The Expressive Power of k-ary Exclusion Logic

In this paper we study the expressive power of k-ary exclusion logic, EXC[k], that is obtained by extending first order logic with k-ary exclusion atoms. It is known that without arity bounds exclusion logic is equivalent with dependence logic. By observing the translations, we see that the expressive power of EXC[k] lies in between k-ary and (k+1)-ary dependence logics. We will show that, at least in the case of k=1, the both of these inclusions are proper. In a recent work by the author it was shown that k-ary inclusion-exclusion logic is equivalent with k-ary existential second order logic, ESO[k]. We will show that, on the level of sentences, it is possible to simulate inclusion atoms with exclusion atoms, and this way express ESO[k]-sentences by using only k-ary exclusion atoms. For this translation we also need to introduce a novel method for "unifying" the values of certain variables in a team. As a consequence, EXC[k] captures ESO[k] on the level of sentences, and we get a strict arity hierarchy for exclusion logic. It also follows that k-ary inclusion logic is strictly weaker than EXC[k]. Finally we will use similar techniques to formulate a translation from ESO[k] to k-ary inclusion logic with strict semantics. Consequently, for any arity fragment of inclusion logic, strict semantics is more expressive than lax semantics.Comment: Preprint of a paper in the special issue of WoLLIC2016 in Annals of Pure and Applied Logic, 170(9):1070-1099, 201

Capturing k-ary Existential Second Order Logic with k-ary Inclusion-Exclusion Logic

In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k]. We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.Comment: Extended version of a paper published in Annals of Pure and Applied Logic 169 (3), 177-21

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.Comment: Preprint of a paper published in ACM Transactions on Computational Logic, 19(3): 17:1-17:38, 201

Inkluusio ja ekskluusio kvantifioinnissa

Tässä tutkielmassa esitelemme uudet loogiset operaatiot, joita kutsumme inkluusio- ja ekskluusiokvanttoreiksi. Kun ensimmäisen kertaluvun logiikkaa laajennetaan inkluusiokvanttoreilla saadaan uusi logiikka, jota nimitämme INF-logiikaksi ('Inclusion Friendly Logic'). Vastaavasti lisäämällä ensimmäisen kertaluvun logiikkaan ekskluusiokvanttorit saadaan EXF-logiikka ('Exclusion Friendly Logic'), ja lisäämällä näistä kvanttoreista molemmat saadaan IEF-logiikka ('Inclusion-Exclusion Friendly Logic'). Esitämme tässä tutkielmassa esimerkkejä näiden kvanttorien käytöstä ilmaisemalla niiden avulla graafien ominaisuuksia, ja määrittelemme niitä käyttäen uusia hyödyllisiä loogisia operaatioita. Vertaamme lisäksi näiden uusien logiikoiden ilmaisuvoimaa inkluusio- ja ekskluusiologiikoihin, riippuvuuslogiikkaan, NDEP-logiikkaan ('Nondependence Logic') sekä lauseiden tasolla EMSO-logiikkaan ('Existential Monadic Second Order Logic'). Osoitamme, että EXF-logiikka on ilmaisuvoimaltaan vahvempi kuin yksipaikkainen riippuvuuslogiikka mutta heikompi kuin kaksipaikkainen riippuvuuslogiikka. Vastaavasti osoitamme, että INF-logiikan ilmaisuvoima sijoittuu aidosti yksi- ja kaksipaikkaisten NDEP-logiikoiden väliin. Lisäksi todistamme, että lauseiden tasolla INF-, EXF- sekä IEF-logiikka sisältyvät kaikki EMSO-logiikkaan. IEF-logiikalle pätee myös käänteinen väite, joten sen ilmaisuvoima on lauseiden tasolla täsmälleen sama kuin EMSO-logiikalla

On definability of team relations with k-invariant atoms

We study the expressive power of logics whose truth is defined over sets of assignments, called teams, instead of single assignments. Given a team X, any k-tuple of variables in the domain of X defines a corresponding k-ary team relation. Thus the expressive power of a logic L with team semantics amounts to the set of properties of team relations which L-formulas can define. We introduce a concept of k-invariance which is a natural semantic restriction on any atomic formulae with team semantics. Then we develop a novel proof method to show that, if L is an extension of FO with any k-invariant atoms, then there are such properties of (k+1)-ary team relations which cannot be defined in L. This method can be applied e.g. for arity fragments of various logics with team semantics to prove undefinability results. In particular, we make some interesting observations on the definability of binary team relations with unary inclusion-exclusion logic.publishedVersionPeer reviewe

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 compare the classes of coordination games that 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.Peer reviewe

Independence-friendly logic without Henkin quantification

We analyze the expressive resources of IF logic that do not stem from Henkin (partially-ordered) 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 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 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.Peer reviewe