51,302 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
Presentation of a Game Semantics for First-Order Propositional Logic
Game semantics aim at describing the interactive behaviour of proofs by
interpreting formulas as games on which proofs induce strategies. In this
article, we introduce a game semantics for a fragment of first order
propositional logic. One of the main difficulties that has to be faced when
constructing such semantics is to make them precise by characterizing definable
strategies - that is strategies which actually behave like a proof. This
characterization is usually done by restricting to the model to strategies
satisfying subtle combinatory conditions such as innocence, whose preservation
under composition is often difficult to show. Here, we present an original
methodology to achieve this task which requires to combine tools from game
semantics, rewriting theory and categorical algebra. We introduce a
diagrammatic presentation of definable strategies by the means of generators
and relations: those strategies can be generated from a finite set of
``atomic'' strategies and that the equality between strategies generated in
such a way admits a finite axiomatization. These generators satisfy laws which
are a variation of bialgebras laws, thus bridging algebra and denotational
semantics in a clean and unexpected way
Constructive Game Logic
Game Logic is an excellent setting to study proofs-about-programs via the
interpretation of those proofs as programs, because constructive proofs for
games correspond to effective winning strategies to follow in response to the
opponent's actions. We thus develop Constructive Game Logic which extends
Parikh's Game Logic (GL) with constructivity and with first-order programs a la
Pratt's first-order dynamic logic (DL). Our major contributions include:
1) a novel realizability semantics capturing the adversarial dynamics of
games, 2) a natural deduction calculus and operational semantics describing the
computational meaning of strategies via proof-terms, and 3) theoretical results
including soundness of the proof calculus w.r.t. realizability semantics,
progress and preservation of the operational semantics of proofs, and Existence
Properties on support of the extraction of computational artifacts from game
proofs.
Together, these results provide the most general account of a Curry-Howard
interpretation for any program logic to date, and the first at all for Game
Logic.Comment: 74 pages, extended preprint for ESO
Some Turing-Complete Extensions of First-Order Logic
We introduce a natural Turing-complete extension of first-order logic FO. The
extension adds two novel features to FO. The first one of these is the capacity
to add new points to models and new tuples to relations. The second one is the
possibility of recursive looping when a formula is evaluated using a semantic
game. We first define a game-theoretic semantics for the logic and then prove
that the expressive power of the logic corresponds in a canonical way to the
recognition capacity of Turing machines. Finally, we show how to incorporate
generalized quantifiers into the logic and argue for a highly natural
connection between oracles and generalized quantifiers.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Inclusion and Exclusion Dependencies in Team Semantics: On Some Logics of Imperfect Information
We introduce some new logics of imperfect information by adding atomic
formulas corresponding to inclusion and exclusion dependencies to the language
of first order logic. The properties of these logics and their relationships
with other logics of imperfect information are then studied. Furthermore, a
game theoretic semantics for these logics is developed. As a corollary of these
results, we characterize the expressive power of independence logic, thus
answering an open problem posed in (Gr\"adel and V\"a\"an\"anen, 2010)
The Structure of First-Order Causality
Game semantics describe the interactive behavior of proofs by interpreting
formulas as games on which proofs induce strategies. Such a semantics is
introduced here for capturing dependencies induced by quantifications in
first-order propositional logic. One of the main difficulties that has to be
faced during the elaboration of this kind of semantics is to characterize
definable strategies, that is strategies which actually behave like a proof.
This is usually done by restricting the model to strategies satisfying subtle
combinatorial conditions, whose preservation under composition is often
difficult to show. Here, we present an original methodology to achieve this
task, which requires to combine advanced tools from game semantics, rewriting
theory and categorical algebra. We introduce a diagrammatic presentation of the
monoidal category of definable strategies of our model, by the means of
generators and relations: those strategies can be generated from a finite set
of atomic strategies and the equality between strategies admits a finite
axiomatization, this equational structure corresponding to a polarized
variation of the notion of bialgebra. This work thus bridges algebra and
denotational semantics in order to reveal the structure of dependencies induced
by first-order quantifiers, and lays the foundations for a mechanized analysis
of causality in programming languages
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