30 research outputs found
McShane-Whitney extensions in constructive analysis
Within Bishop-style constructive mathematics we study the classical
McShane-Whitney theorem on the extendability of real-valued Lipschitz functions
defined on a subset of a metric space. Using a formulation similar to the
formulation of McShane-Whitney theorem, we show that the Lipschitz real-valued
functions on a totally bounded space are uniformly dense in the set of
uniformly continuous functions. Through the introduced notion of a
McShane-Whitney pair we describe the constructive content of the original
McShane-Whitney extension and examine how the properties of a Lipschitz
function defined on the subspace of the pair extend to its McShane-Whitney
extensions on the space of the pair. Similar McShane-Whitney pairs and
extensions are established for H\"{o}lder functions and -continuous
functions, where is a modulus of continuity. A Lipschitz version of a
fundamental corollary of the Hahn-Banach theorem, and the approximate
McShane-Whitney theorem are shown
Direct spectra of Bishop spaces and their limits
We apply fundamental notions of Bishop set theory (BST), an informal theory
that complements Bishop's theory of sets, to the theory of Bishop spaces, a
function-theoretic approach to constructive topology. Within BST we develop the
notions of a direct family of sets, of a direct spectrum of Bishop spaces, of
the direct limit of a direct spectrum of Bishop spaces, and of the inverse
limit of a contravariant direct spectrum of Bishop spaces. Within the extension
of Bishop's informal system of constructive mathematics BISH with inductive
definitions with rules of countably many premises, we prove the fundamental
theorems on the direct and inverse limits of spectra of Bishop spaces and the
duality principle between them
Proof-relevance in Bishop-style constructive mathematics
Bishop's presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory (BST), our minimal extension of Bishop's theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas f of BISH a set Prf(f) of "proofs" or witnesses of f. Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of aMartin-Lof set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional MartinLof type theory (MLTT). Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky's axiom of univalence) can be translated into BISH. While Bishop's theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction
Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory
Bishop's measure theory (BMT) is an abstraction of the measure theory of a
locally compact metric space , and the use of an informal notion of a
set-indexed family of complemented subsets is crucial to its predicative
character. The more general Bishop-Cheng measure theory (BCMT) is a
constructive version of the classical Daniell approach to measure and
integration, and highly impredicative, as many of its fundamental notions, such
as the integration space of -integrable functions , rely on
quantification over proper classes (from the constructive point of view). In
this paper we introduce the notions of a pre-measure and pre-integration space,
a predicative variation of the Bishop-Cheng notion of a measure space and of an
integration space, respectively. Working within Bishop Set Theory (BST), and
using the theory of set-indexed families of complemented subsets and
set-indexed families of real-valued partial functions within BST, we apply the
implicit, predicative spirit of BMT to BCMT. As a first example, we present the
pre-measure space of complemented detachable subsets of a set with the
Dirac-measure, concentrated at a single point. Furthermore, we translate in our
predicative framework the non-trivial, Bishop-Cheng construction of an
integration space from a given measure space, showing that a pre-measure space
induces the pre-integration space of simple functions associated to it.
Finally, a predicative construction of the canonically integrable functions
, as the completion of an integration space, is included.Comment: 29 pages; shortened and corrected versio
Strong negation in the theory of computable functionals TCF
We incorporate strong negation in the theory of computable functionals TCF, a
common extension of Plotkin's PCF and G\"{o}del's system , by
defining simultaneously the strong negation of a formula
and the strong negation of a predicate in TCF. As a
special case of the latter, we get the strong negation of an inductive and a
coinductive predicate of TCF. We prove appropriate versions of the Ex falso
quodlibet and of the double negation elimination for strong negation in TCF,
and we study the so-called tight formulas of TCF i.e., formulas implied from
the weak negation of their strong negation. We present various case-studies and
examples, which reveal the naturality of our definition of strong negation in
TCF and justify the use of TCF as a formal system for a large part of
Bishop-style constructive mathematics
Direct spectra of Bishop spaces and their limits
We apply fundamental notions of Bishop set theory (BST), an informal theory
that complements Bishop's theory of sets, to the theory of Bishop spaces, a
function-theoretic approach to constructive topology. Within BST we develop the
notions of a direct family of sets, of a direct spectrum of Bishop spaces, of
the direct limit of a direct spectrum of Bishop spaces, and of the inverse
limit of a contravariant direct spectrum of Bishop spaces. Within the extension
of Bishop's informal system of constructive mathematics BISH with inductive
definitions with rules of countably many premises, we prove the fundamental
theorems on the direct and inverse limits of spectra of Bishop spaces and the
duality principle between them
Constructive topology of bishop spaces
The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS.
Instead of having a common space-structure on a set X and R, where R denotes the set of
constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F),
where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R.
The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions,
the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a
constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces.
The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient,
the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact,
the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces.
The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively
the Urysohn extension theorem for Bishop spaces.
The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included
Fusion of Knowledge-Based and Data-Driven Approaches to Grammar Induction
Georgiladakis S, Unger C, Iosif E, et al. Fusion of Knowledge-Based and Data-Driven Approaches to Grammar Induction. In: Fifteenth Annual Conference of the International Speech Communication Association. 2014.Using different sources of information for grammar induction results in grammars that vary in coverage and precision. Fusing such grammars with a strategy that exploits their strengths while minimizing their weaknesses is expected to produce grammars with superior performance. We focus on the fusion of grammars produced using a knowledge-based approach using lexicalized ontologies and a data-driven approach using semantic similarity clustering. We propose various algorithms for finding the map- ping between the (non-terminal) rules generated by each gram- mar induction algorithm, followed by rule fusion. Three fusion approaches are investigated: early, mid and late fusion. Results show that late fusion provides the best relative F-measure per- formance improvement by 20%
Embeddings of Bishop spaces
We develop the basic constructive theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of topological spaces. The theory of Bishop spaces is a constructive approach to point-function topology and a natural constructive alternative to the classical theory of the rings of continuous functions. Our most significant result is the translation of the classical Urysohn extension theorem within the theory of Bishop spaces. The related theory of the zero sets of a Bishop topology is also included. We work within BISH*, Bishop's informal system of constructive mathematics BISH equipped with inductive definitions with rules of countably many premises
A Direct Constructive Proof of a Stone-Weierstrass Theorem for Metric Spaces
We present a constructive proof of a Stone-Weierstrass theorem for totally bounded metric spaces (SWtbms) which implies Bishop's Stone-Weierstrass theorem for compact metric spaces (BSWcms) found in [3]. Our proof has a clear computational content, in contrast to Bishop's highly technical proof of BSWcms and his hard to motivate concept of a (Bishop-)separating set of uniformly continuous functions. All corollaries of BSWcms in [3] are proved directly by SWtbms. We work within Bishop's informal system of constructive mathematics BISH