7,266 research outputs found
Epistemic Quantified Boolean Logic
This paper is aimed as a contribution to the use of formal modal languages in Artificial Intelligence. We introduce a multi-modal version of Second-order Propositional Modal Logic (SOPML), an extension of modal logic with propositional quantification, and illustrate its usefulness as a specification language for knowledge representation as well as temporal and spatial reasoning. Then, we define novel notions of (bi)simulation and prove that these preserve the interpretation of SOPML formulas. Finally, we apply these results to assess the expressive power of SOPML
A Semantical Analysis of Second-Order Propositional Modal Logic
International audienceThis paper is aimed as a contribution to the use of formal modal languages in Artificial Intelligence. We introduce a multi-modal version of Second-order Propositional Modal Logic (SOPML), an extension of modal logic with propositional quantification, and illustrate its usefulness as a specification language for knowledge representation as well as temporal and spatial reasoning. Then, we define novel notions of (bi)simulation and prove that these preserve the interpretation of SOPML formulas. Finally, we apply these results to assess the expressive power of SOPML
Axiomatizing modal inclusion logic
Modal inclusion logic is modal logic extended with inclusion atoms. It is the modal variant of first-order inclusion logic, which was introduced by Galliani (2012). Inclusion logic is a main variant of dependence logic (Väänänen 2007). Dependence logic and its variants adopt team semantics, introduced by Hodges (1997). Under team semantics, a modal (inclusion) logic formula is evaluated in a set of states, called a team. The inclusion atom is a type of dependency atom, which describes that the possible values a sequence of formulas can obtain are values of another sequence of formulas. In this thesis, we introduce a sound and complete natural deduction system for modal inclusion logic, which is currently missing in the literature.
The thesis consists of an introductory part, in which we recall the definitions and basic properties of modal logic and modal inclusion logic, followed by two main parts. The first part concerns the expressive power of modal inclusion logic. We review the result of Hella and Stumpf (2015) that modal inclusion logic is expressively complete: A class of Kripke models with teams is closed under unions, closed under k-bisimulation for some natural number k, and has the empty team property if and only if the class can be defined with a modal inclusion logic formula. Through the expressive completeness proof, we obtain characteristic formulas for classes with these three properties. This also provides a normal form for formulas in MIL. The proof of this result is due to Hella and Stumpf, and we suggest a simplification to the normal form by making it similar to the normal form introduced by Kontinen et al. (2014).
In the second part, we introduce a sound and complete natural deduction proof system for modal inclusion logic. Our proof system builds on the proof systems defined for modal dependence logic and propositional inclusion logic by Yang (2017, 2022). We show the completeness theorem using the normal form of modal inclusion logic
On Sub-Propositional Fragments of Modal Logic
In this paper, we consider the well-known modal logics ,
, , and , and we study some of their
sub-propositional fragments, namely the classical Horn fragment, the Krom
fragment, the so-called core fragment, defined as the intersection of the Horn
and the Krom fragments, plus their sub-fragments obtained by limiting the use
of boxes and diamonds in clauses. We focus, first, on the relative expressive
power of such languages: we introduce a suitable measure of expressive power,
and we obtain a complex hierarchy that encompasses all fragments of the
considered logics. Then, after observing the low expressive power, in
particular, of the Horn fragments without diamonds, we study the computational
complexity of their satisfiability problem, proving that, in general, it
becomes polynomial
Inquisitive bisimulation
Inquisitive modal logic InqML is a generalisation of standard Kripke-style
modal logic. In its epistemic incarnation, it extends standard epistemic logic
to capture not just the information that agents have, but also the questions
that they are interested in. Technically, InqML fits within the family of
logics based on team semantics. From a model-theoretic perspective, it takes us
a step in the direction of monadic second-order logic, as inquisitive modal
operators involve quantification over sets of worlds. We introduce and
investigate the natural notion of bisimulation equivalence in the setting of
InqML. We compare the expressiveness of InqML and first-order logic in the
context of relational structures with two sorts, one for worlds and one for
information states. We characterise inquisitive modal logic, as well as its
multi-agent epistemic S5-like variant, as the bisimulation invariant fragment
of first-order logic over various natural classes of two-sorted structures.
These results crucially require non-classical methods in studying bisimulation
and first-order expressiveness over non-elementary classes of structures,
irrespective of whether we aim for characterisations in the sense of classical
or of finite model theory
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
- …