Limitations of Game Comonads via Homomorphism Indistinguishability

Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lov\'asz-type theorems for game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.Comment: Minor corrections in Section

The Expressive Power of CSP-Quantifiers

A generalized quantifier QK is called a CSP-quantifier if its defining class K consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n ≥ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic Lk∞ω(CSP+n ), where CSP+n is the union of the class Q1 of all unary quantifiers and the class CSPn of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n ≥ 2 there exists a CSP-quantifier with template of size n + 1 which is not definable in Lω∞ω(CSP+n ). The proof of this result is based on a new variation of the well-known Cai-Fürer-Immerman construction.publishedVersionPeer reviewe

PTIME and Generalized Quantifiers

We consider the problem of finding a reasonable logical characterization for the complexity class PTIME in the class of all finite models. We approach this problem by adding a set Q_n of n-ary generalized quantifiers to the infinitary finite variable logic L^k_{∞ Ω}. More precisely, we show that it is not possible to characterize PTIME in such a way. This result is obtained by constructing models A(G) and B(G) which are L^k_{∞ Ω}(Q_n)-equivalent and by showing that there is a PTIME computable boolean query q such that q(A(G)) \neq q(B(G)) for any appropriate finite graph G.Työssä käsitellään ongelmaa tyydyttävän loogisen karakterisaation löytämiseksi vaativuusluokalle PTIME kaikkien äärellisten mallien luokassa. Tätä ongelmaa lähestytään lisäämällä kaikki n-paikkaiset yleistetyt kvanttorit infinitaariseen äärellisen monen muuttujan logiikkaan L^{Ω}_{∞ Ω}. Työssä todistetaan, ettei ole mahdollista karakterisoida vaativuusluokkaa PTIME kyseisellä tavalla. Tämä tulos saavutetaan konstruoimalla mallit A(G) ja B(G), jotka ovat L^k_{∞ Ω}(Q_n)-ekvivalentteja ja näyttämällä, että on olemassa polynomiaalisessa ajassa laskettava boolen kysely q, jolle pätee q(A(G)) \neq q(B(G)) kaikilla sopivilla äärellisillä verkoilla G

Some observations about generalized quantifiers in logics of imperfect information

We analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engstrom. comparing them with a more general, higher order definition of team quantifier. We show that Engstrom's definitions (and other quantifiers from the literature) can be identified, by means of appropriate lifts, with special classes of team quantifiers. We point out that the new team quantifiers express a quantitative and a qualitative component, while Engstrom's quantifiers only range over the latter. We further argue that Engstrom's definitions are just embeddings of the first-order generalized quantifiers into team semantics. and fail to capture an adequate notion of team-theoretical generalized quantifier, save for the special cases in which the quantifiers are applied to flat formulas. We also raise several doubts concerning the meaningfulness of the monotone/nonmonotone distinction in this context. In the appendix we develop some proof theory for Engstrom's quantifiers.Peer reviewe

Regular Representations of Uniform TC^0

The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact that AC^0 is not closed under restricting AC^0-computable queries into simple subsequences of the input. Analogously, in descriptive complexity, the logics corresponding to DLOGTIME-uniform AC^0 do not have the relativization property and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt and Th{\'e}rien) was that if a language L has a neutral letter, then L can be defined in first-order logic with the collection of all numerical built-in relations, if and only if L can be already defined in FO with order. In the first part of this article we consider logics in the range of AC^0 and TC^0. First we formulate a combinatorial criterion for a cardinality quantifier C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of some polynomial with positive integer coefficients of degree at least two. In the second part of the paper we first adapt the key properties of abstract logics to accommodate built-in relations. Then we define the regular interior R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of first-order logic with built-in relations B. We show that if B={+}, or B contains only unary relations besides the order, then R-int(FO_B) collapses to FO with order. In contrast, our results imply that if B contains the order and the range of a polynomial of degree at least two, then R-cl(FO_B) includes all languages in DLOGTIME-uniform TC^0

Symmetric Circuits and Model-Theoretic Logics

The question of whether there is a logic that characterises polynomial-time is arguably the most important open question in finite model theory. The study of extensions of fixed-point logic are of central importance to this question. It was shown by Anderson and Dawar that fixed-point logic with counting (FPC) has the same expressive power as uniform families of symmetric circuits over a basis with threshold functions. In this thesis we prove a far-reaching generalisation of their result and establish an analogous circuit characterisation for each from a broad range of extensions of fixed-point logic. In order to do so we fist develop a very general framework for defining and studying extensions of fixed-point logics, which we call generalised operators. These operators generalise Lindström quantifiers as well as the counting and rank operators used to define FPC and fixed-point logic with rank (FPR). We also show that in order to define a symmetric circuit model that goes beyond FPC we need to consider circuits with gates that are allowed to compute non-symmetric functions. In order to do so we develop a far more general framework for studying circuits. We also show that key notions, such as the notion of a symmetric circuit, can be analogously defined in this more general framework. The characterisation of FPC in terms of symmetric circuits, and the treatment of circuits generally, relies heavily on the assumption that the gates in the circuit compute symmetric functions. We develop a broad range of new techniques and approaches in order to study these more general symmetric circuit models. As a corollary of our main result we establish a circuit characterisation of FPR. We also show that the question of whether there is a logic that characterises polynomial-time can be understood as a question about the symmetry property of circuits. We lastly propose a number of new approaches that might exploit this new-found connection between circuit complexity and descriptive complexity.Gates Cambridge Scholarship

Modularity in answer set programs

Answer set programming (ASP) is an approach to rule-based constraint programming allowing flexible knowledge representation in variety of application areas. The declarative nature of ASP is reflected in problem solving. First, a programmer writes down a logic program the answer sets of which correspond to the solutions of the problem. The answer sets of the program are then computed using a special purpose search engine, an ASP solver. The development of efficient ASP solvers has enabled the use of answer set programming in various application domains such as planning, product configuration, computer aided verification, and bioinformatics. The topic of this thesis is modularity in answer set programming. While modern programming languages typically provide means to exploit modularity in a number of ways to govern the complexity of programs and their development process, relatively little attention has been paid to modularity in ASP. When designing a module architecture for ASP, it is essential to establish full compositionality of the semantics with respect to the module system. A balance is sought between introducing restrictions that guarantee the compositionality of the semantics and enforce a good programming style in ASP, and avoiding restrictions on the module hierarchy for the sake of flexibility of knowledge representation. To justify a replacement of a module with another, that is, to be able to guarantee that changes made on the level of modules do not alter the semantics of the program when seen as an entity, a notion of equivalence for modules is provided. In close connection with the development of the compositional module architecture, a transformation from verification of equivalence to search for answer sets is developed. The translation-based approach makes it unnecessary to develop a dedicated tool for the equivalence verification task by allowing the direct use of existing ASP solvers. Translations and transformations between different problems, program classes, and formalisms are another central theme in the thesis. To guarantee efficiency and soundness of the translation-based approach, certain syntactical and semantical properties of transformations are desirable, in terms of translation time, solution correspondence between the original and the transformed problem, and locality/globality of a particular transformation. In certain cases a more refined notion of minimality than that inherent in ASP can make program encodings more intuitive. Lifschitz' parallel and prioritized circumscription offer a solution in which certain atoms are allowed to vary or to have fixed values while others are falsified as far as possible according to priority classes. In this thesis a linear and faithful transformation embedding parallel and prioritized circumscription into ASP is provided. This enhances the knowledge representation capabilities of answer set programming by allowing the use of existing ASP solvers for computing parallel and prioritized circumscription

Grambank reveals the importance of genealogical constraints on linguistic diversity and highlights the impact of language loss

While global patterns of human genetic diversity are increasingly well characterized, the diversity of human languages remains less systematically described. Here we outline the Grambank database. With over 400,000 data points and 2,400 languages, Grambank is the largest comparative grammatical database available. The comprehensiveness of Grambank allows us to quantify the relative effects of genealogical inheritance and geographic proximity on the structural diversity of the world's languages, evaluate constraints on linguistic diversity, and identify the world's most unusual languages. An analysis of the consequences of language loss reveals that the reduction in diversity will be strikingly uneven across the major linguistic regions of the world. Without sustained efforts to document and revitalize endangered languages, our linguistic window into human history, cognition and culture will be seriously fragmented.Genealogy versus geography Constraints on grammar Unusual languages Language loss Conclusio