The circuit complexity class DLOGTIME-uniform AC 0 is known to be a modest subclass of DLOGTIME-u... more 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 Thérien in [BIL + 05]) was that if a language L has a neutral letter, then L can be defined in FOA, first-order logic with the collection of all numerical built-in relations A, if and only if L can be already defined in FO ≤ . 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 CS implying that all languages in DLOGTIME-uniform TC 0 can be defined in FO ≤ (CS). For instance, this criterion is satisfied by CS 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 R-int (FOB). By extending the results of [BIL + 05], we show that if B = {+}, or B contains only unary relations besides ≤, then R-int(FOB) ≡ FO ≤ . In contrast, our results imply that if B contains ≤ and the range of a polynomial of degree at least two, then R-cl(FOB) includes all languages in DLOGTIME-uniform TC 0 .
We establish a connection between term definability of Boolean functions and definability of fini... more We establish a connection between term definability of Boolean functions and definability of finite modal frames. We introduce a bijective translation between functional terms and uniform degree-1 formulas and show that a class of Boolean functions is defined by functional terms if and only if the corresponding class of Scott-Montague frames is defined by the translations of these functional terms, and vice versa. As a special case, we get that the clone Λ 1 of all conjunctions corresponds to the class of all Kripke frames. We also characterize some classes of Scott-Montague frames corresponding to subclones of Λ 1 by restricting the class of Kripke frames in a natural way. Furthermore, by modifying Kripke semantics, we extend our results to correspondences between linear clones and classes of Kripke frames equipped with modified semantics.
We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in ... more We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and nonuniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest.
Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semanti... more Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semantics is defined on Kripke models with teams. A team of a Kripke model is just a subset of its domain. In this paper we give a complete characterisation for the expressive power of modal inclusion logic: a class of Kripke models with teams is definable in modal inclusion logic if and only if it is closed under k-bisimulation for some integer k, it is closed under unions, and it has the empty team property. We also prove that the same expressive power can be obtained by adding a single unary nonemptiness operator to modal logic. Furthermore, we establish an exponential lower bound for the size of the translation from modal inclusion logic to modal logic with the nonemptiness operator.
We provide a sound and complete proof system for an extension of Kleene's ternary logic to predic... more We provide a sound and complete proof system for an extension of Kleene's ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of "is defined" is extended to terms and formulas via a straightforward recursive algorithm. The "is defined" formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the formula, its negation, and the negation of its "is defined" formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to explain, precisely in terms of logic, typical informal methods of reasoning in such applications.
We propose a new version of formula size game for modal logic. The game characterizes the equival... more We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic FO and (basic) modal logic ML. We also present a version of the game for the modal µ-calculus L µ and show that FO is also non-elementarily more succinct than L µ .
We analyze the expressive resources of IF logic that do not stem from Henkin (partiallyordered) q... more We analyze the expressive resources of IF logic that do not stem from Henkin (partiallyordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the gametheoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using "signalling by disjunction" instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.
We consider the length of the longest word definable in FO and MSO via a formula of size n. For b... more We consider the length of the longest word definable in FO and MSO via a formula of size n. For both logics we obtain as an upper bound for this number an exponential tower of height linear in n. We prove this by counting types with respect to a fixed quantifier rank. As lower bounds we obtain for both FO and MSO an exponential tower of height in the order of a rational power of n. We show these lower bounds by giving concrete formulas defining word representations of levels of the cumulative hierarchy of sets. In addition, we consider the Löwenheim-Skolem and Hanf numbers of these logics on words and obtain similar bounds for these as well.
We introduce a new game-theoretic semantics (GTS) for the modal mucalculus. Our so-called bounded... more We introduce a new game-theoretic semantics (GTS) for the modal mucalculus. 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.
This paper is a survery on the technique to prove logics non-finitely generated originated in [H]... more This paper is a survery on the technique to prove logics non-finitely generated originated in [H] and later used in [HL] and [HK]. The basic idea is that many (n + 1)-ary quantifiers Here, quantifier Q is n-ary, if ar(Q ) ≤ n where the arity ar(Q ) = sup{n R : R ∈ τ } is the supremum of the numbers of variables in formulas bounded by the quantifier Q . We say that the strict (strong) dimension of the quantifier Q above is n + 1 (similarly for model-classes). In practice, a back-and-forth characterization and a specific construction are needed to prove the non-redundancy. Back-and-forth systems are developed for logics L ∞ω (Q n ) and L ∞ω (Q G ) where Q n is a collection of n-ary quantifiers and Q G a subcollection of Q n satisfying certain symmetricity conditions (G is a permutation group). Non-trivial pairs of L ∞ω (Q n )-equivalent models are then constructed by means of a game.
We investigate the computational complexity of the satisfiability problem of modal inclusion logi... more We investigate the computational complexity of the satisfiability problem of modal inclusion logic. We distinguish two variants of the problem: one for the strict and another one for the lax semantics. Both problems turn out to be EXPTIME-complete on general structures. Finally, we show how for a specific class of structures NEXPTIMEcompleteness for these problems under strict semantics can be achieved.
We introduce a new variant of dependence logic (D) called Boolean dependence logic (BD). In BD de... more We introduce a new variant of dependence logic (D) called Boolean dependence logic (BD). In BD dependence atoms are of the type =(x 1 , . . . , x n , α), where α is a Boolean variable. Intuitively, with Boolean dependence atoms one can express quantification of relations, while standard dependence atoms express quantification over functions. We compare the expressive power of BD to D and first-order logic enriched by partially-ordered connectives, F O(POC). We show that the expressive power of BD and D coincide. We define natural syntactic fragments of BD and show that they coincide with the corresponding fragments of F O(POC) with respect to expressive power. We then show that the fragments form a strict hierarchy.
Logics with team semantics provide alternative means for logical characterization of complexity c... more Logics with team semantics provide alternative means for logical characterization of complexity classes. Both dependence and independence logic are known to capture non-deterministic polynomial time, and the frontiers of tractability in these logics are relatively well understood. Inclusion logic is similar to these team-based logical formalisms with the exception that it corresponds to deterministic polynomial time in ordered models. In this article we examine connections between syntactical fragments of inclusion logic and different complexity classes in terms of two computational problems: maximal subteam membership and the model checking problem for a fixed inclusion logic formula. We show that very simple quantifier-free formulae with one or two inclusion atoms generate instances of these problems that are complete for (non-deterministic) logarithmic space and polynomial time. Furthermore, we present a fragment of inclusion logic that captures non-deterministic logarithmic space in ordered models.
Propositional and modal inclusion logic are formalisms that belong to the family of logics based ... more Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalising the programme that ultimately aims to classify the complexities of the basic reasoning problems for modal and propositional dependence, independence, and inclusion logics.
We provide a sound and complete proof system for an extension of Kleene's ternary logic to predic... more We provide a sound and complete proof system for an extension of Kleene's ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of "is defined" is extended to terms and formulas via a straightforward recursive algorithm. The "is defined" formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the formula, its negation, and the negation of its "is defined" formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to explain, precisely in terms of logic, typical informal methods of reasoning in such applications.
We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in ... more We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and nonuniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest.
We consider the length of the longest word definable in FO and MSO via a formula of size n. For b... more We consider the length of the longest word definable in FO and MSO via a formula of size n. For both logics we obtain as an upper bound for this number an exponential tower of height linear in n. We prove this by counting types with respect to a fixed quantifier rank. As lower bounds we obtain for both FO and MSO an exponential tower of height in the order of a rational power of n. We show these lower bounds by giving concrete formulas defining word representations of levels of the cumulative hierarchy of sets. In addition, we consider the Löwenheim-Skolem and Hanf numbers of these logics on words and obtain similar bounds for these as well.
We analyze the expressive resources of $$\mathrm {IF}$$ IF logic that do not stem from Henkin (pa... more We analyze the expressive resources of $$\mathrm {IF}$$ IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular $$\mathrm {IF}$$ IF sentences, this amounts to the study of the fragment of $$\mathrm {IF}$$ IF logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes b...
Electronic Proceedings in Theoretical Computer Science, 2020
We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounde... more 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.
Electronic Proceedings in Theoretical Computer Science, 2015
Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semanti... more Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semantics is defined on Kripke models with teams. A team of a Kripke model is just a subset of its domain. In this paper we give a complete characterisation for the expressive power of modal inclusion logic: a class of Kripke models with teams is definable in modal inclusion logic if and only if it is closed under k-bisimulation for some integer k, it is closed under unions, and it has the empty team property. We also prove that the same expressive power can be obtained by adding a single unary nonemptiness operator to modal logic. Furthermore, we establish an exponential lower bound for the size of the translation from modal inclusion logic to modal logic with the nonemptiness operator.
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Papers by Lauri Hella