88 research outputs found
Non-finite axiomatizability of Dynamic Topological Logic
Dynamic topological logic (DTL) is a polymodal logic designed for reasoning
about {\em dynamic topological systems. These are pairs (X,f), where X is a
topological space and f:X->X is continuous. DTL uses a language L which
combines the topological S4 modality [] with temporal operators from linear
temporal logic.
Recently, I gave a sound and complete axiomatization DTL* for an extension of
the logic to the language L*, where is allowed to act on finite sets of
formulas and is interpreted as a tangled closure operator. No complete
axiomatization is known over L, although one proof system, which we shall call
, was conjectured to be complete by Kremer and Mints.
In this paper we show that, given any language L' between L and L*, the set
of valid formulas of L' is not finitely axiomatizable. It follows, in
particular, that KM is incomplete.Comment: arXiv admin note: text overlap with arXiv:1201.5162 by other author
The intuitionistic temporal logic of dynamical systems
A dynamical system is a pair , where is a topological space and
is continuous. Kremer observed that the language of
propositional linear temporal logic can be interpreted over the class of
dynamical systems, giving rise to a natural intuitionistic temporal logic. We
introduce a variant of Kremer's logic, which we denote , and show
that it is decidable. We also show that minimality and Poincar\'e recurrence
are both expressible in the language of , thus providing a
decidable logic expressive enough to reason about non-trivial asymptotic
behavior in dynamical systems
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On the Logic of Belief and Propositional Quantification
We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss
Dynamic Cantor Derivative Logic
Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X,f) consisting of a topological space X equipped with a continuous function f: X ? X.
We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all T_D dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH.
The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation - something known to be impossible over the class of all spaces
Untangled: A Complete Dynamic Topological Logic
Dynamic topological logic () is a trimodal logic designed for
reasoning about dynamic topological systems. It was shown by Fern\'andez-Duque
that the natural set of axioms for is incomplete, but he
provided a complete axiomatisation in an extended language. In this paper, we
consider dynamic topological logic over scattered spaces, which are topological
spaces where every nonempty subspace has an isolated point. Scattered spaces
appear in the context of computational logic as they provide semantics for
provability and enjoy definable fixed points. We exhibit the first sound and
complete dynamic topological logic in the original trimodal language. In
particular, we show that the version of based on the class of
scattered spaces is finitely axiomatisable over the original language, and that
the natural axiomatisation is sound and complete
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