5,048 research outputs found
On the proof-theoretic strength of monotone induction in explicit mathematics
AbstractWe characterize the proof-theoretic strength of systems of explicit mathematics with a general principle (MID) asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ12-axiom of choice and Π12-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems
Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types
Type systems certify program properties in a compositional way. From a bigger
program one can abstract out a part and certify the properties of the resulting
abstract program by just using the type of the part that was abstracted away.
Termination and productivity are non-trivial yet desired program properties,
and several type systems have been put forward that guarantee termination,
compositionally. These type systems are intimately connected to the definition
of least and greatest fixed-points by ordinal iteration. While most type
systems use conventional iteration, we consider inflationary iteration in this
article. We demonstrate how this leads to a more principled type system, with
recursion based on well-founded induction. The type system has a prototypical
implementation, MiniAgda, and we show in particular how it certifies
productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317
Derived rules for predicative set theory: an application of sheaves
We show how one may establish proof-theoretic results for constructive
Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and
the Bar Induction rule for Baire space, by constructing sheaf models and using
their preservation properties
From coinductive proofs to exact real arithmetic: theory and applications
Based on a new coinductive characterization of continuous functions we
extract certified programs for exact real number computation from constructive
proofs. The extracted programs construct and combine exact real number
algorithms with respect to the binary signed digit representation of real
numbers. The data type corresponding to the coinductive definition of
continuous functions consists of finitely branching non-wellfounded trees
describing when the algorithm writes and reads digits. We discuss several
examples including the extraction of programs for polynomials up to degree two
and the definite integral of continuous maps
The topological structure of direct limits in the category of uniform spaces
Let be a sequence of uniform spaces such that each space is
a closed subspace in . We give an explicit description of the topology
and uniformity of the direct limit of the sequence in the
category of uniform spaces. This description implies that a function to a uniform space is continuous if for every the restriction
is continuous and regular at the subset in the sense that for
any entourages U\in\U_Y and V\in\U_X there is an entourage V\in\U_X such
that for each point there is a point with
and . Also we shall compare topologies of
direct limits in various categories.Comment: 10 page
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
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