462 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
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
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
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
Tool support for reasoning in display calculi
We present a tool for reasoning in and about propositional sequent calculi.
One aim is to support reasoning in calculi that contain a hundred rules or
more, so that even relatively small pen and paper derivations become tedious
and error prone. As an example, we implement the display calculus D.EAK of
dynamic epistemic logic. Second, we provide embeddings of the calculus in the
theorem prover Isabelle for formalising proofs about D.EAK. As a case study we
show that the solution of the muddy children puzzle is derivable for any number
of muddy children. Third, there is a set of meta-tools, that allows us to adapt
the tool for a wide variety of user defined calculi
The expressive power of modal logic with inclusion atoms
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.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
Axiomatizing propositional dependence logics
We give sound and complete Hilbert-style axiomatizations for propositional dependence logic (PD), modal dependence logic (MDL), and extended modal dependence logic (EMDL) by extending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for PD, MDL, and EMDL. We prove soundness, completeness and termination for each of the labeled calculi
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