arXiv subjects

Antti Kuusisto

Publications and source records attributed to Antti Kuusisto.

At least 37 records · Page 2Linked to original sources

arXiv2023

Descriptive complexity for distributed computing with circuits

We consider distributed algorithms in the realistic scenario where distributed message passing is operated via circuits. We show that within this setting, modal substitution calculus MSC captures the expressive power of circuits. The translations between circuits and MSC-programs are linear in both directions. Furthermore, we show that the colouring algorithm based on Cole-Vishkin can be specified via logarithmic size programs.

cs.LO↗

arXiv2023

A monotone connection between model class size and description length

This paper links sizes of model classes to the minimum lengths of their defining formulas, that is, to their description complexities. Limiting to models with a fixed domain of size n, we study description complexities with respect to the extension of propositional logic with the ability to count assignments. This logic, called GMLU, can alternatively be conceived as graded modal logic over Kripke models with the universal accessibility relation. While GMLU is expressively complete for defining multisets of assignments, we also investigate its fragments GMLU(d) that can count only up to the integer threshold d. We focus in particular on description complexities of equivalence classes of GMLU(d). We show that, in restriction to a poset of type realizations, the order of the equivalence classes based on size is identical to the order based on description complexities. This also demonstrates a monotone connection between Boltzmann entropies of model classes and description complexities. Furthermore, we characterize how the relation between domain size n and counting threshold d determines whether or not there exists a dominating class, which essentially means a model class with limit probability one. To obtain our results, we prove new estimates on r-associated Stirling numbers. As another crucial tool, we show that model classes split into two distinct cases in relation to their description complexity.

math.LO↗

arXiv2023

Algebraic classifications for fragments of first-order logic and beyond

Complexity and decidability of logics is a major research area involving a huge range of different logical systems. This calls for a unified and systematic approach for the field. We introduce a research program based on an algebraic approach to complexity classifications of fragments of first-order logic (FO) and beyond. Our base system GRA, or general relation algebra, is equiexpressive with FO. It resembles cylindric algebra but employs a finite signature with only seven different operators. We provide a comprehensive classification of the decidability and complexity of the systems obtained by limiting the allowed sets of operators. We also give algebraic characterizations of the best known decidable fragments of FO. Furthermore, to move beyond FO, we introduce the notion of a generalized operator and briefly study related systems.

math.LO↗

arXiv2022

Explainability via Short Formulas: the Case of Propositional Logic with Implementation

We conceptualize explainability in terms of logic and formula size, giving a number of related definitions of explainability in a very general setting. Our main interest is the so-called special explanation problem which aims to explain the truth value of an input formula in an input model. The explanation is a formula of minimal size that (1) agrees with the input formula on the input model and (2) transmits the involved truth value to the input formula globally, i.e., on every model. As an important example case, we study propositional logic in this setting and show that the special explainability problem is complete for the second level of the polynomial hierarchy. We also provide an implementation of this problem in answer set programming and investigate its capacity in relation to explaining answers to the n-queens and dominating set problems.

cs.AI↗

arXiv2022

Relating description complexity to entropy

We demonstrate some novel links between entropy and description complexity, a notion referring to the minimal formula length for specifying given properties. Let MLU be the logic obtained by extending propositional logic with the universal modality, and let GMLU be the corresponding extension with the ability to count. In the finite, MLU is expressively complete for specifying sets of variable assignments, while GMLU is expressively complete for multisets. We show that for MLU, the model classes with maximal Boltzmann entropy are the ones with maximal description complexity. Concerning GMLU, we show that expected Boltzmann entropy is asymptotically equivalent to expected description complexity multiplied by the number of proposition symbols considered. To contrast these results, we show that this link breaks when we move to considering first-order logic FO over vocabularies with higher-arity relations. To establish the aforementioned result, we show that almost all finite models require relatively large FO-formulas to define them. Our results relate to links between Kolmogorov complexity and entropy, demonstrating a way to conceive such results in the logic-based scenario where relational structures are classified by formulas of different sizes.

math.LO↗

arXiv2022

First-order logic with self-reference

We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two closely related but different logics. We provide a natural deduction system that is complete for validities for both of these logics, and we also investigate a range of related basic decision problems. For example, the validity problems of the two-variable fragments of the logics are shown coNexpTime-complete, which is in stark contrast with the high undecidability of two-variable logic extended with least fixed points. We also argue for the naturalness and benefits of the investigated approach to recursion and self-reference by, for example, relating the new logics to Lindstrom's Second Theorem.

math.LO↗

arXiv2022

On expressive rule-based logics

We investigate a family of rule-based logics. The focus is on very expressive languages. We provide a range of characterization results for the expressive powers of the logics and relate them with corresponding game systems.

cs.LO↗

arXiv2021

One-Dimensional Fragment over Words and Trees

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential (universal) quantifiers that leave at most one variable free. We investigate this fragment over words and trees, presenting a complete classification of the complexity of its satisfiability problem for various navigational signatures, and comparing its expressive power with other important formalisms. These include the two-variable fragment with counting and the unary negation fragment.

cs.LO↗

arXiv2021

The Optimal Way to Play the Most Difficult Repeated Coordination Games

This paper investigates repeated win-lose coordination games (WLC-games). We analyse which protocols are optimal for these games, covering both the worst case and average case scenarios, i,e., optimizing the guaranteed and expected coordination times. We begin by analysing Choice Matching Games (CM-games) which are a simple yet fundamental type of WLC-games, where the goal of the players is to pick the same choice from a finite set of initially indistinguishable choices. We give a fully complete classification of optimal expected and guaranteed coordination times in two-player CM-games and show that the corresponding optimal protocols are unique in every case - except in the CM-game with four choices, which we analyse separately. Our results on CM-games are essential for proving a more general result on the difficulty of all WLC-games: we provide a complete analysis of least upper bounds for optimal expected coordination times in all two-player WLC-games as a function of game size. We also show that CM-games can be seen as the most difficult games among all two-player WLC-games, as they turn out to have the greatest optimal expected coordination times.

cs.GT↗

arXiv2021

Description logics as polyadic modal logics

We study extensions of standard description logics to the framework of polyadic modal logic. We promote a natural approach to such logics via general relation algebras that can be used to define operations on relations of all arities. As a concrete system to illustrate our approach, we investigate the polyadic version of ALC extended with relational permutation operators and tuple counting. The focus of the paper is conceptual rather than technical.

cs.LO↗

arXiv2021

Rational coordination with no communication or conventions

We study pure coordination games where in every outcome, all players have identical payoffs, 'win' or 'lose'. We identify and discuss a range of 'purely rational principles' guiding the reasoning of rational players in such games and analyze which classes of coordination games can be solved by such players with no preplay communication or conventions. We observe that it is highly nontrivial to delineate a boundary between purely rational principles and other decision methods, such as conventions, for solving such coordination games.

cs.MA↗

arXiv2020

Bounded Game-Theoretic Semantics for Modal Mu-Calculus and Some Variants

We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounded GTS replaces parity games with alternative evaluation games where only finite paths arise; infinite paths are not needed even when the considered transition system is infinite. The novel games offer alternative approaches to various constructions in the framework of the mu-calculus. For example, they have already been successfully used as a basis for an approach leading to a natural formula size game for the logic. While our main focus is introducing the new GTS, we also consider some applications to demonstrate its uses. For example, we consider a natural model transformation procedure that reduces model checking games to checking a single, fixed formula in the constructed models, and we also use the GTS to identify new alternative variants of the mu-calculus with PTime model checking.

cs.LO↗

arXiv2020

Bounded game-theoretic semantics for modal mu-calculus

math.LO↗

arXiv2020

Optimal protocols for the most difficult repeated coordination games

This paper investigates repeated win-lose coordination games (WLC-games). We analyse which protocols are optimal for these games covering both the worst case and average case scenarios, i,e., optimizing the guaranteed and expected coordination times. We begin by analysing Choice Matching Games (CM-games) which are a simple yet fundamental type of WLC-games, where the goal of the players is to pick the same choice from a finite set of initially indistinguishable choices. We give a complete classification of optimal expected and guaranteed coordination times in two-player CM-games and show that the corresponding optimal protocols are unique in every case - except in the CM-game with four choices, which we analyse separately. Our results on CM-games are also essential for proving a more general result on the difficulty of all WLC-games: we provide a complete analysis of least upper bounds for optimal expected coordination times in all two-player WLC-games as a function of game size. We also show that CM-games can be seen as the most difficult games among all two-player WLC-games, as they turn out to have the greatest optimal expected coordination times.

cs.GT↗

arXiv2019

On Games and Computation

We introduce and investigate a range of general notions of a game. Our principal notion is based on a set of agents modifying a relational structure in a discrete evolution sequence. We also introduce and study a variety of ways to model partial and erroneous information in the setting. We discuss the connection of the related general setting to logic and computation formalisms, with emphasis on the recently introduced Turing-complete logic based on game-theoretic semantics.

cs.LO↗

arXiv2019

Game-Theoretic Semantics for Alternating-Time Temporal Logic

We introduce versions of game-theoretic semantics (GTS) for Alternating-Time Temporal Logic (ATL). In GTS, truth is defined in terms of existence of a winning strategy in a semantic evaluation game, and thus the game-theoretic perspective appears in the framework of ATL on two semantic levels: on the object level in the standard semantics of the strategic operators, and on the meta-level where game-theoretic logical semantics is applied to ATL. We unify these two perspectives into semantic evaluation games specially designed for ATL. The game-theoretic perspective enables us to identify new variants of the semantics of ATL based on limiting the time resources available to the verifier and falsifier in the semantic evaluation game. We introduce and analyse an unbounded and (ordinal) bounded GTS and prove these to be equivalent to the standard (Tarski-style) compositional semantics. We show that in these both versions of GTS, truth of ATL formulae can always be determined in finite time, i.e., without constructing infinite paths. We also introduce a non-equivalent finitely bounded semantics and argue that it is natural from both logical and game-theoretic perspectives.

math.LO↗

arXiv2019

Game-Theoretic Semantics for ATL+ with Applications to Model Checking

We develop game-theoretic semantics (GTS) for the fragment ATL+ of the full Alternating-time Temporal Logic ATL*, essentially extending a recently introduced GTS for ATL. We first show that the new game-theoretic semantics is equivalent to the standard semantics of ATL+ (based on perfect recall strategies). We then provide an analysis, based on the new semantics, of the memory and time resources needed for model checking ATL+. Based on that, we establish that strategies that use only a very limited amount of memory suffice for ATL+. Furthermore, using the GTS we provide a new algorithm for model checking of ATL+ and identify a natural hierarchy of tractable fragments of ATL+ that extend ATL.

math.LO↗

arXiv2019

Rewritability in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics

We study rewritability of monadic disjunctive Datalog programs, (the complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and on conjunctive queries. We show that rewritability into FO and into monadic Datalog (MDLog) are decidable, and that rewritability into Datalog is decidable when the original query satisfies a certain condition related to equality. We establish 2NExpTime-completeness for all studied problems except rewritability into MDLog for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze the shape of rewritings, which in the MMSNP case correspond to obstructions, and give a new construction of canonical Datalog programs that is more elementary than existing ones and also applies to formulas with free variables.

cs.LO↗