15 research outputs found
Completeness for Flat Modal Fixpoint Logics
This paper exhibits a general and uniform method to prove completeness for
certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form
\gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the
language L\sharp (\Gamma) is obtained by adding to the language of polymodal
logic a connective \sharp\_\gamma for each \gamma \epsilon. The term
\sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the
least fixed point of the functional interpretation of the term \gamma(x,
\varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma,
construct an axiom system which is sound and complete with respect to the
concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We
prove two results that solve this problem. First, let K\sharp (\Gamma) be the
logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint
axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma.
Provided that each indexing formula \gamma satisfies the syntactic criterion of
being untied in x, we prove this axiom system to be complete. Second,
addressing the general case, we prove the soundness and completeness of an
extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an
effective procedure that, given an indexing formula \gamma as input, returns a
finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded
by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever
\Gamma is finite
Games for topological fixpoint logic
Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we introduce games characterising clopen fixpoints of monotone operators on Stone spaces. These fixpoint games allow us to characterise the semantics for our topological fixpoint logic using a two-player graph game. Adequacy of this game is the main result of our paper. Finally, we define bisimulations for the topological structures under consideration and use our game semantics to prove that the truth of a formula of our topological fixpoint logic is bisimulation-invariant
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Completeness of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
A Functional (Monadic) Second-Order Theory of Infinite Trees
This paper presents a complete axiomatization of Monadic Second-Order Logic
(MSO) over infinite trees. MSO on infinite trees is a rich system, and its
decidability ("Rabin's Tree Theorem") is one of the most powerful known results
concerning the decidability of logics. By a complete axiomatization we mean a
complete deduction system with a polynomial-time recognizable set of axioms. By
naive enumeration of formal derivations, this formally gives a proof of Rabin's
Tree Theorem. The deduction system consists of the usual rules for second-order
logic seen as two-sorted first-order logic, together with the natural
adaptation In addition, it contains an axiom scheme expressing the (positional)
determinacy of certain parity games. The main difficulty resides in the limited
expressive power of the language of MSO. We actually devise an extension of
MSO, called Functional (Monadic) Second-Order Logic (FSO), which allows us to
uniformly manipulate (hereditarily) finite sets and corresponding labeled
trees, and whose language allows for higher abstraction than that of MSO