88 research outputs found
Hybrid realizability for intuitionistic and classical choice
International audienceIn intuitionistic realizability like Kleene's or Kreisel's, the axiom of choice is trivially realized. It is even provable in Martin-Löf's intu-itionistic type theory. In classical logic, however, even the weaker axiom of countable choice proves the existence of non-computable functions. This logical strength comes at the price of a complicated computational interpretation which involves strong recursion schemes like bar recursion. We take the best from both worlds and define a realizability model for arithmetic and the axiom of choice which encompasses both intuitionistic and classical reasoning. In this model two versions of the axiom of choice can co-exist in a single proof: intuitionistic choice and classical countable choice. We interpret intuitionistic choice efficiently, however its premise cannot come from classical reasoning. Conversely, our version of classical choice is valid in full classical logic, but it is restricted to the countable case and its realizer involves bar recursion. Having both versions allows us to obtain efficient extracted programs while keeping the provability strength of classical logic
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
A functional interpretation for nonstandard arithmetic
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop âMathematical Logic: Proof Theory, Constructive Mathematicsâ was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)
The Workshop "Mathematical Logic: Proof Theory,
Constructive Mathematics" focused on
proofs both as formal derivations in deductive systems as well as on
the extraction of explicit computational content from
given proofs in core areas of ordinary mathematics using proof-theoretic
methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory
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
Computational Interpretations of Classical Linear Logic
Abstract. We survey several computational interpretations of classical linear logic based on two-player one-move games. The moves of the games are higher-order functionals in the language of finite types. All interpretations discussed treat the exponential-free fragment of linear logic in a common way. They only differ in how much advantage one of the players has in the exponentials games. We dis-cuss how the several choices for the interpretation of the modalities correspond to various well-known functional interpretations of intuitionistic logic, including GoÌdelâs Dialectica interpretation and Kreiselâs modified realizability.
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