Results 1  10
of
10
A Semantical Storage Operator Theorem For All Types
, 1997
"... Storage operators are terms which simulate callbyvalue in callbyname for a given set of terms. Krivine's storage operator theorem shows that any term of type :D ! :D , where D is the Godel translation of D, is a storage operator for the terms of type D when D is a datatype or a formula ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Storage operators are terms which simulate callbyvalue in callbyname for a given set of terms. Krivine's storage operator theorem shows that any term of type :D ! :D , where D is the Godel translation of D, is a storage operator for the terms of type D when D is a datatype or a formula with only positive second order quantifiers. We prove that a new semantical version of Krivine's theorem is valid for every types. This also gives a simpler proof of Krivine's theorem using the properties of datatypes. Key words: calculus. Types. AF 2 type system. Storage operators. Godel translations. 1 Introduction. The notion of storage operator was introduced by Krivine in [3]. A storage operator for a set of terms D is a term T simulating callbyvalue in headreduction for all elements in D: for t in D, (T ' t) headreduces to (' t 0 ) where t 0 fi t only depends on the normal form of t (the actual definition is slightly more complex). The storage operator theorem is valid for a typ...
Simple Proof of the Completeness Theorem for Second Order Classical and Intuitionistic Logic
"... We present a simpler way than usual to deduce the completeness theorem for the secondoder classical logic from the rstorder one. We also extend our method to the case of secondorder intuitionnistic logic. 1 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We present a simpler way than usual to deduce the completeness theorem for the secondoder classical logic from the rstorder one. We also extend our method to the case of secondorder intuitionnistic logic. 1
Getting results from programs extracted from classical proofs.
, 2002
"... We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is verified on a non trivial example: a proof of Dickson’s lemma. 1 Introduction. Since Griffin and Felleisen [7, 8], we know a relation between classical proofs and programs. However, it is not true that from a classical proof of the existence of an object you can compute this object. This would clearly be a contradiction with the existence of provably total but non computable functions (such as a function saying
System ST  Toward A Type System for Extraction AND Proof of Programs
, 2001
"... We introduce a new type system called \System ST" (ST stands for SubTyping), based on subtyping, and prove the basic property of the system. We show the extraordinary expressive power of the system which leads us to think that it could be a good candidate for doing both proof and extraction of pr ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We introduce a new type system called \System ST" (ST stands for SubTyping), based on subtyping, and prove the basic property of the system. We show the extraordinary expressive power of the system which leads us to think that it could be a good candidate for doing both proof and extraction of programs.
System ST, βreduction and completeness
 In Logic In Computer Science
, 2003
"... We prove that system ST (introduced in a previous work) enjoys subject reduction and is complete for realizability semantics. As far as the author knows, this is the only type system enjoying the second property. System ST is a very expressive type system, whose principle is to use two kinds of form ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We prove that system ST (introduced in a previous work) enjoys subject reduction and is complete for realizability semantics. As far as the author knows, this is the only type system enjoying the second property. System ST is a very expressive type system, whose principle is to use two kinds of formulae: types (formulae with algorithmic content) and propositions (formulae without algorithmic content). The fact that subtyping is used to build propositions and that propositions can be used in types trough a special implication gives its great expressive power to the system: all the operators you can imagine are definable (union, intersection, singleton,...).
Completeness, Minimal Logic and Programs Extraction
"... We formalize, in the AF 2 type system, a completeness and correctness theorem for a modied Kripke semantics for minimal logic. We show that any program extracted from a proof of the completeness theorem translates an higherorder encoding of a term into a Debruijn encoding of another term with the ..."
Abstract
 Add to MetaCart
We formalize, in the AF 2 type system, a completeness and correctness theorem for a modied Kripke semantics for minimal logic. We show that any program extracted from a proof of the completeness theorem translates an higherorder encoding of a term into a Debruijn encoding of another term with the same type. We also show that any program extracted from the correctness theorem performs the reverse translation. Moreover in both cases, there is one proof which corresponds to a program that does not change the underlying term. Key words: calculus. Types. AF 2 type system. Intuitionnistic Logic. Program Extraction. 1 Introduction The CurryHoward isomorphism [2] establishes a correspondence between proofs and programs. We can extract a correct program from a proof of its specication. But to what specication corresponds a mathematical theorem like the completeness theorem ? More precisely, is there a property characterizing any program extracted from a proof of the completene...
Type Checking in System . . .
"... The main contribution of this paper is a partial typechecking algorithm for the system F and its use in a programming language like ML. We dene this system as an extension of the secondorder calculus (system F) verifying the preservation of type during computation (subjectreduction) for red ..."
Abstract
 Add to MetaCart
The main contribution of this paper is a partial typechecking algorithm for the system F and its use in a programming language like ML. We dene this system as an extension of the secondorder calculus (system F) verifying the preservation of type during computation (subjectreduction) for reduction (this result fails for reduction in system F). Our presentation is based on an original notion of subtyping which includes all the handling of quantication rules. 1 Introduction. Motivation. Type systems have proved to be useful for many modern functional programming languages such as ML, Miranda, Haskell, . . . . In most cases, the basis of the type system is Milner's algorithm [12]. The main characteristic of these type systems is polymorphism which allows the programmer to write generic functions that can work on arguments of dierent types. However it is often insucient: polymorphic recursion, existential types or the state monad of Haskell are treated using specic exte...
unknown title
, 905
"... Simple proof of the completeness theorem for second order classical and intuitionistic logic by reduction to firstorder monosorted logic ∗ ..."
Abstract
 Add to MetaCart
Simple proof of the completeness theorem for second order classical and intuitionistic logic by reduction to firstorder monosorted logic ∗
unknown title
"... Simple proof of the completeness theorem for second order classical and intuitionistic logic by reduction to firstorder monosorted logic ..."
Abstract
 Add to MetaCart
Simple proof of the completeness theorem for second order classical and intuitionistic logic by reduction to firstorder monosorted logic
unknown title
, 2009
"... Simple proof of the completeness theorem for second order classical and intuitionistic logic by reduction to firstorder monosorted logic ∗ ..."
Abstract
 Add to MetaCart
Simple proof of the completeness theorem for second order classical and intuitionistic logic by reduction to firstorder monosorted logic ∗