On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella

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

Team building in dependence

Hintikka and Sandu’s Independence-Friendly Logic was introduced as a logic for partially ordered quantification, in which the independence of (existential) quantifiers from previous (universal) quantifiers is written by explicit syntax. It was originally given a semantics by games of imperfect information; Hodges then gave a (necessarily) second-order Tarskian semantics. More recently, Väänänen (2007) has proposed that the many curious features of IF logic can be better understood in his Dependence Logic, in which the (in)dependence of variables is stated in atomic formula, rather than by changing the definition of quantifier; he gives semantics in Tarskian form, via imperfect information games, and via a routine second-order perfect information game. He then defines Team Logic, where classical negation is added to the mix, resulting in a full second-order expressive logic. He remarks that no game semantics appears possible (other than by playing at second order). In this article, we explore an alternative approach to game semantics for DL, where we avoid imperfect information, yet stay locally apparently first-order, by sweeping the second-order information into longer games (infinite games in the case of countable models). Extending the game to Team Logic is not possible in standard games, but we conjecture a move to transfinite games may achieve a ‘natural ’ game for Team Logic

Dependence Logic : Investigations into Higher-Order Semantics Defined on Teams

This thesis is a study of a rather new logic called dependence logic and its closure under classical negation, team logic. In this thesis, dependence logic is investigated from several aspects. Some rules are presented for quantifier swapping in dependence logic and team logic. Such rules are among the basic tools one must be familiar with in order to gain the required intuition for using the logic for practical purposes. The thesis compares Ehrenfeucht-Fraïssé (EF) games of first order logic and dependence logic and defines a third EF game that characterises a mixed case where first order formulas are measured in the formula rank of dependence logic. The thesis contains detailed proofs of several translations between dependence logic, team logic, second order logic and its existential fragment. Translations are useful for showing relationships between the expressive powers of logics. Also, by inspecting the form of the translated formulas, one can see how an aspect of one logic can be expressed in the other logic. The thesis makes preliminary investigations into proof theory of dependence logic. Attempts focus on finding a complete proof system for a modest yet nontrivial fragment of dependence logic. A key problem is identified and addressed in adapting a known proof system of classical propositional logic to become a proof system for the fragment, namely that the rule of contraction is needed but is unsound in its unrestricted form. A proof system is suggested for the fragment and its completeness conjectured. Finally, the thesis investigates the very foundation of dependence logic. An alternative semantics called 1-semantics is suggested for the syntax of dependence logic. There are several key differences between 1-semantics and other semantics of dependence logic. 1-semantics is derived from first order semantics by a natural type shift. Therefore 1-semantics reflects an established semantics in a coherent manner. Negation in 1-semantics is a semantic operation and satisfies the law of excluded middle. A translation is provided from unrestricted formulas of existential second order logic into 1-semantics. Also game theoretic semantics are considerd in the light of 1-semantics.Väitöskirja käsittelee uudehkoa logiikkaa nimeltään riippuvuuslogiikkaa (dependence logic) sekä sen laajennusta nimeltään tiimilogiikka (team logic). Riippuvuuslogiikan tunnusomainen piirre on mahdollisuus ilmaista tavallista monipuolisempia riippuvuussuhteita loogisten kaavojen muuttujien välillä. Riippuvuuslogiikka onkin aidosti ilmaisuvoimaisempi kuin tavallinen predikaattilogiikka. Tiimilogiikka on riippuvuuslogiikan sulkeuma klassisen negaation suhteen. Väitöskirjan käytännöllisempiin tuloksiin kuuluvat kvanttorien vaihtosäännöt riippuvuuslogiikan kaavoissa. Lisäksi väitöskirja tutkii Ehrenfeucht-Fraïssé-pelien suhteita riippuvuuslogiikan ja predikaattilogiikan välillä. Nämä tulokset auttavat osaltaan ymmärtämään riippuvuuslogiikkaa ja edesauttavat intuition kehittämistä sen käytössä. Abstraktimpiin tuloksiin kuuluvat useat yksityiskohtaiset käännöstulokset riippuvuuslogiikan, tiimilogiikan, toisen kertaluvun logiikan ja sen eksistentiaalisen fragmentin välillä. Käännösten avulla voidaan vertailla logiikoiden ilmaisuvoimia keskenään. Lisäksi tarkastelemalla käännettyjä kaavoja voidaan havaita, miten yhden logiikan tietyt piirteet siirtyvät toiseen logiikkaan. Väitöskirjassa tutkitaan myös riippuvuuslogiikan todistusteoriaa. Tutkimus keskittyy riippuvuuslogiikan erääseen yksinkertaiseen fragmenttiin, joka muistuttaa klassista propositiologiikkaa. Tälle fragmentille määritellään klassisen propositiologiikan todistusjärjestelmää muistuttava järjestelmä, jossa supistussääntöä joudutaan rajoittamaan. Lopuksi väitöskirja pureutuu riippuvuuslogiikan ytimeen ja määrittelee sille vaihtoehtoisen semantiikan nimeltään 1-semantiikka. 1-semantiikka eroaa aiemmista semantiikoista useilla tavoilla. Ensinnäkin se perustuu luontevaan tyypinnosto-operaatioon, joka johtaa riippuvuuslogiikan semantiikan yksiselitteisesti predikaattilogiikasta. Tämän seurauksena negaatiolle voidaan antaa puhtaasti semanttinen määritelmä. 1-semantiikan negaatio toteuttaa lisäksi kolmannen poissuljetun lain. 1-semantiikan ilmaisuvoiman todistetaan käännöksen avulla kattavan täsmälleen eksistentiaalinen toisen kertaluvun logiikan siinä, missä aiemmat semantiikat ovat kattaneet vain sen alaspäin suljetun fragmentin. Myös peliteoreettista semantiikkaa tarkastellaan 1-semantiikan valossa