1,258 research outputs found
A logic with temporally accessible iteration
Deficiency in expressive power of the first-order logic has led to developing
its numerous extensions by fixed point operators, such as Least Fixed-Point
(LFP), inflationary fixed-point (IFP), partial fixed-point (PFP), etc. These
logics have been extensively studied in finite model theory, database theory,
descriptive complexity. In this paper we introduce unifying framework, the
logic with iteration operator, in which iteration steps may be accessed by
temporal logic formulae. We show that proposed logic FO+TAI subsumes all
mentioned fixed point extensions as well as many other fixed point logics as
natural fragments. On the other hand we show that over finite structures FO+TAI
is no more expressive than FO+PFP. Further we show that adding the same
machinery to the logic of monotone inductions (FO+LFP) does not increase its
expressive power either
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
A Type-Directed Negation Elimination
In the modal mu-calculus, a formula is well-formed if each recursive variable
occurs underneath an even number of negations. By means of De Morgan's laws, it
is easy to transform any well-formed formula into an equivalent formula without
negations -- its negation normal form. Moreover, if the formula is of size n,
its negation normal form of is of the same size O(n). The full modal
mu-calculus and the negation normal form fragment are thus equally expressive
and concise.
In this paper we extend this result to the higher-order modal fixed point
logic (HFL), an extension of the modal mu-calculus with higher-order recursive
predicate transformers. We present a procedure that converts a formula into an
equivalent formula without negations of quadratic size in the worst case and of
linear size when the number of variables of the formula is fixed.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Succinctness in subsystems of the spatial mu-calculus
In this paper we systematically explore questions of succinctness in modal
logics employed in spatial reasoning. We show that the closure operator,
despite being less expressive, is exponentially more succinct than the
limit-point operator, and that the -calculus is exponentially more
succinct than the equally-expressive tangled limit operator. These results hold
for any class of spaces containing at least one crowded metric space or
containing all spaces based on ordinals below , with the usual
limit operator. We also show that these results continue to hold even if we
enrich the less succinct language with the universal modality
Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets
In the context of strategic games, we provide an axiomatic proof of the
statement Common knowledge of rationality implies that the players will choose
only strategies that survive the iterated elimination of strictly dominated
strategies. Rationality here means playing only strategies one believes to be
best responses. This involves looking at two formal languages. One is
first-order, and is used to formalise optimality conditions, like avoiding
strictly dominated strategies, or playing a best response. The other is a modal
fixpoint language with expressions for optimality, rationality and belief.
Fixpoints are used to form expressions for common belief and for iterated
elimination of non-optimal strategies.Comment: 16 pages, Proc. 11th International Workshop on Computational Logic in
Multi-Agent Systems (CLIMA XI). To appea
The modal logic of set-theoretic potentialism and the potentialist maximality principles
We analyze the precise modal commitments of several natural varieties of
set-theoretic potentialism, using tools we develop for a general
model-theoretic account of potentialism, building on those of Hamkins, Leibman
and L\"owe, including the use of buttons, switches, dials and ratchets. Among
the potentialist conceptions we consider are: rank potentialism (true in all
larger ); Grothendieck-Zermelo potentialism (true in all larger
for inaccessible cardinals ); transitive-set potentialism
(true in all larger transitive sets); forcing potentialism (true in all forcing
extensions); countable-transitive-model potentialism (true in all larger
countable transitive models of ZFC); countable-model potentialism (true in all
larger countable models of ZFC); and others. In each case, we identify lower
bounds for the modal validities, which are generally either S4.2 or S4.3, and
an upper bound of S5, proving in each case that these bounds are optimal. The
validity of S5 in a world is a potentialist maximality principle, an
interesting set-theoretic principle of its own. The results can be viewed as
providing an analysis of the modal commitments of the various set-theoretic
multiverse conceptions corresponding to each potentialist account.Comment: 36 pages. Commentary can be made about this article at
http://jdh.hamkins.org/set-theoretic-potentialism. Minor revisions in v2;
further minor revisions in v
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