Computational Aspects of Dependence Logic

In this thesis (modal) dependence logic is investigated. It was introduced in 2007 by Jouko V\"a\"aan\"anen as an extension of first-order (resp. modal) logic by the dependence operator =(). For first-order (resp. propositional) variables x_1,...,x_n, =(x_1,...,x_n) intuitively states that the value of x_n is determined by those of x_1,...,x_n-1. We consider fragments of modal dependence logic obtained by restricting the set of allowed modal and propositional connectives. We classify these fragments with respect to the complexity of their satisfiability and model-checking problems. For satisfiability we obtain complexity degrees from P over NP, Sigma_P^2 and PSPACE up to NEXP, while for model-checking we only classify the fragments with respect to their tractability, i.e. we either show NP-completeness or containment in P. We then study the extension of modal dependence logic by intuitionistic implication. For this extension we again classify the complexity of the model-checking problem for its fragments. Here we obtain complexity degrees from P over NP and coNP up to PSPACE. Finally, we analyze first-order dependence logic, independence-friendly logic and their two-variable fragments. We prove that satisfiability for two-variable dependence logic is NEXP-complete, whereas for two-variable independence-friendly logic it is undecidable; and use this to prove that the latter is also more expressive than the former.Comment: PhD thesis; 138 pages (110 main matter

Canonical Models and the Complexity of Modal Team Logic

We study modal team logic MTL, the team-semantical extension of classical modal logic closed under Boolean negation. Its fragments, such as modal dependence, independence, and inclusion logic, are well-understood. However, due to the unrestricted Boolean negation, the satisfiability problem of full MTL has been notoriously resistant to a complexity theoretical classification. In our approach, we adapt the notion of canonical models for team semantics. By construction of such a model, we reduce the satisfiability problem of MTL to simple model checking. Afterwards, we show that this method is optimal in the sense that MTL-formulas can efficiently enforce canonicity. Furthermore, to capture these results in terms of computational complexity, we introduce a non-elementary complexity class, TOWER(poly), and prove that the satisfiability and validity problem of MTL are complete for it. We also show that the fragments of MTL with bounded modal depth are complete for the levels of the elementary hierarchy (with polynomially many alternations)

Complexity of validity for propositional dependence logics

We study the validity problem for propositional dependence logic, modal dependence logic and extended modal dependence logic. We show that the validity problem for propositional dependence logic is NEXPTIME-complete. In addition, we establish that the corresponding problem for modal dependence logic and extended modal dependence logic is NEXPTIME-hard and in NEXPTIME^NP.Comment: In Proceedings GandALF 2014, arXiv:1408.556

A Team Based Variant of CTL

We introduce two variants of computation tree logic CTL based on team semantics: an asynchronous one and a synchronous one. For both variants we investigate the computational complexity of the satisfiability as well as the model checking problem. The satisfiability problem is shown to be EXPTIME-complete. Here it does not matter which of the two semantics are considered. For model checking we prove a PSPACE-completeness for the synchronous case, and show P-completeness for the asynchronous case. Furthermore we prove several interesting fundamental properties of both semantics.Comment: TIME 2015 conference version, modified title and motiviatio

The Expressive Power of Modal Dependence Logic

We study the expressive power of various modal logics with team semantics. We show that exactly the properties of teams that are downward closed and closed under team k-bisimulation, for some finite k, are definable in modal logic extended with intuitionistic disjunction. Furthermore, we show that the expressive power of modal logic with intuitionistic disjunction and extended modal dependence logic coincide. Finally we establish that any translation from extended modal dependence logic into modal logic with intuitionistic disjunction increases the size of some formulas exponentially.Comment: 19 page

Parametrised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of V\"a\"an\"anen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture.Comment: Update includes refined result

On Extensions and Variants of Dependence Logic : A study of intuitionistic connectives in the team semantics setting

Dependence logic is a new logic which incorporates the notion of dependence , as well as independence between variables into first-order logic. In this thesis, we study extensions and variants of dependence logic on the first-order, propositional and modal level. In particular, the role of intuitionistic connectives in this setting is emphasized. We obtain, among others, the following results: 1. First-order intuitionistic dependence logic is proved to have the same expressive power as the full second-order logic. 2. Complete axiomatizations for propositional dependence logic and its variants are obtained. 3. The complexity of model checking problem for modal intuitionistic dependence logic is analyzed.Riippuvuus ja riippumattomuus ovat yleisiä ilmiöitä monella alalla aina tietojenksittelytieteestä (tietokannat, ohjelmistotekniikka, tiedon esitys, tekoäly) valtiotieteisiin (historia, osakemarkkinat). 1960-luvulta lähtien matemaatikot ja filosofit ovat olleet tietoisia klassisen ensimmäisen kertaluvun logiikan rajoitteista muuttujien riippuvuuden ja riippumattomuuden ilmaisemisessa. Ongelman ratkaisemiseksi Henkin (1961) laajensi ensimmäisen kertaluvun logiikkaa haarautuvilla kvanttoreilla ja Hintikka ja Sandu (1989) määrittelivät IF-logiikan. Väänäsen (2007) kehittämä riippuvuuslogiikka on uusi suunta lähestymistavoissa. Riippuvuuslogiikan käsitteellinen uutuus on lisätä vaatimukset riippuvuudesta ja riippumattomuudesta atomaariselle tasolle, eikä kvanttoritasolle, kuten aiemmissa lähestymistavoissa. Lisäksi logiikan metodologia on täysin uusi: tavanomaisesta yhteen tulkintafunktioon perustuvasta Tarksin semantiikasta poiketen riippuvuuslogiikan toteutuvuusrelaatio määrtellään tulkintafunktiojoukon suhteen (alunperin Hodgesilta, 1997). Riippuvuuslogiikka on luonteeltaan hyvin monitieteinen ja siksi logiikalla, ja sen monilla laajennuksilla ja muunnelmilla, on mahdollisia sovelluksia mm. tietokantateorian, kielifilosofian ja valtiotieteiden aloilla. Tämä väitöskirja tutkii riippuvuuslogiikan laajennuksia ja muunnelmia. Erityisesti painotetaan intuitionististen konnektiivien roolia tässä lähestymistavassa. Päätuloksia ovat: 1. Ensimmäisen kertaluvun intuitionistisen riippuvuuslogiikan ilmaisuvoima osoitetaan yhtä vahvaksi kuin täyden toisen kertaluvun logiikan. 2. Annetaan täydellisiä aksiomatisointeja propositionaaliselle riippuvuuslogiikalle ja sen variaatioille. 3. Analysodaan modaalisen intuitionsitisen riippuvuuslogiikan mallintarkastusongelman kompleksisuutta