Results 1  10
of
85
Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
Abstract

Cited by 165 (4 self)
 Add to MetaCart
Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Practical type inference for arbitraryrank types
 Journal of Functional Programming
, 2005
"... Note: This document accompanies the paper “Practical type inference for arbitraryrank types ” [6]. Prior reading of the main paper is required. 1 Contents ..."
Abstract

Cited by 98 (22 self)
 Add to MetaCart
Note: This document accompanies the paper “Practical type inference for arbitraryrank types ” [6]. Prior reading of the main paper is required. 1 Contents
MultiStage Programming: Its Theory and Applications
, 1999
"... MetaML is a statically typed functional programming language with special support for program generation. In addition to providing the standard features of contemporary programming languages such as Standard ML, MetaML provides three staging annotations. These staging annotations allow the construct ..."
Abstract

Cited by 89 (18 self)
 Add to MetaCart
MetaML is a statically typed functional programming language with special support for program generation. In addition to providing the standard features of contemporary programming languages such as Standard ML, MetaML provides three staging annotations. These staging annotations allow the construction, combination, and execution of objectprograms. Our thesis is that MetaML's three staging annotations provide a useful, theoretically sound basis for building program generators. This dissertation reports on our study of MetaML's staging constructs, their use, their implementation, and their formal semantics. Our results include an extended example of where MetaML allows us to produce efficient programs, an explanation of why implementing these constructs in traditional ways can be challenging, two formulations of MetaML's semantics, a type system for MetaML, and a proposal for extending ...
The Essence of Principal Typings
 In Proc. 29th Int’l Coll. Automata, Languages, and Programming, volume 2380 of LNCS
, 2002
"... Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meanin ..."
Abstract

Cited by 87 (12 self)
 Add to MetaCart
Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meaning that M has result type # when assuming the types of free variables are given by A. Then (A, #) is a typing for M .
Recursive abstract state machines
 J. of Universal Computer Science
, 1997
"... Abstract: As introduced in the Lipari guide, Abstract State Machines (abbreviated as ASMs) are untyped. This is useful for many purposes. However, typed languages have their own advantages. Types structure the data, type checking uncovers errors. Here we propose a typed version of ASMs. ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
Abstract: As introduced in the Lipari guide, Abstract State Machines (abbreviated as ASMs) are untyped. This is useful for many purposes. However, typed languages have their own advantages. Types structure the data, type checking uncovers errors. Here we propose a typed version of ASMs.
Lambda Terms for Natural Deduction, Sequent Calculus and Cut Elimination
"... It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequ ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a oneone map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (N) and the other to sequent calculus (L). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the manyone correspondence mentioned above. Moreover, the second type assignment system has a `cutfree' fragment (L cf ). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form.
Parsing and Generation as Datalog Queries
"... We show that the problems of parsing and surface realization for grammar formalisms with “contextfree ” derivations, coupled with Montague semantics (under a certain restriction) can be reduced in a uniform way to Datalog query evaluation. As well as giving a polynomialtime algorithm for computing ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
We show that the problems of parsing and surface realization for grammar formalisms with “contextfree ” derivations, coupled with Montague semantics (under a certain restriction) can be reduced in a uniform way to Datalog query evaluation. As well as giving a polynomialtime algorithm for computing all derivation trees (in the form of a shared forest) from an input string or input logical form, this reduction has the following complexitytheoretic consequences for all such formalisms: (i) the decision problem of recognizing grammaticality (surface realizability) of an input string (logical form) is in LOGCFL; and (ii) the search problem of finding one logical form (surface string) from an input string (logical form) is in functional LOGCFL. Moreover, the generalized supplementary magicsets rewriting of the Datalog program resulting from the reduction yields efficient Earleystyle algorithms for both parsing and generation. 1
Generating Control Languages with Abstract Categorial Grammars
, 2007
"... We show that the class of levelk control languages, as defined by Weir, is properly included in the class of 2 k−1multiple contextfree languages for each k ≥ 2. The proof of inclusion uses a representation of the set of derivation trees for a levelk control language in terms of a secondorder ab ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We show that the class of levelk control languages, as defined by Weir, is properly included in the class of 2 k−1multiple contextfree languages for each k ≥ 2. The proof of inclusion uses a representation of the set of derivation trees for a levelk control language in terms of a secondorder abstract categorial grammar.
Parallel Reduction in Type Free λµCalculus
 KYUSHU UNIVERSITY
, 2000
"... Typed λµcalculus is known to be strongly normalizing and weakly ChurchRosser, and hence confluent. In fact, Parigot formulated a parallel reduction to prove confluency of typed λµcalculus by "TaitandMartinLöf" method. However, the diamond property does not hold for his parallel reduc ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Typed λµcalculus is known to be strongly normalizing and weakly ChurchRosser, and hence confluent. In fact, Parigot formulated a parallel reduction to prove confluency of typed λµcalculus by "TaitandMartinLöf" method. However, the diamond property does not hold for his parallel reduction. The confluency for typefree λµcalculus cannot be derived from that of typed λµcalculus and is not known. We analyzed granualities of the reduction rules. We consider a renaming and consecutive structural reductions as one step parallel reduction, and show that the new formulation of parallel reduction has the diamond property, which yields the correct proof of confluency of type free λµcalculus. The diamond property of new parallel reduction is also shown for the callbyvalue version of λµcalculus contains the symmetric structural reduction rule.
Strict Intersection Types for the Lambda Calculus
, 2010
"... This paper will show the usefulness and elegance of strict intersection types for the Lambda Calculus; these are strict in the sense that they are the representatives of equivalence classes of types in the BCDsystem [15]. We will focus on the essential intersection type assignment; this system is a ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
This paper will show the usefulness and elegance of strict intersection types for the Lambda Calculus; these are strict in the sense that they are the representatives of equivalence classes of types in the BCDsystem [15]. We will focus on the essential intersection type assignment; this system is almost syntax directed, and we will show that all major properties hold that are known to hold for other intersection systems, like the approximation theorem, the characterisation of (head/strong) normalisation, completeness of type assignment using filter semantics, strong normalisation for cutelimination and the principal pair property. In part, the proofs for these properties are new; we will briefly compare the essential system with other existing systems.