Leivant, D. Reasoning About Functional Programs and Complexity Classes Associated with Type Disciplines. in: 24th Annual Symposium on Foundations of Computer Science, IEEE, Tucson, Arizona, November 7--9. 1983, pp. 460--469.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Types will be formulas of second order predicate calculus, and not only, as in system F, second order propositional calculus [5, 6] In a certain sense, this is a harmless extension, since the # terms which are typable are the same. This kind of extension has already been considered by D. Leivant [11]. A much more serious extension is the following: the underlying logic will be classical logic, and not only, as in system F, intuitionistic logic. Extraction of programs from classical proofs has been considered, since two or three years by several people (C. Murthy [12] J.Y. Girardapproach ....

D. Leivant. Reasoning about functional programs and complexity classes associated with type disciplines. 24th Annual Symp. on Found. of Comp. Sc. p. 460-469 (1983).

....This proposition states that the trees we considered are representable as lambda terms in such a way that the constructors ( leaf and ) are lambda definable. In fact, the lambda terms involved can be typed in #2. A nice connection between these terms and proofs in second order logic is given in [79]. Now we will show that iterative functions over these trees, like f mir , are lambda definable. THE IMPACT OF THE LAMBDA CALCULUS IN LOGIC AND COMPUTER SCIENCE 191 Proposition 3.3 (Iteration) Given lambda terms A 0 , A 1 , A 2 there exists a lambda term F such that (for variables n, t 1 , t 2 ....

D. Leivant, Reasoning about functional programs and complexity classes associated with type disciplines, 24th annual symposium on foundations of computer science, IEEE, 1983, pp. 460--469.

....programs requiring too much type information. Related work. A lot of work has been devoted to type systems and particularly to system F. The undecidability of both type checking and type inference has been establish by Wells [18] The decidability problem of many strati cations has been studied [3, 6, 7, 9, 10, 11]. It is our opinion that the algorithms developed so far are not as simple as the one we present. However, they are usually complete for well de ned fragments of system F. On the other hand our algorithm works for the complete system F (and therefore system F which is a sub system) 2 but is ....

D. Leivant. Reasoning about functional programs and complexity classes associated with type disciplines. In 24th Annual Symposium on Foundations of Computer Science, volume 44, pages 460-469, 1983.

.... slightly more complex) The storage operator theorem is valid for a type D, if any term of type :D :D is a storage operator for the elements of D (D being the Godel translation of D) In [4] Krivine proves the storage operator theorem for the type of Church integer in the AF 2 type system [5,8,9]. This result has been 1 I shall thank Jean Louis Krivine for his comments and the fruitful discussions we had. Preprint submitted to Elsevier Preprint 9 July 1997 extended to any 8 positive type D (a type with only positive 8 quantifiers) by Krivine [6] using a semantical proof) and Nour ....

D. Leivant. Reasoning about functional programs and complexity classes associated with type disciplines. In 24th Annual Symposium on Foundations of Computer Science, volume 44, pages 460--469, 1983.

....may be viewed as codata types, their introductions as corecursors and eliminations as codata destructors. In the literature, a fairly large number of axiomatizations and encodings of both particular [co]inductively de ned types and general [co]inductive de nition operators can be found, see e.g. [1,14,19,20,24,25,15,7]. The paper grew out of a wish to better understand their individual properties and their relations to each other. The contribution of the paper consists in a coordinated analysis of eight intensional semantically distinct pairs of [co]inductive de nition operators, arranged into a cube shaped ....

....of a monotonic function is its least [greatest] pre xed [post xed] point. In its general form, the encoding seems to be a piece of folklore. For the special case of polynomial proposition functions (such as N) essentially the same encoding was rst given by B ohm and Berarducci [1] and Leivant [14]. For naturals, our encoding specializes to the following: Nat ] 8 2 ( X) X X) X) zero ] inl(hi) succ ] c) inr(c ) natcata ] c; e z ; e s ) c ( case( e z ; e s ( In B ohm and Berarducci s encoding, Nat ] 8 2 ( X)X (X X) ....

[Article contains additional citation context not shown here]

D. Leivant, Reasoning about functional programs and complexity classes associated with type disciplines, in: Proceedings 24th Annual IEEE Symp. on Foundations of Computer Science, FOCS'83 (Tucson, AZ, USA, Nov. 1983) (IEEE CS Press, Los Alamitos, CA, 1983) 460-469.

....relation whenever the relations corresponding to the open type variables are taken to be the identity relation. Both Girard s and Reynold s results are sufficiently interesting to have spawned a large body of related work. Girard s representation theorem has been further explored by Leivant [Lei83] and by Krivine and Parigot [KP90] among others, with tutorials by Girard, Taylor, and Lafont [GLT89] and by Leivant [Lei90] The representation of algebraic data types in polymorphic lambda calculus was proposed by Bohm and Berarducci [BB85] Reynolds s parametricity has been further explored by ....

....parametricity. There is nothing built into the logic about types or relations, with the exception of the parametricity postulate itself. There is also another sense in which Reynold s embedding is inverse to Girard s projection, which plays a central role in the proof of the representation theorem [Gir72, GLT89, KP90, Lei83, Lei90]. If one takes the normal form proof that an untyped Church numeral belongs to d Nat and applies the Girard projection, the result is the corresponding typed Church numeral in Nat; applying the Reynolds embedding then yields the original proof. This paper can be viewed as expository in nature. The ....

[Article contains additional citation context not shown here]

D. Leivant, Reasoning about functional programs and complexity classes associated with type disciplines, 24'th Symposium on Foundations of Computer Science, Washington D.C., IEEE, 460--469.

....relation whenever the relations corresponding to the open type variables are taken to be the identity relation. Both Girard s and Reynold s results are sufficiently interesting to have spawned a large body of related work. Girard s representation theorem has been further explored by Leivant [Lei83] and by Krivine and Parigot [KP90] among others, with tutorials by Girard, Taylor, and Lafont [GLT89] and by Leivant [Lei90] The representation of algebraic data types in polymorphic lambda calculus was proposed by Bohm and Berarducci [BB85] Reynolds s parametricity has been further explored by ....

....There is nothing built into the logic about types or relations, with the exception of the parametricity postulate itself. There is also another sense in which Reynold s embedding is inverse to Girards s projection, which plays a central role in the proof of the representation theorem [Gir72, GLT89, KP90, Lei83, Lei90]. If one takes the normal form proof that an untyped Church numeral belongs to Nat and applies the Girard projection, the result is the corresponding typed Church numeral in Nat; applying the Reynolds embedding then yields the original proof. This paper can be viewed as expository in nature. The ....

[Article contains additional citation context not shown here]

D. Leivant, Reasoning about functional programs and complexity classes associated with type disciplines, 24'th Symposium on Foundations of Computer Science, Washington D.C., IEEE, 460--469.

....the # term begins by a #. Thus, the rule for call by name reduction is: #x u)tt 1 . t k # u[t x]t 1 . t k . We consider a second order type assignment system for this # calculus, in which types are formulas of second order predicate logic. Such a type system has already been used in [9, 4, 8]. It allows to get # terms from proofs in second order intuitionistic logic, by means of the well known Curry Howard isomorphism. The notion of storage operator defined in [5, 6] appears as an important tool in the study of this second order call by name # calculus. They are closed # terms which ....

....typed terms i.e. expressions of the following form: x 1 : A 1 , x k : A k # E # : A, where A 1 , A k , A are formulas of L, x 1 , x k are # variables, # is a # term, and E a system of equations for integers. The rules of construction of typed terms are as follows ([4, 8, 9]) 1. x 1 : A 1 , x k : A k # E x i : A i . 2. If x 1 : A 1 , x k : A k , x : A # E # : B, then x 1 : A 1 , x k : A k # E #x # : A # B. 3. If x 1 : A 1 , x k : A k # E # : A # B, # # : A then x 1 : A 1 , x k : A k # E ## # : B. 4. If x 1 : A ....

D. Leivant. Reasoning about functional programs and complexity classes associated with type disciplines. 24th Annual Symp. on Found. of Comp. Sc. p. 460-469 (1983).

....system [4] namely propositional logic over implication only. The most fruitful ones are the intuitionistic restrictions of Second order propositional logic (Girard s System F ( 7] 8] 9] rediscovered by Reynolds [30] in a computer science context) and Second order predicate logic (Leivant [22], Krivine s System FA 2 [15, p.141] and extensions (Parigot [27] and Raoeali [29] 0.2.4 Programming with classical proofs. At the beginning of the 90 s it was realized by two computer scientists, Grif n [11] followed by Murthy ( 23] 24] that, surprisingly, classical logic could also be ....

D. Leivant, Reasoning about functional programs and complexity classes associated with type discipline, 24th Annual Symp. on Foundations of Computer Science, p.460-469, 1983.

....For the Calculus of Constructions, this has a precise statement in the result of Berardi and Mohring [81,7] Theorem CC is conservative over F . Consequently, CC can represent, in the standard representation of functions on inductive types due variously to Church, Girard, Leivant and others [32,56,9], no more functions than F . The proof is based on a syntactic mapping, the so called Berardi Mohring projection. Mohring used this mapping as an extraction function, which, coupled with the associated realisability predicate, allows a powerful and flexible approach to program development from ....

.... Gamma nat Phi:nat Prop: z: Phi(0) s:8k:nat : Phi(k) Phi(S(Sk) z : 8 Phi:nat Prop: Phi(0) 8k:nat: Phi(k) Phi(S(Sk) Phi(0) is a proof that 0 is even. Such definitions arise in much the same way as Church s representation of the datatypes in second order calculus [32, 56,9]; Universes and cumulativity As simple examples of the use of universes, we may derive, for each i; j 2 with i j, Type i Type i : Type j and :Type i :x: x : Type i Type i ; Chapter 2. Type theoretic preliminaries: ECC and LEGO 20 defining a polymorphic identity function, one for ....

D.Leivant, Reasoning about functional programs and complexity classes associated with type disciplines, in: Proceedings of the 24th IEEE symposium on Foundations of Computer Science, 1983.

.... The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard s LC and Parigot s , FD ( 10, 12, 29, 33] delineates other viable systems as well, and gives means to extend the Krivine Leivant paradigm of programming with proofs ([24, 25]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non additive proof nets, to be precise) using appropriate embeddings (so called decorations) it is unifying: it organizes known solutions in a simple pattern that ....

.... Curry Howard correspondence ( 16] between proofs in intuitionistic logic and terms (providing, on the theoretical side, a mathematical foundation for functional programming languages, and, from a more practical point of view, a method of programming which ensures correctness of programs, cf. [24, 25]) Classical logic , it was said, is nonconstructive , also in this, highly specific, sense. This, however, turned out to be no more than a, somewhat longstanding, prejudice (as witnessed by our programming theorem in the last section) nurtured by the fact that nothing better than a mere ....

[Article contains additional citation context not shown here]

Leivant, D. (1983) Reasoning about functional programs and complexity classes associated to type disciplines. FOCS 83, 460--469.

....This proposition states that the trees we considered are representable as lambda terms in such a way that the constructors (ffl,leaf and ) are lambda definable. In fact, the lambda terms involved can be typed in 2. A nice connection between these terms and proofs in second order logic is given in Leivant [1983]. Now we will show that iterative functions over these trees, like f mir , are lambda definable. 3.3. Proposition (Iteration) Given lambda terms A 0 ; A 1 ; A 2 there exists a lambda term F such that (for variables n; t 1 ; t 2 ) FB 1 = A 0 ; F (L 1 n) A 1 n; F (P 1 t 1 t 2 ) A 2 (F t 1 ) F ....

Leivant, Daniel [1983] Reasoning about functional programs and complexity classes associated with type disciplines, 24th Annual Symposium on Foundations of Computer Science, IEEE, 460--469.

....on the outside and s do not nest more than two deep on the left have meanings that are sorts of initial anarchic many sorted algebras. The analogous syntactic result, that the corresponding sets of closed normal forms are such algebras, was shown by Bohm and Berarducci [42] and by Leivant [43]. The basic idea is anticipated in the work of Martin Lof [44] and was probably known to early proof theorists such as Takeuti. The simplest nontrivial case is that the meanings of type Delta0: 0 0) 0 0) are in a one to one correspondence (given by the Church numerals) with the natural ....

Leivant, D. Reasoning About Functional Programs and Complexity Classes Associated with Type Disciplines. in: 24th Annual Symposium on Foundations of Computer Science, IEEE, Tucson, Arizona, November 7--9. 1983, pp. 460--469.

....solution in F 2 and C in the case of sums. Now, many more types than sums and pairs are definable in F 2 e.g. numerals, lists, trees, etc. see [Gir89] Pie89] In fact, all these types are special instances of a general notion, viz. inductive types ( Pie89] Gir89] or data systems ( Boh85] [Lei83]) and all these can be represented in F 2 as shown by Bohm and Beraducci [Boh85] and Leivant [Lei83] The question then naturally arises whether lists, trees, etc. or in general all data systems or inductive types can be defined in C too. It turns out that this is not possible with the rules ....

....in F 2 e.g. numerals, lists, trees, etc. see [Gir89] Pie89] In fact, all these types are special instances of a general notion, viz. inductive types ( Pie89] Gir89] or data systems ( Boh85] Lei83] and all these can be represented in F 2 as shown by Bohm and Beraducci [Boh85] and Leivant [Lei83]. The question then naturally arises whether lists, trees, etc. or in general all data systems or inductive types can be defined in C too. It turns out that this is not possible with the rules (2 4) for Delta. We are currently investigating the possibility of adding rules to allow such ....

Daniel Leivant. Reasoning about Functional Programs and Complexity Classes Associated with type disciplines.. In Proceedings of the 24th Annual Symposiom on the Foundations of Computer Science. 160-169, IEEE, 1983.

....of Girard and Reynold s system F [Gir71, Rey74] a nice simply typed calculus. The codomain of the isomorphism for the full NIP , in contrast, is a dependently typed calculus, too heavy to serve as a practical functional programming language. A more general approach, first put forth by Leivant [Lei83], is to consider a certain part of the structure present in sentences and derivations to be essential for typing and computation, and to relate a typed calculus to the n.d. calculus by means of an appropriate contracting homomorphism. In this paper, we choose a contraction of the following ....

.... given in [Geu92] The homomorphism from NIP ; to NIP ] j (x)8 2 ( X)8 1 ( y)F (X) y) oe X(y) oe X(x) This definition of the and and the associating inference rules of NIP ; via the 2nd order quantifiers and the associating inference rules is well known; it is due to Leivant [Lei83] and Bohm and Berarducci [BB85] NIP enjoy the property of strong normalizability. Therefore, the n.d. calculi of the cube also enjoy this property, since for each of them, there exists a homomorphism to NIP . 4 Discussion The calculi of the cube are equally powerful in terms of ....

Daniel Leivant. Reasoning about functional programs and complexity classes associated with type disciplines. In Proceedings 24th Annual IEEE Symp. on Foundations of Computer Science, FOCS'83, Tucson, AZ, USA, 7--9 Nov 1983, pages 460--469. IEEE Computer Society Press, Los Alamitos, CA, 1983.

.... 8R ( R(u 0 ; u 1 ) Prog 2 [R] R(v 0 ; v 1 ) where Prog 2 [R] j df 8w 0 ; w 1 ; z 0 ; z 1 (R(w 0 ; w 1 ) g(w 0 ) w 1 MP [w 0 ; w 1 ; z 0 ; z 1 ] R(z 0 ; z 1 ) It is easy to exhibit similar definitions for most basic programming constructs, including recursive procedures and modules [ Leivant, 1983 ] 2.5 Expressing convergence using second order validity The definability of free algebras by second order formulas can be extended to a characterization of the total computable functions. We summarize the argument of [ Leivant, 1983; Leivant, 1994 ] Of the many computation calculi that ....

....constructs, including recursive procedures and modules [ Leivant, 1983 ] 2.5 Expressing convergence using second order validity The definability of free algebras by second order formulas can be extended to a characterization of the total computable functions. We summarize the argument of [ Leivant, 1983; Leivant, 1994 ] Of the many computation calculi that generate the total recursive functions one that lends itself naturally to this model theoretic setting are equational programs; a rudimentary form of such programs would serve us best. 14 Fix a free algebra A . We posit an unlimited ....

D. Leivant. Reasoning about functional programs and complexity classes associated with type disciplines. In Twenty-fourth Annual Symposium on Foundations of Computer Science, pages 460--469, 1983.

No context found.

Leivant, D. Reasoning About Functional Programs and Complexity Classes Associated with Type Disciplines. in: 24th Annual Symposium on Foundations of Computer Science, IEEE, Tucson, Arizona, November 7--9. 1983, pp. 460--469.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC