81,088 research outputs found
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
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
Modal Logics that Bound the Circumference of Transitive Frames
For each natural number we study the modal logic determined by the class
of transitive Kripke frames in which there are no cycles of length greater than
and no strictly ascending chains. The case is the G\"odel-L\"ob
provability logic. Each logic is axiomatised by adding a single axiom to K4,
and is shown to have the finite model property and be decidable.
We then consider a number of extensions of these logics, including
restricting to reflexive frames to obtain a corresponding sequence of
extensions of S4. When , this gives the famous logic of Grzegorczyk, known
as S4Grz, which is the strongest modal companion to intuitionistic
propositional logic. A topological semantic analysis shows that the -th
member of the sequence of extensions of S4 is the logic of hereditarily
-irresolvable spaces when the modality is interpreted as the
topological closure operation. We also study the definability of this class of
spaces under the interpretation of as the derived set (of limit
points) operation.
The variety of modal algebras validating the -th logic is shown to be
generated by the powerset algebras of the finite frames with cycle length
bounded by . Moreover each algebra in the variety is a model of the
universal theory of the finite ones, and so is embeddable into an ultraproduct
of them
Propositional dynamic logic for searching games with errors
We investigate some finitely-valued generalizations of propositional dynamic
logic with tests. We start by introducing the (n+1)-valued Kripke models and a
corresponding language based on a modal extension of {\L}ukasiewicz many-valued
logic. We illustrate the definitions by providing a framework for an analysis
of the R\'enyi - Ulam searching game with errors.
Our main result is the axiomatization of the theory of the (n+1)-valued
Kripke models. This result is obtained through filtration of the canonical
model of the smallest (n+1)-valued propositional dynamic logic
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