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62
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 ..."
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Cited by 162 (4 self)
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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 ..."
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Note: This document accompanies the paper “Practical type inference for arbitraryrank types ” [6]. Prior reading of the main paper is required. 1 Contents
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 ..."
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Cited by 85 (12 self)
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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 .
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 ..."
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Cited by 85 (18 self)
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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 ...
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. ..."
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Cited by 17 (3 self)
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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 ..."
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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 ..."
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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
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 ..."
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Cited by 6 (5 self)
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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.
Weaker DComplete Logics
 University of Wollongong Department
"... BB 0 IW logic (or T! ) is known to be Dcomplete. This paper shows that there are infinitely many weaker Dcomplete logics and it also examines how certain Dincomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction ..."
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BB 0 IW logic (or T! ) is known to be Dcomplete. This paper shows that there are infinitely many weaker Dcomplete logics and it also examines how certain Dincomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction The condensed detachment rule, first proposed by C. A. Meredith in Lemmon et al [5], is a form of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification algorithm [8]. Roughly, a system of implicational logic is Dcomplete if the system with the same axioms, but with condensed detachment (D) instead of modus ponens and substitution, has the same theorems. To show that a logic is Dcomplete it is sufficient to show that all the substitution instances of its axioms are deducible in the corresponding condensed logic (i.e. the logic with rule D only). It is well know...
Polymorphic type inference for the relational algebra
 Journal of Computer and System Sciences
"... We give a polymorphic account of the relational algebra. We introduce a formalism of “type formulas ” specifically tuned for relational algebra expressions, and present an algorithm that computes the “principal ” type for a given expression. The principal type of an expression is a formula that spec ..."
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We give a polymorphic account of the relational algebra. We introduce a formalism of “type formulas ” specifically tuned for relational algebra expressions, and present an algorithm that computes the “principal ” type for a given expression. The principal type of an expression is a formula that specifies, in a clear and concise manner, all assignments of types (sets of attributes) to relation names, under which a given relational algebra expression is welltyped, as well as the output type that expression will have under each of these assignments. Topics discussed include complexity and polymorphic expressive power. 1