E. Engeler. Algebras and combinators. Algebra Universalis, 13(3):389--392, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theory of computation there exist models of computation which are given as algebraic structures, e.g. combinatory algebras. A popular model of combinatory algebras is calculus [Barendregt, 1977] In this work we consider other models of combinatory algebras, namely graph models [Engeler, 1981A, Engeler, 1981B] It was shown that any algebraic structure can be embedded in a graph model [Engeler, 1988] Hence graph models give rise to an algebraic model of computation in algebraic structures. So it appears appropriate to use them as models for symbolic computation. On the other hand, graph models have, ....

Engeler, E.(1981). Algebras and Combinators. Algebra Universalis, 389-392.

.... has already been obtained for the simple semantics [4,17] and the Fsemantics [12] We obtain completeness for two more general classes of type interpretations, where there is a pleasing correspondence between the type expression models and the set theoretic models of Plotkin and Engeler [30, 14]. It would be interesting to have a formal definition of a notion of a static type inference system. Such a system provides the basis for static type inference (or type checking) algorithms. It is desirable that the system admits a reduction rule. That implies that the types of terms after ....

....some interest in pursuing such systems. First the availability of non simple type interpretations allows completeness proofs for other, perhaps simpler, type inference systems. Secondly these systems are very closely connected to the models of the k calculus given by Plotkin [30] and Engeler [14]. A natural basic system with intersection types is obtained by adding to the system k binary and empty intersections of types: and co. The rules we take are (I) F [ M: x, F [ M:3 F M:c3 (E) 1. F [ M:ct 2. F M:3 (mI) F I M:m This system has not been considered in the literature ....

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Engeler, E. Algebras and combinators, Algebra Universalis 13, no. 3, pp. 389-392. (1981)

....recall the models of the lazy calculus studied in the literature and we will compare their local structures with the models M 1 and M 2 defined in the present paper. Longo model The Longo s model ML (defined in [14] as D A and called D lazy A in [15] is based on the free PSEalgebra (see [6]) built over a given set A, namely the pair DA = P(B) ffl) where B = S n Bn with B 0 = A and Bn 1 = Bn [ f(b; fi) j b 2 P fin (Bn ) fi 2 Bn g, and d 1 ffl d 2 = fff j 9a 2 P fin (d 2 ) a; ff) 2 d 1 g. Defining the set V as P(B) n ; it is easy to show that ML = P(B) ffl; V; Delta] A ....

E. Engeler. Algebras and combinators. Algebra Universalis, 13:389 -- 392, 1981.

....of suitable partially ordered models. Scott s continuous semantics [42] is the class of the partially ordered models whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see [43] [18] [33] 34] 9, Section 5.5] is a subclass of the K semantics isolated by Krivine (see [28] 9, Section 5.6.2] within the continuous semantics. The filter model semantics was defined by Coppo, Dezani, Honsell and Longo in [15] see also [4] within the continuous semantics. The stable semantics ....

Engeler E., Algebras and combinators, Algebra Universalis 13 (1981), 389--392.

....as monotone functions over such sets (see [47] Scott s continuous semantics [45] is the class of pomodels whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see [46] [19], 37] 38] 10, Section 5.5] is a subclass of the K semantics isolated by Krivine (see [30] 10, Section 5.6.2] within the continuous semantics. The filter model semantics was defined by Coppo, Dezani, Honsell and Longo in [16] see also [4] within the continuous semantics. The stable ....

E. Engeler, "Algebras and combinators", Algebra Universalis, 13 (1981), pp. 389-392

....14. Let A = hA; s; k; Deltai be a pca. A is irreducible if E OE fha; ai j a 2 Ag for every collapse OE of A. There are prominent pca s which share this property. We give two examples. Example 1. The first example uses only elementary properties of sets, and is directly taken from Engeler [Eng81]. It is in fact a notational variant of one of several ca s first described in Plotkin [Plo72] which in turn are nearly the same as the better known P construction of Scott [Sco76] Let A be any nonempty set, and let B be the least set containing A and all ordered pairs consisting of a finite ....

E. Engeler. Algebras and combinators. Algebra Universalis, 13:389--392, 1981.

....Barbanera and Berardi [2] called here the BB model for short, which was shown to be complete for F in [5] and was indeed the rst nonsyntactic complete model exhibited for this system. It also contains simpler models. The simplest model, called E 2 here, is based on Engeler Plotkin s model E [12], 28] which will be called here Engeler s model for short. We then compare the present class with the models proposed previously for F , at least with those models for which, from our point of view, comparison 2 makes sense. This comparison supposes some familiarity with those previously ....

E. Engeler, Algebras and combinators, Alg. Univ. 13 (3), p.289-371, 1981.

.... semantics that has an especially clear structure with respect to approximation as well to algebraization are Engeler graph models [Engeler, 1981A, Maeder, 1986] Graph models were already used as programming semantics for several mathematical structures, like varieties, geometries and analysis [Engeler, 1981B, Engeler, 1984, Engeler, 1988, Fehlmann, 1981, Seeland, 1978] We will show how to bring these graph models in a form where they allow the incorporation of the algebraic aspects of analysis as delivered by the differential fields theory. This goes back to an approach that was first outlined in ....

Engeler, E. (1981). Algebras and Combinators. Algebra Universalis, 389-392.

....abstraction result for . Theorem 5 Let = be a free lazy PSE model; then M N iff = M = N . 2 We refer to [9, 12] for a detailed presentation of free lazy PSE models. The family of combinatory algebras called Plotkin Scott Engeler (PSE) Algebra arises from the study conducted by Engeler in [6] (which followed earlier ideas by Plotkin and Scott) Its theoretical properties were later investigated by Longo in [9] For any non empty set A, the free PSE algebra on A is built inductively from A in a very natural set theoretic way, the notion of application generalizing the classical ....

E., Engeler, "Algebras and Combinators", Algebra Universalis, Vol. 13, 389-392, 1981.

.... head trees represent the local structure of Scott s P model as defined in [27] a discussion on this topic can be found in [4] Chapter 19) ffl the weak trees were introduced by Longo in [22] following [21] who proved that they represent the local structure of Engeler s models as defined in [17]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms incorporated in the tree representations. For instance, in [29] Wadsworth shows that two terms M;N have the same infinite eta tree if and only if, for all ....

Erwin Engeler. Algebras and combinators. Algebra Universalis, 13(3):389--392, 1981.

.... head trees represent the local structure of Scott s P model as defined in [27] a discussion on this topic can be found in [4] Chapter 19) ffl the weak trees were introduced by Longo in [22] following [21] who proved that they represent the local structure of Engeler s models as defined in [17]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms incorporated in the tree representations. For instance, in [29] Wadsworth showed that two terms M;N have the same infinite eta tree if and only if, for all ....

Erwin Engeler. Algebras and combinators. Algebra Universalis, 13(3):389--392, 1981.

.... head trees represent the local structure of Scott s P model as defined in [27] a discussion on this topic can be found in [4] Chapter 19) ffl the weak trees were introduced by Longo in [22] following [21] who proved that they represent the local structure of Engeler s models as defined in [17]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms encorporated in the tree representations. For instance, in [29] Wadsworth shows that two terms M;N have the same infinite eta tree if and only if, for all ....

Erwin Engeler. Algebras and combinators. Algebra Universalis, 13(3):389--392, 1981.

....Barbanera and Berardi [2] called here the BB model for short, which was shown to be complete for Fj in [5] and was indeed the rst non syntactical complete model exhibited for this system. It also contains simpler models. The simplest model, called E 2 here, is based on Engeler Plotkin s model E [12], 27] which will be called here iEngeler s modelj for short. We then compare the present class with the models proposed previously for F , at least with those models for which, from our point of view, comparison makes sense. This comparison supposes some familiarity with those previously known ....

E. Engeler, Algebras and combinators, Alg. Univ. 13 (3), p.289-371, 1981.

.... head trees represent the local structure of Scott s P model as defined in [30] a discussion on this topic can be found in [5] Chapter 19) ffl the weak trees were introduced by Longo in [25] following [24] who proved that they represent the local structure of Engeler models as defined in [19]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms encorporated in the tree representations. For instance in [33] Wadsworth shows that two terms M;N have the same infinite eta tree if and only if for all ....

Engeler E., "Algebra and Combinators", Algebra Universalis 13(3), 1981, 389392.

....original ; the above example shows that the containment is strict. Theorem 5.4 has an important consequence in terms of models, more precisely the free lazy Plotkin Scott Engeler (PSE) models. Briefly, a PSE model is defined on top of a PSE algebra. These are combinatory algebras introduced by Engeler (1981), which followed earlier ideas by Plotkin and Scott. PSE algebras are defined in a very natural set theoretic way, the notion of application generalising the classical Myhill Shepherdson Roger definition of application in the graph model P . There are two canonical ways to expand a PSE algebra ....

Engeler, E. (1981) Algebras and Combinators, Algebra Universalis, 13, 389-392.

....translations to and from pointed lazy first order lazy model whose definition we have omitted. See [Ong88b, Ch. 3] for details. 3 Plotkin Scott Engeler Algebras and Models We introduce a family of combinatory algebras called Plotkin Scott Engeler (pse) algebras. First formalized by Engeler [Eng81], pse algebras may be seen as a code free variant of the well known graph model P( due independently to Plotkin [Plo72] and Scott [Sco76] Let B be a non empty set satisfying (io) fi fin B b 2 B ( fi; b) 2 B; where the meta variable fi ranges over finite sets (similarly for ....

E. Engeler. Algebras and combinators. Algebra Universalis, 13:389--392, 1981.

.... head trees represent the local structure of Scott s P model as defined in [30] a discussion on this topic can be found in [5] Chapter 19) ffl the weak trees were introduced by Longo in [25] following [24] who proved that they represent the local structure of Engeler models as defined in [19]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms encorporated in the tree representations. For instance in [33] Wadsworth shows that two terms M;N have the same infinite eta tree if and only if for all ....

Engeler E., "Algebra and Combinators", Algebra Universalis 13(3), 1981, 389-392.

....in the notation of applicative algebra which has no variables a combination. Any combination F defines an n ary operation: F (x 1 ) x 2 ) Delta Delta Delta (x n ) What we have been remarking is that the algebras so obtained from combinations can be very rich. In a series of papers [Eng81, Eng] Engeler discussed just how rich these algebras can be. A representative result, following Engeler, will be exhibited here. Theorem 26 Given a signature (s 1 ; s 2 ; s n ) there are combinations F 1 ; F 2 ; F n defining operations on D of these arities such that whenever a ....

E. Engeler. Algebra and combinators. Algebra Universalis, 13:389--392, 1981.

....that the strict system is not closed for j reduction, whereas the BCD system is. The strict system gives rise to a strict filter model F S , that satisfies all major properties of the filter model F as presented in [6] but is an essentially different model, equivalent to Engeler s model D A [18]. In [1] was shown that soundness for the notion of type assignment of [6] is lost if instead of simple type semantics, the inference type semantics is used. Take, for example, the statement x:x: oe oe) oe ) oe: this statement is derivable in the system , but it is not valid in F S . With ....

E. Engeler. Algebras and combinators. Algebra universalis, 13(3):389--392, 1981.

....domains in [1] framing it into the paradigm of Stone duality. A strikingly short, but good introduction to this subject is chapter 10 of [7] We also suggest the reading of [20] it is not exactly about filter models, but it gives a perspective to the webbed models (originated with Engeler models [48]) which are an alternative, though related, finitary description of models. See also [75] The use of trees to study the structure of models dates to the 70 s. Beside Bohm trees, introduced in Barendregt book [15] and recently considered in [41] L evy Longo trees have been known and studied ....

E. Engeler, "Algebras and Combinators", Algebra Universalis 13(3), 1981, 289-- 371.

....WORK In this section we will recall the models of the lazy calculus studied in the literature and we will compare their local structures with the models M 1 and M 2 defined in the present paper. Longo model The Longo s model ML defined in Longo (1983) is based on the free PSE algebra (see Engeler (1981)) built over a given set A, namely the pair DA = P(B) ffl) where B = S n Bn with B 0 = A and Bn 1 = Bn [ f(b; fi) j b 2 P fin (Bn ) fi 2 Bn g and d 1 ffl d 2 = fff j 9a 2 P fin (d 2 ) a; ff) 2 d 1 g. Defining the set V as P(B) n ; it is easy to show that ML = P(B) ffl; V; Delta] A ....

Engeler, E. (1981), `Algebras and combinators.' Algebra Universalis 13, 389 -- 392.

.... structure of Scott s P model as defined in (Scott 1976) a discussion on this topic can be found in (Barendregt 1984) Chapter 19) the weak trees were introduced by Longo in (Longo 1983) following (Levy 1976) who proved that they represent the local structure of Engeler models as defined in (Engeler 1981). Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms encorporated in the tree representations. For instance in (Wadsworth 1976) Wadsworth shows that two terms M;N have the same infinite Bohm tree iff for all ....

Engeler E., "Algebra and Combinators", Algebra Universalis 13(3), 1981, 389-392.

.... head trees represent the local structure of Scott s P model as defined in [26] a discussion on this topic can be found [4] Chapter 19) the weak trees were introduced by Longo in [22] following L evy [21] who proved that they represent the local structure of Engeler models as defined in [16]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamically evolving meanings of terms encorporated in the tree representations. For instance in [28] Wadsworth shows that two terms M;N have the same infinite Bohm tree iff for all contexts C[ the ....

Engeler E., Algebra and Combinators, Algebra Universalis 13(3), 1981, 389-392.

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E. Engeler. Algebras and combinators. Algebra Universalis, 13(3):389--392, 1981.

No context found.

E. Engeler. Algebras and combinators. Algebra Universalis, 13(3):389-392, 1981.

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E. Engeler, "Algebras and Combinators", Algebra Universalis 13(3), 1981, 289-- 371.

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Engeler E., "Algebra and Combinators", Algebra Universalis 13(3), 1981, 389-392.

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Erwin Engeler. Algebras and combinators. Algebra Universalis, 13(3):389--392, 1981.

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E. Engeler, "Algebra and Combinators", Algebra Universalis 13(3), 1981, 389-392.

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E. Engeler [1981]. Algebras and combinators, Algebra Universalis 13(3), pp. 389--392.

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