Finite Model Theory Research Articles

Game Comonads & Generalised Quantifiers

Game comonads, introduced by Abramsky, Dawar and Wang and developed by Abramsky and Shah, give an interesting categorical semantics to some Spoiler-Duplicator games that are common in finite model theory. In particular they expose connections between one-sided and two-sided games, and parameters such as treewidth and treedepth and corresponding notions of decomposition. In the present paper, we expand the realm of game comonads to logics with generalised quantifiers. In particular, we introduce a comonad graded by two parameters $n \leq k$ such that isomorphisms in the resulting Kleisli category are exactly Duplicator winning strategies in Hella's $n$-bijection game with $k$ pebbles. We define a one-sided version of this game which allows us to provide a categorical semantics for a number of logics with generalised quantifiers. We also give a novel notion of tree decomposition that emerges from the construction.

The generalised distribution semantics and projective families of distributions

We generalise the distribution semantics underpinning probabilistic logic programming by distilling its essential concept, the separation of a free random component and a deterministic part. This abstracts the core ideas beyond logic programming as such to encompass frameworks from probabilistic databases, probabilistic finite model theory and discrete lifted Bayesian networks. To demonstrate the usefulness of such a general approach, we completely characterise the projective families of distributions representable in the generalised distribution semantics and we demonstrate both that large classes of interesting projective families cannot be represented in a generalised distribution semantics and that already a very limited fragment of logic programming (acyclic determinate logic programs) in the deterministic part suffices to represent all those projective families that are representable in the generalised distribution semantics at all.

A Short Note on the Early History of the Spectrum Problem and Finite Model Theory

Finite model theory is currently not one of the hot topics in the philosophy and history of mathematics, not even in the philosophy and history of mathematical logic. The philosophy of mathematics and mathematical logic has concentrated on infinite structures, closely related to foundational issues. In that context, finite models deserved only marginal attention because it was taken for granted that the study of finite structures is trivial compared to the study of infinite structures. In retrospect, research on finite structures turned out to be neither trivial nor irrelevant but had an important impact on theoretical computer science. A closer look at the early history of the spectrum problem shows that the birth of finite model theory has roots in formal logic, at a time when computer science did not yet exist.

Twin-width IV: ordered graphs and matrices

We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory. This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. ’06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles the small conjecture [SODA ’21] in the case of ordered graphs.

Report on Logic in Computer Science (LICS'23)

Th 38th Annual ACM/ IEEE Symposium on Logic in Computer Science (LICS), took place at Boston University, Boston, USA, from June 26 to June 29, 2023, with co-located events taking place on June 24 and June 25. Five workshops were co-located with LICS 2023: combinatorial games in finite model theory, the decision problem in first order logic, international workshop on quantitative logical method, structure meets power, and the logic mentoring workshop.

The umbilical cord of finite model theory

Abstract Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but it didn’t turn out that way. There is one application domain, relational database management, that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.

Arboreal Categories: An Axiomatic Theory of Resources

Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, we introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fra\"iss\'e and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.

Pollution threshold assessment and risk area delineation of heavy metals in soils through the finite mixture distribution model and Bayesian maximum entropy theory

Pollution threshold and high-risk area determination for heavy metals is important for effectively developing pollution control strategies. Based on heavy metal contents in 3627 dense samples, an integrated framework combining the finite mixture distribution model and Bayesian maximum entropy (BME) theory was proposed to assess pollution thresholds, contamination levels and risk areas in an uncertain environment for soil heavy metals. The results showed that the average heavy metal contents were in the order Zn > Cr > Pb > Cu > Ni > As > Cd > Hg, with strong/moderate variation, and the corresponding pollution thresholds were 158.39, 84.29, 47.84, 49.75, 28.95, 18.01, 0.49 and 0.16 mg/kg, respectively. The thresholds were consistently greater than the Zhejiang Province backgrounds but lower than the national risk screening values, except for Cd. Approximately 27.9% of the samples were classified as contaminated at various levels, and they were distributed in the northern, northwestern and eastern regions of the study area. Additionally, 3.73%, 5.34% and 8.22% of the total area were classified as at-risk areas under confidence levels of 95%, 90% and 75%, respectively, through BME theory. The findings provide a reasonable classification system and suggestions for heavy metal pollution management and control.

The seismic damage mechanism of Daliang tunnel by fault dislocation during the 2022 Menyuan Ms6.9 earthquake based on unidirectional velocity pulse input

Fault dislocation may cause severe damage to tunnel structures. A representative example is the severe damage of the Daliang Tunnel caused by the dislocation of the Lenglongling Fault in the 2022 Menyuan, Qinghai, China, Ms6.9 earthquake. The relative displacement on both sides of the fault during an earthquake is often assumed to be a quasi-static dislocation load, ignoring the dynamic effect. In this study, the dislocation load was considered as a unidirectional harmonic velocity pulse and the damage mechanism of the Daliang tunnel was studied. First, the background of this earthquake and the damage of Daliang tunnel is introduced. The characteristics of surface displacement in this earthquake are analyzed by finite fault model and Okada dislocation theory model, and the dislocation vector of Daliang tunnel is calculated. On this basis, a fault-tunnel FEM nonlinear dynamic model is established, in which unidirectional velocity pulse is used as input of dislocation displacement to consider the dynamic effect of fault dislocation. The study found that the Daliang tunnel was located in one of the regions with the largest dislocation displacement in this earthquake, and the tunnel structure was damaged as a result. The analysis of FEM model reveals that when a unidirectional velocity pulse is used as the load, the tunnel damage is consistent with reality, and the conventional method using the quasi-static dislocation as the load only reflects the dislocation effect of the fault displacement and ignores the impact effect. When the dislocation displacement is certain, the duration of the unidirectional velocity pulse is an important parameter; the shorter the duration, the greater the amplitude of unidirectional velocity pulse, resulting in more serious tunnel damage. The tunnel’s failure mechanism near the fault can be divided into dislocation failure and impact failure. The dislocation failure is manifested by non-uniform dislocation displacement along the longitudinal direction of tunnel, which is mainly related to the amount of dislocation. The impact failure is manifested by crushing and invert uplift of the lining in cross-section, and it is related to the pulse duration.

Quillen Model Categories-Based Notions of Locality of Logics over Finite Structures

AbstractLocality is a property of logics, whose origins lie in the works of Hanf and Gaifman, having their utility in the context of finite model theory. Such a property is quite useful in proofs of inexpressibility, but it is also useful in establishing normal forms for logical formulas. There are generally two forms of locality: (i’) if two structures $\mathfrak {A}$ and $\mathfrak {B}$ realize the same multiset of types of neighborhoods of radius d, then they agree on a given sentence $\Phi $ . Here d depends only on $\Phi $ ; (ii’) if the d-neighborhoods of two tuples $\vec {a}_1$ and $\vec {a}_2$ in a structure $\mathfrak {A}$ are isomorphic, then $\mathfrak {A} \models \Phi (\vec {a}_1) \Leftrightarrow \Phi (\vec {a}_2)$ . Again, d depends on $\Phi $ , and not on $\mathfrak {A}$ . Form (i’) originated from Hanf’s works. Form (ii’) came from Gaifman’s theorem. There is no doubt about the usefulness of the notion of locality, which as seen applies to a huge number of situations. However, there is a deficiency in such a notion: all versions of the notion of locality refer to isomorphism of neighborhoods, which is a fairly strong property. For example, where structures simply do not have sufficient isomorphic neighborhoods, versions of the notion of locality obviously cannot be applied. So the question that immediately arises is: would it be possible to weaken such a condition and maintain Hanf/Gaifman-localities? Arenas, Barceló, and Libkin establish a new condition for the notions of locality, weakening the requirement that neighborhoods should be isomorphic, establishing only the condition that they must be indistinguishable in a given logic. That is, instead of requiring $N_d(\vec {a}) \cong N_d(\vec {b})$ , you should only require $N_d(\vec {a}) \equiv _k N_d(\vec {b})$ , for some $k \geq 0$ . Using the fact that logical equivalence is often captured by Ehrenfeucht–Fraïssé games, the authors formulate a game-based framework in which logical equivalence-based locality can be defined. Thus, the notion defined by the authors is that of game-based locality. Although quite promising as well as easy to apply, the game-based framework (used to define locality under logical equivalence) has the following problem: if a logic $\mathcal {L}$ is local (Hanf-, or Gaifman-, or weakly) under isomorphisms, and $\mathcal {L}'$ is a sub-logic of $\mathcal {L}$ , then $\mathcal {L}'$ is local as well. The same, however, is not true for game-based locality: properties of games guaranteeing locality need not be preserved if one passes to weaker games. The question that immediately arises is: is it possible to define the notion of locality under logical equivalence without resorting to game-based frameworks? In this thesis, I present a homotopic variation for locality under logical equivalence, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k.Abstract prepared by Hendrick Maia.E-mail: hendrickmaia@gmail.comURL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334956

Structure and Power: an Emerging Landscape

In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for this work come from Computer Science, but there may also be something of interest for model theorists and other logicians. The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into tree-like forms, allowing natural resource parameters to be assigned to these unfoldings. One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way: the Ehrenfeucht-Fraïssé game; the quantifier rank fragments of first-order logic; the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction of this fragment to the existential positive part, and (iii) the extension with counting quantifiers; and the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth. Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters. This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers. Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages. Examples include semantic characterisation and preservation theorems, and Lovász-type results on counting homomorphisms.

Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens

We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. To showcase an application of Trakhtenbrot's theorem, we continue our reduction chain with a many-one reduction from FSAT to separation logic. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.

Piezoresistive/piezoelectric intrinsic sensing properties of carbon nanotube cement-based smart composite and its electromechanical sensing mechanisms: A review

Abstract Structural health monitoring (SHM) technology based on the mechanical–electrical sensing effect of various intrinsic smart materials has a good application prospect. Carbon nanotube (CNT) has excellent electromechanical properties and hence can be doped into cement by appropriate dispersive means to produce CNT-modified cement-based smart material (CNTCS) with excellent electromechanical (piezoresistive/piezoelectric) capacity. CNTCS can be developed into a static/dynamic intrinsic sensor for SHM after effective packaging and calibration. Based on the characteristics of CNT, the dispersion methods and the dispersity characterization techniques of CNT in the water/cement matrix are summarized, and then the influence laws of various factors on piezoresistive and piezoelectric sensing behaviors of the corresponding CNTCS are also discussed. The full-frequency domain sensing mechanism of CNTCS is analyzed by combining its finite element model and electromechanical coupling theory, and the practicability of applying CNTCS as an SHM static/dynamic intrinsic sensor is further investigated.

A polynomial excluded-minor approximation of treedepth

Treedepth is a minor-monotone graph invariant in the family of “width measures” that includes treewidth and pathwidth. The characterization and approximation of these invariants in terms of excluded minors has been a topic of interest in the study of sparse graphs. A celebrated result of Chekuri and Chuzhoy (2014) shows that treewidth is polynomially approximated by the largest k \times k grid minor in a graph. In this paper, we give an analogous polynomial approximation of treedepth via three distinct obstructions: grids, balanced binary trees, and paths. Namely, we show that there is a constant c such that every graph with treedepth \Omega(k^c) has at least one of the following minors (each of treedepth at least k ): Moreover, given a graph G we can, in randomized polynomial time, find an embedding of one of these minors or conclude that treedepth of G is at most O(k^c) . This result has applications in various settings where bounded treedepth plays a role. In particular, we describe one application in finite model theory, an improved homomorphism preservation theorem over finite structures [Rossman, 2017], which was the original motivation for our investigation of treedepth.

Relating structure and power: Comonadic semantics for computational resources

AbstractCombinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht–Fraïssé games, pebble games and bisimulation games play a central role. We show how each of these types of games can be described in terms of an indexed family of comonads on the category of relational structures and homomorphisms. The index $k$ is a resource parameter that bounds the degree of access to the underlying structure. The coKleisli categories for these comonads can be used to give syntax-free characterizations of a wide range of important logical equivalences. Moreover, the coalgebras for these indexed comonads can be used to characterize key combinatorial parameters: tree depth for the Ehrenfeucht–Fraïssé comonad, tree width for the pebbling comonad and synchronization tree depth for the modal unfolding comonad. These results pave the way for systematic connections between two major branches of the field of logic in computer science, which hitherto have been almost disjoint: categorical semantics and finite and algorithmic model theory.

Some thoughts on knowledge-enhanced machine learning

How can we employ theoretical insights and practical tools from knowledge representation and reasoning to enhance machine learning, and when is it worthwhile to do so? This paper is based on an invited talk delivered at ECSQARU2019 around this question. It emphasizes the knowledge representation and reasoning side of knowledge-enhanced machine learning, looking at a few case studies: the finite model theory of probabilistic languages, the generation of explanations for embeddings, and an “explainable” version of the Winograd Challenge.

A Proof of the Algebraic Tractability Conjecture for Monotone Monadic SNP

The logic MMSNP is a restricted fragment of existential second-order logic which can express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and ...

Classes of graphs with low complexity: The case of classes with bounded linear rankwidth

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths – a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. (1) The number of unlabeled graphs of order n with linear rank-width at most r is at most [(2r+1)(r+1)!2r23r+1]n (2) Graphs with linear rankwidth at most r are linearly χ-bounded. Actually, they have bounded c-chromatic number, meaning that they can be colored with f(r) colors, each color inducing a cograph. (3) To the contrary, based on a Ramsey-like argument, we prove for every proper hereditary family ℱ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in ℱ.From the model theoretical side we obtain the following results: (1) A direct short proof that graphs with linear rankwidth at most r are first-order transductions of linear orders. This result could also be derived from Colcombet’s theorem on first-order transduction of linear orders and the equivalence of linear rankwidth with linear cliquewidth. (2) For a class C with bounded linear rankwidth the following conditions are equivalent: (a) C is stable, (b) C excludes some half-graph as a semi-induced subgraph, (c) C is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

Database Theory Column Report on PODS 2019

The 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS 2019) took place in Amsterdam, The Netherlands, from July 1 to July 3, 2019, in conjunction with the 2019 ACM SIGMOD International Conference on Management of Data. PODS focuses on theoretical aspects of data management, and the co-location with SIGMOD stimulates interaction between theory-oriented and systems-oriented research. Since the first edition of the symposium in 1982, the PODS papers are distinguished by a rigorous approach to widely diverse problems in data management, often bringing to bear techniques from a variety of different areas, including computational logic, finite model theory, computational complexity, algorithm design and analysis, programming languages, and artificial intelligence.

Whither semantics?

We discuss how mathematical semantics has evolved, and suggest some new directions for future work. As an example, we discuss some recent work on encapsulating model comparison games as comonads, in the context of finite model theory.