37,396 research outputs found
Inductive types in the Calculus of Algebraic Constructions
In a previous work, we proved that an important part of the Calculus of
Inductive Constructions (CIC), the basis of the Coq proof assistant, can be
seen as a Calculus of Algebraic Constructions (CAC), an extension of the
Calculus of Constructions with functions and predicates defined by higher-order
rewrite rules. In this paper, we prove that almost all CIC can be seen as a
CAC, and that it can be further extended with non-strictly positive types and
inductive-recursive types together with non-free constructors and
pattern-matching on defined symbols.Comment: Journal version of TLCA'0
Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion
We study the semantics of a resource-sensitive extension of the lambda
calculus in a canonical reflexive object of a category of sets and relations, a
relational version of Scott's original model of the pure lambda calculus. This
calculus is related to Boudol's resource calculus and is derived from Ehrhard
and Regnier's differential extension of Linear Logic and of the lambda
calculus. We extend it with new constructions, to be understood as implementing
a very simple exception mechanism, and with a "must" parallel composition.
These new operations allow to associate a context of this calculus with any
point of the model and to prove full abstraction for the finite sub-calculus
where ordinary lambda calculus application is not allowed. The result is then
extended to the full calculus by means of a Taylor Expansion formula. As an
intermediate result we prove that the exception mechanism is not essential in
the finite sub-calculus
On computational interpretations of the modal logic S4. I. Cut elimination
A language of constructions for minimal logic is the
-calculus, where cut-elimination is encoded as
-reduction. We examine corresponding languages for the
minimal version of the modal logic S4, with notions of reduction
that encodes cut-elimination for the corresponding sequent system.
It turns out that a natural interpretation of the latter
constructions is a -calculus extended by an idealized
version of Lisp\u27s \verb/eval/ and \verb/quote/ constructs.
In this first part, we analyze how cut-elimination works in the
standard sequent system for minimal S4, and where problems arise.
Bierman and De Paiva\u27s proposal is a natural language of constructions
for this logic, but their calculus lacks a few rules that are
essential to eliminate all cuts. The -calculus,
namelyBierman and De Paiva\u27s proposal extended with all needed rules,
is confluent. There is a polynomial-time algorithm to compute
principal typings of given terms, or answer that the given terms are
not typable. The typed -calculus terminates, and
normal forms are exactly constructions for cut-free proofs. Finally,
modulo some notion \sqeq of equivalence, there is a natural
Curry-Howard style isomorphism between typed
-terms and natural deduction proofs in minimal S4.
However, the -calculus has a non-operational
flavor, in that the extra rules include explicit garbage collection,
contraction and exchange rules. We shall propose another language of
constructions to repair this in Part II
Second-order subdifferential calculus with applications to tilt stability in optimization
The paper concerns the second-order generalized differentiation theory of
variational analysis and new applications of this theory to some problems of
constrained optimization in finitedimensional spaces. The main attention is
paid to the so-called (full and partial) second-order subdifferentials of
extended-real-valued functions, which are dual-type constructions generated by
coderivatives of frst-order subdifferential mappings. We develop an extended
second-order subdifferential calculus and analyze the basic second-order
qualification condition ensuring the fulfillment of the principal secondorder
chain rule for strongly and fully amenable compositions. The calculus results
obtained in this way and computing the second-order subdifferentials for
piecewise linear-quadratic functions and their major specifications are applied
then to the study of tilt stability of local minimizers for important classes
of problems in constrained optimization that include, in particular, problems
of nonlinear programming and certain classes of extended nonlinear programs
described in composite terms
Noncommutative generalization of SU(n)-principal fiber bundles: a review
This is an extended version of a communication made at the international
conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007.
In this proceeding, we make a review of some noncommutative constructions
connected to the ordinary fiber bundle theory. The noncommutative algebra is
the endomorphism algebra of a SU(n)-vector bundle, and its differential
calculus is based on its Lie algebra of derivations. It is shown that this
noncommutative geometry contains some of the most important constructions
introduced and used in the theory of connections on vector bundles, in
particular, what is needed to introduce gauge models in physics, and it also
contains naturally the essential aspects of the Higgs fields and its associated
mechanics of mass generation. It permits one also to extend some previous
constructions, as for instance symmetric reduction of (here noncommutative)
connections. From a mathematical point of view, these geometrico-algebraic
considerations highlight some new point on view, in particular we introduce a
new construction of the Chern characteristic classes
Constructions, inductive types and strong normalization
This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the..
Unbased calculus for functors to chain complexes
Recently, the Johnson-McCarthy discrete calculus for homotopy functors was
extended to include functors from an unbased simplicial model category to
spectra. This paper completes the constructions needed to ensure that there
exists a discrete calculus tower for functors from an unbased simplicial model
category to chain complexes over a fixed commutative ring. Much of the
construction of the Taylor tower for functors to spectra carries over to this
context. However, one of the essential steps in the construction requires
proving that a particular functor is part of a cotriple. For this, one needs to
prove that certain identities involving homotopy limits hold up to isomorphism,
rather than just up to weak equivalence. As the target category of chain
complexes is not a simplicial model category, the arguments for functors to
spectra need to be adjusted for chain complexes. In this paper, we take
advantage of the fact that we can construct an explicit model for iterated
fibers, and prove that the functor is a cotriple directly. We use related ideas
to provide concrete infinite deloopings of the first terms in the resulting
Taylor towers when evaluated at the initial object in the source category.Comment: 20 page
A calculus of constructions with explicit subtyping
International audienceThe calculus of constructions can be extended with an infinite hierarchy of universes and cumulative subtyping. In this hierarchy, each universe is contained in a higher universe. Subtyping is usually left implicit in the typing rules. We present an alternative version of the calculus of constructions where subtyping is explicit. This new system avoids problems related to coercions and dependent types by using the Tarski style of universes and by introducing additional equations to reflect equality
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