Results 1  10
of
18
Type Theory and Programming
, 1994
"... This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents MartinLof's monomorphic type theory and an im ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents MartinLof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.
A short and flexible proof of Strong Normalization for the Calculus of Constructions
, 1994
"... this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through is a slight strengthening of the Stripping property (also called Generation). This property says, for example, that if \Gamma ` v:T:M : U has a derivation D, then one can find a subderivation of
Parameterised multiparty session types
 In FOSSACS, LNCS
, 2010
"... Abstract. For many applicationlevel distributed protocols and parallel algorithms, the set of participants, the number of messages or the interaction structure are only known at runtime. This paper proposes a dependent type theory for multiparty sessions which can statically guarantee typesafe, d ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
Abstract. For many applicationlevel distributed protocols and parallel algorithms, the set of participants, the number of messages or the interaction structure are only known at runtime. This paper proposes a dependent type theory for multiparty sessions which can statically guarantee typesafe, deadlockfree multiparty interactions among processes whose specifications are parameterised by indices. We use the primitive recursion operator from Gödel’s System T to express a wide range of communication patterns while keeping type checking decidable. We illustrate our type theory through nontrivial programming and verification examples taken from parallel algorithms and Web services usecases. 1
Kreisel's `Unwinding Program
 In Odifreddi [53
, 1996
"... Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give
The "Hardest" Natural Decidable Theory
 12th Annual IEEE Symp. on Logic in Computer Science (LICS'97)', IEEE
, 1997
"... We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linea ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2's and since midseventies it was an open problem whether natural decidable theories requiring more than that exist [12, 2]. We give the affirmative answer. As an application of this today's strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus. 1. Introduction In his survey paper [12] A. Meyer mentioned (p. 479), as a curious empirical observation, that all known natural decidable nonelementary problems require at most (upper bound) F (1; n) = exp 1 (n) = 2 2 \Delta \Delta \Delta 2 oe n Turing machine steps on inputs of length n to decide 1 . Until now the highest known lower bounds for natu...
The Calculus of Constructions and Higher Order Logic
 In preparation
, 1992
"... The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calcul ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calculus, and a first order dependent typed lambda calculus in the style of de Bruijn's Automath ([de Bruijn 1980]) or MartinLof's intuitionistic theory of types ([MartinLof 1984]). Using the impredicative coding of data types in F! , the Calculus of Constructions thus becomes a higher order language for the typing of functional programs. We shall introduce and try to explain CC by exploiting especially the first point of view, by introducing a typed lambda calculus that faithfully represent higher order predicate logic (so for this system the CurryHoward `formulasastypes isomorphism' is really an isomorphism.) Then we discuss some propositions that are provable in CC but not in the higher or...
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Classifying Recursive Functions
, 1997
"... computability. With the preparations done it is now rather straightforward to define computability in our iterated function spaces C ae (based on N viewed as a flat information system). The tokens and finite sets of tokens are encodable by integers using sequencecoding. It is easy to see that the ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
computability. With the preparations done it is now rather straightforward to define computability in our iterated function spaces C ae (based on N viewed as a flat information system). The tokens and finite sets of tokens are encodable by integers using sequencecoding. It is easy to see that the notions X 2 Con ae of consistency and X ` ae a of entailment correspond to recursive (in fact elementary) relations. Definition. A partial continuous functional ' of type ae is said to be computable if  when viewed as a set of (codes of) tokens  it is \Sigma 0 1 definable. 5. Computability in higher types 27 Lemma. For all types ae; oe; ø the functionals eval ae;oe : (C ae ! C oe ) \Theta C ae ! C oe curry ae;oe;ø : (C ae \Theta C oe ! C ø ) ! (C ae ! (C oe ! C ø )) are computable. Proof. The tokens of eval ae;oe are of the form (W; X; a) with W 2 Con ae!oe , X 2 Con ae and a 2 C oe , and we have (W; X; a) 2 eval j a 2 W (X) j a 2 WX j WX ` a: Here WX is the application of fini...
Type Inference and Reconstruction for First Order Dependent Types
, 1995
"... x 1 Introduction 1 1.1 Dependent Types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Dependent Type Inference and Reconstruction : : : : : : : : : : : : : : : : 8 2 Primitive Recursive Functionals with Dependent Types 17 2.1 A Dependent Type System for T : : : : : : : : : ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
x 1 Introduction 1 1.1 Dependent Types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Dependent Type Inference and Reconstruction : : : : : : : : : : : : : : : : 8 2 Primitive Recursive Functionals with Dependent Types 17 2.1 A Dependent Type System for T : : : : : : : : : : : : : : : : : : : : : : : 17 2.1.1 Terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 2.1.2 Types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.1.3 Typing Rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 2.1.4 Strong Normalization of T Terms : : : : : : : : : : : : : : : : : : : 28 2.2 Dependent Typing Examples : : : : : : : : : : : : : : : : : : : : : : : : : : 29 2.3 A Term Model Semantics for T : : : : : : : : : : : : : : : : : : : : : : : : 34 3 Principal Types and Dependent Type Reconstruction 58 3.1 Type Subsumption and Unification : : : : : : : : : : : : : : : : : : : : : : : 58 3.2 Matching : : : : : :...