Model-theoretic Point Of View Research Articles

Generic length functions on countable groups

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

Existence of log canonical modifications and its applications

The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some suitable assumptions. Then we recover Kawakita’s inversion of adjunction on log canonicity in full generality. We also discuss the existence of semi-log canonical modifications for demi-normal pairs and construct dlt blow-ups with several extra good properties. As an application, we study lengths of extremal rational curves.

The model theory of commutative near-vector spaces

In this paper we study near-vector spaces over a commutative F from a model theoretic point of view. In this context we show regular near-vector spaces are in fact vector spaces. We find that near-vector spaces are not first-order axiomatisable, but that finite block near-vector spaces are. In the latter case we establish quantifier elimination, and that the theory is “controlled” by which elements of the pointwise additive closure of F are automorphisms of the near-vector space.

A logician's view of graph polynomials

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. Graph polynomials appear in the literature either as generating functions, as generalized chromatic polynomials, or as polynomials derived via determinants of adjacency or Laplacian matrices. We show that these forms are mutually incomparable, and propose a unified framework based on definability in Second Order Logic. We show that this comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view. It gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature.

HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY

AbstractThe article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations.

Belief revision, minimal change and relaxation: A general framework based on satisfaction systems, and applications to description logics

Belief revision of knowledge bases represented by a set of sentences in a given logic has been extensively studied but for specific logics, mainly propositional, and also recently Horn and description logics. Here, we propose to generalize this operation from a model-theoretic point of view, by defining revision in the abstract model theory of satisfaction systems. In this framework, we generalize to any satisfaction system the characterization of the AGM postulates given by Katsuno and Mendelzon for propositional logic in terms of minimal change among interpretations. In this generalization, the constraint on syntax independence is partially relaxed. Moreover, we study how to define revision, satisfying these weakened AGM postulates, from relaxation notions that have been first introduced in description logics to define dissimilarity measures between concepts, and the consequence of which is to relax the set of models of the old belief until it becomes consistent with the new pieces of knowledge. We show how the proposed general framework can be instantiated in different logics such as propositional, first-order, description and Horn logics. In particular for description logics, we introduce several concrete relaxation operators tailored for the description logic ALC and its fragments EL and ELU, discuss their properties and provide some illustrative examples.

On some dynamical aspects of NIP theories

We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems and also to the zero entropy property of automorphisms. These results give a correspondence between some notions of tameness in model theory and ergodic theory. Moreover, we study the concept of symbolic representation and consider it in some well known mathematical objects such as the circle group, Bohr sets, Sturmian sequences, the structure $$(\mathbb {Z},+,U)$$ , and random graphs with a model theoretic point of view in mind. We establish certain characterizations for stability theoretic dividing lines, such as independence property, order property and strict order property in terms of associated symbolic representations. At the end, we propose some applications of symbolic representations and these characterizations by giving a proof for a classical theorem by Shelah and also introducing some invariants associated to the types and elements of models.

Transitivity degrees of countable groups and acylindrical hyperbolicity

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely $1$ and $\infty$. Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group $G$ admitting a highly transitive faithful action, we prove the following dichotomy: Either $G$ contains a normal subgroup isomorphic to the infinite alternating group or $G$ resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.

Classical realizability and arithmetical formulæ

In this paper, we treat the specification problem in Krivine classical realizability (Krivine 2009Panoramas et synthèses27), in the case of arithmetical formulæ. In the continuity of previous works from Miquel and the first author (Guillermo 2008Jeux de réalisabilité en arithmétique classique, Ph.D. thesis, Université Paris 7; Guillermo and Miquel 2014Mathematical Structures in Computer Science, Epub ahead of print), we characterize the universal realizers of a formula as being the winning strategies for a game (defined according to the formula). In the first sections, we recall the definition of classical realizability, as well as a few technical results. In Section 5, we introduce in more details the specification problem and the intuition of the game-theoretic point of view we adopt later. We first present a game1, that we prove to be adequate and complete if the language contains no instructions ‘quote’ (Krivine 2003Theoretical Computer Science308259–276), using interaction constants to do substitution over execution threads. We then show that as soon as the language contain ‘Quote,’ the game is no more complete, and present a second game2that is both adequate and complete in the general case. In the last Section, we draw attention to a model-theoretic point of view and use our specification result to show that arithmetical formulæ are absolute for realizability models.

Constructing regular ultrafilters from a model-theoretic point of view

This paper contributes to the set-theoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality l c f ( ℵ 0 , D ) \mathrm {lcf}(\aleph _0, \mathcal {D}) of ℵ 0 \aleph _0 modulo D \mathcal {D} , saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters and thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming κ > ℵ 0 \kappa > \aleph _0 is measurable, we construct a regular ultrafilter on λ ≥ 2 κ \lambda \geq 2^\kappa which is flexible but not good, and which moreover has large l c f ( ℵ 0 ) \mathrm {lcf}(\aleph _0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, OK need not imply good, addressing a problem from Dow (1985). Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for D \mathcal {D} regular on κ \kappa and M M a model of an unstable theory, M κ / D M^\kappa /\mathcal {D} is not ( 2 κ ) + (2^\kappa )^+ -saturated; and for M M a model of a non-simple theory and λ = λ > λ \lambda = \lambda ^{>\lambda } , M λ / D M^\lambda /\mathcal {D} is not λ + + \lambda ^{++} -saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler’s order, and by recent work of the authors on S O P 2 SOP_2 . We prove that if D \mathcal {D} is a κ \kappa -complete ultrafilter on κ \kappa , any ultrapower of a sufficiently saturated model of linear order will have no ( κ , κ ) (\kappa , \kappa ) -cuts, and that if D \mathcal {D} is also normal, it will have a ( κ + , κ + ) (\kappa ^+, \kappa ^+) -cut. We apply this to prove that for any n > ω n > \omega , assuming the existence of n n measurable cardinals below λ \lambda , there is a regular ultrafilter D D on λ \lambda such that any D D -ultrapower of a model of linear order will have n n alternations of cuts, as defined below. Moreover, D D will λ + \lambda ^+ -saturate all stable theories but will not ( 2 κ ) + (2^{\kappa })^+ -saturate any unstable theory, where κ \kappa is the smallest measurable cardinal used in the construction.

Establishing and preserving protocol security goals

We take a model-theoretic viewpoint on security goals and how to establish them. The models are (possibly fragmentary) executions. Security goals such as authentication and confidentiality are geometric sequents, i.e. implications Φ → Ψ where Φ and Ψ

Constraint satisfaction tractability from semi-lattice operations on infinite sets

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.

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima's representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.

Grades of Probability Modality in the Law of Evidence

The paper presents an infinite hierarchy PR m [m = 1, 2, . . . ] of sound and complete axiomatic systems for modal logic with graded probabilistic modalities, which are to reflect what I have elsewhere called the Bolding-Ekelof degrees of evidential strength as applied to the establishment of matters of fact in law-courts. Our present approach is seen to differ from earlier work by the author in that it treats the logic of these graded modalities not only from a semantical or model-theoretic viewpoint but from a prooftheoretical and axiomatic stance as well. A paramount feature of the approach is its use of so-called systematic frame constants as labels of diverse grades of probability. Apart from this novel feature our approach can be seen to go back to pioneering work by Lou Goble in 1970.

Implicit function theorem over free groups

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lifting solutions of equations into generic points. From the model theoretic view-point we claim the existence of simple Skolem functions for particular ∀∃-formulas over free groups. Proving these theorems we describe in details a new version of the Makanin–Razborov process for solving equations in free groups. We also prove a weak version of the implicit function theorem for NTQ systems which is one of the key results in the solution of the Tarski's problems about the elementary theory of a free group. We call it the parametrization theorem.

Constrained abstract representation problems in semigroups and partial groupoids

In this paper dieren t constrained abstract representa- tion theorems for partial groupoids and semigroups are proved. Methods for improving the retract properties of the structures are also developed. These have strong class-theoretical implications for many types of gener- alized periodic semigroups and related partial semigroups. The results are signican t in a model-theoretical setting. In this paper dieren t new methods for improving the retract properties of semigroups and partial semigroups are developed. The results are of much signicance in model-theoretical settings too. Given an ideal or a generalised ideal in a semigroup and two or more subsemigroups subject to certain restric- tions, new semigroups are derived over the same base set via multiple derived partial semigroups. The method is signican t from the classication and em- beddability viewpoints too. The connections between the retract properties of the structures is then shown to be signican t. The results are extended to CSM-partial semigroups -these being dened by the constraints on their process of generation (12). Within the classes of semigroups/partial semigroups for which the method is denable a classication theory appears possible. This directly relates to the automorphism group of the structures. New class-closure operators are introduced in the light of the above. From the model-theoretic viewpoint

Some characterization theorems for infinitary universal Horn logic without equality

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.

The Theory of $\kappa$-like Models of Arithmetic

A model $(M,<,\ldots)$ is said to be $\kappa$-like if $\mbox{card $\;$} M =\kappa$ but for all $a\in M$, $\mbox{card}\{x \in M: x< a\}< \kappa$. In this paper, we shall study sentences true in $\kappa$-like models of arithmetic, especially in the cases when $\kappa$ is singular. In particular, we identify axiom schemes true in such models which are particularly `natural' from a combinatorial or model-theoretic point of view and investigate the properties of models of these schemes.

The s-semantics approach: Theory and applications

This paper is a general overview of an approach to the semantics of logic programs whose aim is to find notions of models which really capture the operational semantics, and are, therefore, useful for defining program equivalences and for semantics-based program analysis. The approach leads to the introduction of extended interpretations which are more expressive than Herbrand interpretations. The semantics in terms of extended interpretations can be obtained as a result of both an operational (top-down) and a fixpoint (bottom-up) construction. It can also be characterized from the model-theoretic viewpoint, by defining a set of extended models which contains standard Herbrand models. We discuss the original construction modeling computed answer substitutions, its compositional version, and various semantics modeling more concrete observables. We then show how the approach can be applied to several extensions of positive logic programs. We finally consider some applications, mainly in the area of semantics-based program transformation and analysis.

Abelian-by-G Groups, forG Finite, from the Model Theoretic Point of View

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ℤ[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree. Mathematics Subject Classification: 03C60, 03B25.