P.-L. Curien. Categorical combinators, sequential algorithms and functional programming. Birkhauser, Boston, second edition, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... from a 2 RO ( then (a) is defined and a is reachable from (a) The idea is that if Player, when playing according to some strategy , is prepared to reach a position a 2 RP ( The terminology hints at a connection with concrete data structures [9] and event structures [13] see also [3]. then he should be prepared to do so from every position that might come up in a play in accord with . We briefly describe a concrete data structure intuition. A P position a can be thought of as representing a situation in which certain specified cells have all been filled by Player (in an ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Birkhauser, 2nd edn. edition, 1993.

....process. This paper presents an explicit substitution calculus which is able to handle local substitution. There are two main styles of explicit substitution: the and the s e styles. The calculus [1] re ects in its choice of operators and rules the calculus of categorical combinators [3]. The main innovation of the calculus is the division of terms in two sorts: sort term and sort substitution. s e [11] departs from this style of explicit substitutions in two ways. First, it keeps the classical and unique sort term of the calculus. Second, it does not use some of the ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

....improvement over Definition 1.3. But just imagine that in the calculus you had not only and ffi as internal operators but also oe for substitution, for typing and so on. In fact, internalising substitution (i.e. making it explicit) has been a topic of research in the last decade (see [1] 8] [7], 9] Now, internalising extra operators means that in classical notation, in Definition 1.3, two extra rules are added for each new operator. In item notation on the other hand, Definition 2.3 does not depend on the number of operators. Simply, the set of operators to which belongs will ....

....this as its root. 8 Describing normal forms in a substitution calculus Lambda calculi with explicit substitutions attempt to close the gap between the classical calculus and concrete implementations. Recently, there has been various attempts at providing calculi of explicit substitution ( 6] [7], 9] 13] 14] Most of the above mentioned work (except [9] uses classical notation. 13] provided s, a calculus of substitution a la de Bruijn, which remains as close as possible to the classical calculus. Here is a descrition of s (we assume familiarity with de Bruijn indices) ....

P.-L. Curien, Categorical Combinators, Sequential Algorithms and Functional Programming, Pitman, 1986.

....a theoretical framework for the implementation of functional programming languages. Several calculi including new operators to denote substitution and new rules to handle these operators have been proposed. Amongst these calculi we mention COE (cf. 6] the calculi of categorical combinators (cf. [4]) oe, oe , oe SP (cf. 1, 5, 14] referred to as the oe family; oeBLT (cf. 7] AE (cf. 3] and i (cf. 13] which are descendants of the oe family; s (cf. 8] s e (cf. 11] and t (cf. 10] This article will focus on oe, oe , AE, s, t and u which is an efficient version of s ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

....explicit substitutions [Hur96b, Mag95] The last 15 years have seen an increasing interest in formalizing substitution explicitly; various calculi, including new operators to denote substitution, have been proposed. Among these calculi we mention COE [dB78] the calculi of categorical combinators [Cur86]; oe [ACCL91] oe [CHL96] and oe SP [R io93] referred to as the oe family; AE [BBLRD96] a descendant of the oe family; oeBLT [KN93] exp [Blo95] s [KR95a] s e [KR97] and i [Hur96a] All of these calculi (except exp) are described in a de Bruijn setting, where natural numbers play the role ....

.... # (W ) cgggg) 2.6) Wffi cgg(a 2 ff1 U (a 1 )ggffi # (W ) cgg) 3. 3) W (a 2 ff1 U 0 (a 1 )gg) ffi cgg = bffi cgg Proof of Lemma 5 2 3 The s and gs Calculi The oe calculus (cf. ACCL91] reflects in its choice of operators and rules the calculus of categorical combinators (cf. [Cur86]) The main innovation of the oe calculus is the division of terms into two sorts: sort term and sort substitution. We depart from this style of explicit substitutions in two ways. First, we keep the classical and unique sort term of the calculus. Second, 16 we do not use some of the categorical ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Wiley, New York, 1986.

....adjoining a top element to G . The set of all such closure operators S = Cl(G is our universal space of strategies. Various sub classes of S are important. In particular, there are constraints on strategies which correspond to important computational properties. One of these is sequentiality [Cur86]. A closure operator 2 S has an output function f : G G if 8x 2 dom( x) f(x) x 25 dom( fx 2 G j (x) 6= g: We recall the domain theoretic conditions of stability and sequentiality [AC98] These make sense because our underlying domain G is simple and concrete it is in fact a ....

....) 8y x: f(y) c) 9c : x(c ) 8y x: f(y) c)# ) y(c )#) There says that there is some particular cell in the input which must get lled before we can ll the given output cell. Classically, this condition excludes the parallel or function, Berry s Gustave function , etc. [Vui73, KP78, Cur86]. We take the constraint of having a sequential output function as picking out those strategies which are to be regarded as computing in a deterministic fashion, and hence sequentially implementable. In support of this contention, we have the fact that, when we make our universe of sequential ....

P.-L. Curien. Categorical combinators, sequential algorithms and functional programming. Pitman

....by a simple case analysis. But (f(g 1 ) f(g 2 ) f(g 3 ) 62 unit(T; S) S ; so f is not uniform. Thus, f cannot be a morphism, and hence it cannot be definable. 4. 2 Sieber s last example is not definable The second example is essentially the last example in [33] an example modified from [4]. Let ffl bool be the object (unit Phi unit ) and ffl A = unit ) bool) Omega (unit ) bool) Let true denote inj 1 ( 2 bool and false denote inj 2 ( 2 bool. Consider the morphisms g 1 ; g 2 ; g 3 ; g 4 : A unit defined by if h 2 ( true if h 1 ( true and h 2 ( ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. John Wiley & Sons, 1986.

....of #, e 1 x and we have a contradiction. So e 1 e 1 . Symmetry: e 1 , e 2 E, e 1 e 2 iff (e 1 # e 2 ) iff (e 2 # e 1 ) iff e 2 e 1 . 4 What we call here Drostes event domain, Droste [DRO 89, p. 40] calls event domain, and what we call event domain, Droste calls conflict event domain (see [CUR 86, p. 133] and [DRO 89, p. 43] 5 Girard [GIR 89, p. 56] defines a web as = e; e , where is a coherence space and a A b iff a, b . From the results above we have that (E, is a web of a coherence space. 3.3 Proposition Let (D, be an event domain. Then, D, is a ....

....that (E, is isomorphic to (D(M) in the following way: 1. eE, e 2. the transitive closure of the negation if is transitive 3. X,e 1 ,e 2 , X e 1 , e 1 e 2 ) X e 2 . Proof) Given (E, we can obtain (by theorem 3. 11) an event structure (E, #, and by Curien s assertion [CUR 86, section 2.2 p. 139] we have that, given an event structure (E, #, we can obtain a cds such that D(M) and D(E) be isomorphic, given the following conditions: the transitive closure of the negation of is transitive, X,e 1 ,e 2 , X e 1 , e 1 # e 2 X e 2 . 4 Conclusion We have ....

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CURIEN, P.L. Categorical Combinators, Sequential Algorithms and Functional Programming. In: Research Notes in Theoretical Computer Science, Pitman, London, 1993. 300 p.

....a top element to G . The set of all such closure operators S = Cl(G ) is our universal space of strategies. Various sub classes of S are important. In particular, there are constraints on strategies which correspond to important computational properties. One of these is sequentiality [Cur86]. A closure operator 2 S has an output function f : G G if 8x 2 dom( x) f(x) x 25 where dom( fx 2 G j (x) 6= g: We recall the domain theoretic conditions of stability and sequentiality [AC98] These make sense because our underlying domain G is simple and concrete it is in ....

....) 8y x: f(y) c) 9c 0 : x(c 0 ) 8y x: f(y) c)# ) y(c 0 )#) There says that there is some particular cell in the input which must get lled before we can ll the given output cell. Classically, this condition excludes the parallel or function, Berry s Gustave function , etc. [Vui73, KP78, Cur86]. We take the constraint of having a sequential output function as picking out those strategies which are to be regarded as computing in a deterministic fashion, and hence sequentially implementable. In support of this contention, we have the fact that, when we make our universe of sequential ....

P.-L. Curien. Categorical combinators, sequential algorithms and functional programming. Pitman

....compute f. We define a strategy of computation so that one can use it to give the semantics of segments of programs, and this gives a sort of compositional operational semantics. 2 Concrete Data Structures In this section we define the concrete data structure (cds) and their states, following [CUR 86] 2.1 Definition A concrete data structure or cds (C, V, E, is given by three sets C, V, E of cells, of values and of events such that E C V and cC, vV s.t. c, v)E ( any cell may be filled ) 3 The name is due to the fact that the algorithms thus obtained compute linear functions. ....

....a compositional operational semantics for program segments would be a function mapping program segments to computation strategies, which preserves program composition. 8 Conclusion The definition of the linear algorithms can be considered as an extension of the sequential algorithms of Curien ( CUR 86] One advantage of the former is that its includes not only the sequential strategies but the parallel ones too. We had proved (see [SCH 95] that for all sequential algorithm there exists a linear algorithm that contains one sequential strategy equivalent to the Curien s algorithm. We have the ....

CURIEN, P.L. Categorical Combinators, Sequential Algorithms and Functional Programming. In: Research Notes in Theoretical Computer Science, Pitman, London, 1993. 300 p.

....of firstorder programs. It has been introduced independently by Vuillemin and Milner (cf. V] and [Mi] The problem of extending this notion to higher order programs, like those of Godel system T, has led to various definitions: ffl Sequential algorithms on concrete data structures (see [C1]) and more recently various gametheoretic models (see [AJ, C2, HO, L] inspired by the work of Blass [Bl1, Bl2] which are quite intentional models where programs of functional type are not simply interpreted by functions, but by more complicated objects ( algorithms or strategies ) which ....

P-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. (

....continues in that component. Exponential structure. Play in A is in e ect play in an in nite sequence of copies of A. At any stage Opponent can play in any game already begun or can choose to start the next version of A in the sequence. Sequential algorithms. Filiform concrete data structures [21] can be regarded as games. Let FCDS be the category of games Gam but equipped with a more sophisticated structure making it a model of intuitionistic linear logic. We take the same multiplicative and additive structure as before but change the exponential to the following. Curien exponential. ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Birkhauser, 2nd edn. edition, 1993.

....is either empty or contains a least element. 3.4. Stable Functions and Demand Driven Computations The failure in example 3. 5 is obviously due to the use of a truely parallel function (parallel or) It seems reasonable, therefore, to conjecture that if we employ only sequential functions (see [Cur85, KaP, Mil77, Pin86, Vui73]) then such a phenomenon will not happen. Formally, a function f : Omega D oe 1 : Omega D oe n Omega D oe n 1 is sequential iff for every X = X 1 ; Xn ) and every k such that f(X) k] there exist some 1 i n and j such that X i [j] and 8Y = Y 1 ; Yn ) X; ....

Curien P.: Categorical Combinators, Sequential Algorithms and Functional Programming. Ph.D Thesis, Univesite Paris 7, Paris, 1985.

....arguments. A good definition of sequentiality for a function f : D D 0 doesn t seem possible, with this technique. The search for a good definition of sequentiality was also inspired by the study of the syntactic properties of the evaluator for PCF, which is a 3 sequential process; see [5] and [3] for a review of these properties. As a consequence of the sequential nature of the evaluator, the following property was identified. Consider PCF extended with a new constant Omega at each type, for the undefined term, and define M Omega N iff N can be obtained from M by ....

....included in E(x) or else we would have c 2 E(x) and c i c. Continuing, we find an infinite descending chain of cells. 2 In this report, we have chosen to state all properties and make all proofs for arbitrary cds, without relying on well foundness. That this is indeed possible is stated in [5] and [2] Definition 5 A cds M is called stable or deterministic, if for any state x and any cell c, if x 0 c, x 0 0 c and x 0 x, x 0 0 x, then x 0 = x 0 0 . A deterministic cds is abbreviated dcds. 8 So any state of a dcds contains at most one enabling for any cell c. It seems ....

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P.L. Curien, Categorical Combinators, Sequential Algorithms and Functional Programming, Pitman 1986

....is that they preserve strong normalisation. The other approach, which we propose to call the oe family, has been proposed by Abadi, Cardelli, Curien, Hardin, L#vy and Field around 1989 [1, 2, 15, 19, 10, 24] It follows previous research by Curien who proposed in 1983 categorical combinators [6, 8, 7] a combinatory logic more intuitive than the classical one. Hardin in 1987 [17, 18] studied conAEuence on open terms for that calculus. Categorical combinators are more intuitive in the sense that they are based on calculus, more precisely on calculus with Cartesian products and keep its ....

P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

....LAA I s. LAMBDA ABSTRACTION ALGEBRAS 39 Related work. There have been several attempts to reformulate the lambda calculus as a purely algebraic theory within the context of category theory: Obtu lowicz and Wieger [31] via the algebraic theories of Lawvere; Adachi [2] via monads; Curien [11] via categorical combinators. There have also been several works that present an algebraic theory of the lambda calculus very close to lambda calculus in spirit. Locally finite functional LAA s are very similar to the functional models of the lambda calculus developed in Krivine [26] However, ....

P.L. Curien, Categorical combinators, sequential algorithms and functional programming, Pitman, 1986.

.... s one has to cross between this occurrence and its binder. Definition 1.7. The set DB (X ) of terms in de Bruijn s notation, is defined inductively as: a : n j X j a j (a a) where n is an integer greater than or equal to 1 and X 2 X (some authors use 0 as the first de Bruijn number, see [10] for example. Notice that we replace bound variables and constants by indices, but we keep the names for metavariables. This distinction is not done in the original de Bruijn calculus. In the original notation of de Bruijn, free variables of a term a are ordered into a list (x 0 : xn ) ....

P.-L. Curien, Categorical Combinators, Sequential Algorithms and Functional Programming, Birkhauser, 1993. 2nd edition.

....matching algorithm is proposed in [10] If we choose the combinatory expressions presented above, the matching is now firstorder modulo the theory P of categorical products. The theory P is elegant in that it creates a canonical (Con uent, Terminating and Interreduced) Term Rewriting System for P [5, 2]. f ffi g) ffi h f ffi (g ffi h) f ffi Id f Id ffi f f Fst ffi hf; gi f Snd ffi hf; gi g hFst ffi h; Snd ffi hi h x ffi y x hf; gi ffi h hf ffi h; g ffi hi hFst; Sndi Id This Term Rewriting System allows us to normalize terms and to decide that two terms are P ....

P. L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

....reader may well verify. Moreover, as discussed already, this is not an assumption that is valid in all contexts. In our knowledge, the first serious consideration of a notation and a calculus that incorporate a fine grained control over substitutions appears in the work of P L. Curien [Cur86a, Cur86b] In this work, a categorical combinatory logic called CCL is described. The language underlying this logic is not the lambda calculus, but bears a close relationship to it: there is a translation from the (pure) lambda calculus augmented with the pairing function to CCL and vice versa that ....

P-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

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P.-L. Curien, Categorical combinators, sequential algorithms and functional programming,Pitman (1986.

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P.-L. Curien. Categorical combinators, sequential algorithms and functional programming. Birkhauser, Boston, second edition, 1993.

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Curien P.-L., Categorical Combinators, Sequential algorithms and Functional Programming. Second Edition, Birkauser Boston, 1993.

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P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

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P.-L. Curien, Categorical combinators, sequential algorithms and functional programming, Birkhauser (1993).

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P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986.

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