12,188 research outputs found
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
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