Logic Journal of IGPL Advance Access originally published online on January 27, 2007
Logic Journal of IGPL 2007 15(1):77-107; doi:10.1093/jigpal/jzl036
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Dynamic Topological Completeness for 
Mathematics Department, 450 Serra Mall Bldg 380, Stanford, CA, 94305. E-mail: dfd{at}math.stanford.edu.
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Abstract |
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Dynamic topological logic (DTL) combines topological and temporal modalities to express asymptotic properties of dynamic systems on topological spaces. A dynamic topological model is a triple 
X ,f , V 
, where X is a topological space, f : X 
X a continuous function and V a truth valuation assigning subsets of X to propositional variables. Valid formulas are those that are true in every model, independently of X or f. A natural problem that arises is to identify the logics obtained on familiar spaces, such as 
. It [9] it was shown that any satisfiable formula could be satisfied in some 
for n large enough, but the question of how the logic varies with n remained open.
In this paper we prove that any fragment of DTL that is complete for locally finite Kripke frames is complete for 
. This includes DTL
; it also includes some larger fragments, such as DTL1, where "henceforth" may not appear in the scope of a topological operator. We show that satisfiability of any formula of our language in a locally finite Kripke frame implies satisfiability in 
by constructing continuous, open maps from the plane into arbitrary locally finite Kripke frames, which give us a type of bisimulation. We also show that the results cannot be extended to arbitrary formulas of DTL by exhibiting a formula which is valid in 
but not in arbitrary topological spaces.
Key Words: Dynamic topological logic • spatial logic • temporal logic
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