6,774 research outputs found
Game semantics for first-order logic
We refine HO/N game semantics with an additional notion of pointer
(mu-pointers) and extend it to first-order classical logic with completeness
results. We use a Church style extension of Parigot's lambda-mu-calculus to
represent proofs of first-order classical logic. We present some relations with
Krivine's classical realizability and applications to type isomorphisms
Sequentiality vs. Concurrency in Games and Logic
Connections between the sequentiality/concurrency distinction and the
semantics of proofs are investigated, with particular reference to games and
Linear Logic.Comment: 35 pages, appeared in Mathematical Structures in Computer Scienc
Propositional computability logic I
In the same sense as classical logic is a formal theory of truth, the
recently initiated approach called computability logic is a formal theory of
computability. It understands (interactive) computational problems as games
played by a machine against the environment, their computability as existence
of a machine that always wins the game, logical operators as operations on
computational problems, and validity of a logical formula as being a scheme of
"always computable" problems. The present contribution gives a detailed
exposition of a soundness and completeness proof for an axiomatization of one
of the most basic fragments of computability logic. The logical vocabulary of
this fragment contains operators for the so called parallel and choice
operations, and its atoms represent elementary problems, i.e. predicates in the
standard sense. This article is self-contained as it explains all relevant
concepts. While not technically necessary, however, familiarity with the
foundational paper "Introduction to computability logic" [Annals of Pure and
Applied Logic 123 (2003), pp.1-99] would greatly help the reader in
understanding the philosophy, underlying motivations, potential and utility of
computability logic, -- the context that determines the value of the present
results. Online introduction to the subject is available at
http://www.cis.upenn.edu/~giorgi/cl.html and
http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi
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
Introduction to Cirquent Calculus and Abstract Resource Semantics
This paper introduces a refinement of the sequent calculus approach called
cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.)
sequents are disjoint sequences of formulas, in cirquent calculus they are
permitted to share elements. Explicitly allowing or disallowing shared
resources and thus taking to a more subtle level the resource-awareness
intuitions underlying substructural logics, cirquent calculus offers much
greater flexibility and power than sequent calculus does. A need for
substantially new deductive tools came with the birth of computability logic
(see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed
formal theory of computational resources, which has stubbornly resisted any
axiomatization attempts within the framework of traditional syntactic
approaches. Cirquent calculus breaks the ice. Removing contraction from the
full collection of its rules yields a sound and complete system for the basic
fragment CL5 of computability logic. Doing the same in sequent calculus, on the
other hand, throws out the baby with the bath water, resulting in the strictly
weaker affine logic. An implied claim of computability logic is that it is CL5
rather than affine logic that adequately materializes the resource philosophy
traditionally associated with the latter. To strengthen this claim, the paper
further introduces an abstract resource semantics and shows the soundness and
completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
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