential–universal sentences. Some results about completeness and model completeness are also given.

Out of order quantifier elimination for Standard Quantified Linear Programs

In this paper, we present an out of order quantifier elimination algorithm for a class of Quantified Linear Programs (QLPs) called Standard Quantified Linear Programs (SQLPs). QLPs in general and SQLPs in particular are extremely useful constraint logic programming structures that find wide applicability in the modeling of real-time schedulability specifications; see Subramani [Subramani, K., 2005a. A comprehensive framework for specifying clairvoyance, constraints and periodicity in real-time scheduling. The Computer Journal 48 (3), 259–272]. Consequently any algorithmic advance in their solution has a strong practical impact. Prior to this work, the only known approaches to the solution of QLPs involved sequential variable elimination; see Subramani [Subramani, K., 2003b. An analysis of quantified linear programs. In: Calude, C.S. et al. (Eds.), Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science. DMTCS. In: Lecture Notes in Computer Science, vol. 2731. Springer-Verlag, pp. 265–277]. In the sequential approach, the innermost quantified variable is eliminated first, followed by the variable which then becomes the innermost quantified variable and so on, until we are left with a single variable from which the satisfiability of the original formula is easily deduced. This approach is applicable in both discrete and continuous domains; however, it is to be noted that the logic demanding the sequential approach requires that the variables are discrete-valued. To the best of our knowledge, the necessity for sequential elimination over continuous-valued variables has not been investigated in the literature. The techniques used in the development of our elimination algorithm may find applications in domains such as classical logic and finite model theory. The final aspect of our research concerns the structure-preserving nature of the algorithm that we introduce here; in general, it is not known whether discrete domains admit such elimination procedures.