5,945 research outputs found
Realizability, Covers, and Sheaves I. Application to the Simply-Typed Lambda-Calculus
We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part I of this paper applies the above approach to the simply-typed λ-calculus (with types →, ×, +, and ⊥). Part II of this paper deals with the second-order (polymorphic) λ-calculus (with types → and ∀)
Proving Properties of Typed Lambda-Terms Using Realizability, Covers, and Sheaves (Preliminary Version)
We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed λ-calculus (with types →, x, +, and ⊥), and to the second-order (polymorphic λ-calculus (with types → and ∀2), for which it yields a new theorem
Comparing cohomology obstructions
We show that three different kinds of cohomology - Baues-Wirsching
cohomology, the (S,O)-cohomology of Dwyer-Kan, and the Andre-Quillen cohomology
of a Pi-algebra - are isomorphic, under certain assumptions.
This is then used to identify the cohomological obstructions in three general
approaches to realizability problems: the track category version of
Baues-Wirsching, the diagram rectifications of Dwyer-Kan-Smith, and the
Pi-Algebra realization of Dwyer-Kan-Stover. Our main tool in this
identification is the notion of a mapping algebra: a simplicially enriched
version of an algebra over a theory
Realizability, Covers, and Sheaves II. Applications to the Second-Order Lambda-Calculus
We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part II of this paper applies the above approach to the second-order (polymorphic) λ-calculus λ→,∀2 (with types → and ∀)
Interactive Learning-Based Realizability for Heyting Arithmetic with EM1
We apply to the semantics of Arithmetic the idea of ``finite approximation''
used to provide computational interpretations of Herbrand's Theorem, and we
interpret classical proofs as constructive proofs (with constructive rules for
) over a suitable structure \StructureN for the language of
natural numbers and maps of G\"odel's system \SystemT. We introduce a new
Realizability semantics we call ``Interactive learning-based Realizability'',
for Heyting Arithmetic plus \EM_1 (Excluded middle axiom restricted to
formulas). Individuals of \StructureN evolve with time, and
realizers may ``interact'' with them, by influencing their evolution. We build
our semantics over Avigad's fixed point result, but the same semantics may be
defined over different constructive interpretations of classical arithmetic
(Berardi and de' Liguoro use continuations). Our notion of realizability
extends intuitionistic realizability and differs from it only in the atomic
case: we interpret atomic realizers as ``learning agents''
Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic
We present a new syntactical proof that first-order Peano Arithmetic with
Skolem axioms is conservative over Peano Arithmetic alone for arithmetical
formulas. This result - which shows that the Excluded Middle principle can be
used to eliminate Skolem functions - has been previously proved by other
techniques, among them the epsilon substitution method and forcing. In our
proof, we employ Interactive Realizability, a computational semantics for Peano
Arithmetic which extends Kreisel's modified realizability to the classical
case.Comment: In Proceedings CL&C 2012, arXiv:1210.289
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