M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Math. Annalen, 92:305--316, 1924. 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

....prevent variables in lambda calculus from operating as real algebraic variables. However, there have been several attempts to reformulate the lambda calculus as a purely algebraic theory. The earliest, and best known, algebraic models are the combinatory algebras of Curry and Schonfinkel (see [7] [24]) Combinatory algebras have a simple purely equational characterization. Curry also specified (by a considerably less natural set of axioms) a purely equational subclass of combinatory algebras, the algebras (see Barendregt [1] 5.2.5) that he viewed as algebraic models of the lambda calculus. ....

Schonfinkel, M., "Uber die bausteine der Mathematischen Logik", Mathematischen Annalen (english translation in J. van Heijenoort ed.'s book"From Frege to Godel, a source book in Mathematical Logic, 1879-

....lambda theory if T = T (see [3, Def. 4.1.1] As a matter of notation, T t = u stands for T t = u. 2. 2 Combinatory algebras and models An algebra C = C; Delta; k; s) where Delta is a binary operation and k; s are constants, is called a combinatory algebra (Curry [16] Schonfinkel [40]) if it satisfies the following identities (as usual the symbol Delta is omitted, and association is to the left) kxy = x; sxyz = xz(yz) In the equational language of combinatory algebras the derived combinator 1 is defined as 1 j s(ki) A function f : C C is called representable if there ....

Schonfinkel M.,Uber die bausteine der Mathematischen Logik, Mathematischen Annalen (english translation in J. van Heijenoort ed.'s book"From Frege to Godel, a source book in Mathematical Logic, 1879-

....T is called semisensible [3, Def. 4.1.7(iii) if T 6 M = N whenever M is solvable and N is unsolvable. 2.2. Combinatory algebras and models An algebra C = C; Delta; k; s) where Delta is a binary operation and k; s are constants, is called a combinatory algebra (Curry [17] Schonfinkel [43]) if it satisfies the following identities (as usual the symbol Delta is omitted, and association is to the left) kxy = x; sxyz = xz(yz) In the equational language of combinatory algebras the derived combinator 1 is defined as 1 j s(ki) A function f : C C is called representable if there ....

M. Schonfinkel,"Uber die bausteine der Mathematischen Logik", Mathematischen Annalen (english translation in J. van Heijenoort ed.'s book"From Frege to Godel, a source book in Mathematical Logic, 1879-

....of these cases a rudimentary form of typed calculus is used to organize the distribution of data within a process expression. In this paper we intend to clarify in detail the type structure of data dependent actions and processes. In addition we propose to employ typed combinators (cf. e.g. [Sch24], HS86] in order to stay entirely within typed equational logic. As an illustration SABP, the Simple Alternating Bit Protocol, taken from [Par85] and adopted to ACP syntax in [BKO87] is verified in a purely equational way. This improves all previous verifications, e.g. the ones in [Bae90] by not ....

M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Mathematische Annalen, 92:305-316, 1924.

....the start as not fulfilling this purpose, because it is not clear that there is a bound on the amount of work involved in the substitution of N for x in M since M can contain an arbitrary number of xs. This was recognized as a problem and a first solution was found: combinatory logic introduced by Schonfinkel (1924) and (see Curry Feys (1958) for a discussion Curry 1930) gives a truly stepwise realization of the calculus and hence assigns a complexity to reduction of terms. However, this measure is far too pessimistic for realistic use for two reasons: first it duplicates computation, second it strictly ....

Schonfinkel, M. (1924), ` Uber die Bausteine der mathematischen Logik', Math. Ann. 92, 305--316.

....are downsides to using KOLA as a query representation, but a lack of expressive power for denoting queries is not one of them. Combinators can appear to lack expressivity to those who first use them, but the right set of combinators can have rich expressive power. For example, 48 Schoenfinkel [Sch24] established that three combinators (S, K and I) were all that were required as an alphabet for a free algebra over which one could translate all of the calculus. It was even shown later that I was superfluous ) Within the context of querying, the rich expressivity of KOLA has been established ....

....according to whether the combinator set is fixed or variable. Fixed combinator sets consist of a finite set of combinators that are used as the target for all expression translations. The best known of the fixed sets of combinators are the SKI combinator set, first introduced by Schonfinkel [Sch24] It has been shown that this small set is sufficient to translate all of calculus (in fact I is superfluous) but the size of the resulting code is too large to be of practical use [JRSB85] Variations of the SKI combinator sets add additional, redundant combinators (e.g. B and Y) to reduce the ....

M. Schonfinkel. Uber die bausteine der mathematischen logik. Math. Annalen, 92:305-- 316, 1924.

.... ( eq plus (first expression) let ( x (second expression) y (third expression) consequent 0 ) eq sqrt (first expression) let ( x (second expression) consequent 1 ) else (let ( f (first expression) a (rest expression) consequent 2 ) We believe that a curried [9, 25] programming style best expresses the structure of some functions, and so in the definition of eval dest in Figure 4, and eval dest# in Figure 6, we use an extended form of define, which mimics the pattern of the curried function call. In Scheme, define ( f x) y) body) define (f x) lambda (y) ....

Schonfinkel, M., (1924), Uber die Bausteine der mathematischen Logik, Mathematische Annalen 92, pp. 305-316.

.... symbol i means reduces to and in the traditional calculus it is replaced by = The right hand side column in Figure 3 shows that combinators can be defined without abstraction and, in this sense, they become proper combinators dissociated from the calculus, as in their original formulation [22]. There are also compound combinators obtained from the primitive combinators in Figure 3 using functional application: C 1 C 2 reads C 1 applied to C 2 ; application is also left associative. For example, combinators W and S are interdefinable in 1 in the absence of commutativity, and are ....

A. Schonfinkel. Uber die Bausteine der Mathematischen Logik. In J. van Heijenoort, editor, From Frege to Godel. Harvard Univ. Press, Cambridge, Mass., 1924. Reprinted.

....Rule 1 is sound by the definition of I. Rule 2 is sound by KA = fdef. Kg (xy:x)A = ffi reductiong y:A = fff conversion, x 62 Free(A)g x:A Rule 3 is sound by S(x:A) x:B) fdef. Sg (fgx:fx(gx) x:A) x:B) ffi reduction (twice)g x: x:A)x( x:B)x) fj conversion (twice)g x:AB Schonfinkel [Sch24] noted that I can be constructed from S and K: Theorem 2. fS; Kg is a basis. Proof. By theorem 1, fS; K; Ig is a basis. Replace all occurrences of I by SKA, where A is an arbitrary term, e.g. K. This rule is sound by SKA = fdef. Sg (fgx:fx(gx) KA = ffi reduction (twice)g x:Kx(Ax) fdef. Kg ....

....= S Of course A and B have to be distinct. As a second design decision we choose A and B to be the two simplest distinct expressions that can be made with X, and as K is simpler than S we choose A to be simpler than B: A = X B = XX The specification now reads XX = K X(XX) S (Schonfinkel [Sch24] has already given this specification, but with A and B reversed. However, he didn t calculate a closed form for X. By using the first line in the second one we obtain: XX = K XK = S As X is applied to functions (X and K) it is of the form f:M . Now we must construct the body M out of f . It ....

Schonfinkel, M.: Uber die Bausteine der mathematischen Logik. Mathematische Annalen, 92, 307--316 (1924).

....completeness to define x:ff(x) as the coefficient a of the normal form a Delta x of the polynomial ff(x) The abstraction thus appears as an inductively derivable operation. The idea of hiding away the variables and eliminating the substitution for the sake of function application goes back to (Schonfinkel 1924) and actually predates the calculus. However, Schonfinkel s applicative algebras were properly understood only when Curry (Curry 1930) had displayed them as the combinatorially complete kernel of the untyped calculus. The abstraction seemed easier to understand as an operation, than as a ....

M. Schonfinkel (1924) Uber die Bausteine der Mathematischen Logik, Math. Annalen 92, 305-- 316.

....see (1) From these types we can construct complex types by applying type constructors. type type defines function types, see (2) We consider only functions with one argument. It is obvious that this does not pose any restriction on the generality of the type system as argued by Schonfinkel [11] and later used by Curry [2] Records can be built by joining tuples of names and types, see (3) 6) type : INT j REAL j BOOL (1) j type type (2) j f components g (3) components : 4) j comp ; components (5) comp : name : type (6) For structured types, different notions of type ....

M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Math. Ann., 92:305--316, 1924.

....short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schonfinkel [9]. The algorithm introduces at most one combinator for each abstraction in the initial lambda term. This provides the system with quite good properties in terms of memory space as well as in terms of length of evaluations. We prove the correctness of the algorithm and establish some relations ....

....due to substitution and congruence that arise in implementations of systems of Lambda Calculus. In fact it is possible to translate any term into a combinatory term using for this a practical algorithm for carrying out abstraction on combinatory terms. This algorithm was defined by Schonfinkel [9] in 1924. However, when the abstraction algorithm is used repeatedly, it tends to produce expressions of disastrous size with bad consequences in practical applications. First the large size of terms increases the storage space required to represent them and in addition to that there are usually ....

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Mathematische Annalen, 92:305--316, 1924.

....there may meaningfully be multiple tabulation statements mentioning the same predicate symbol, for example, if the predicate symbol denotes a bijective function. Let us define three metafunctions of mode atoms that we shall use in our proof procedure below. All three functions are curried [28, 29], because the intention is that applying them to particular mode atoms should yield particular tests and selectors on atoms. ffl Let applicable be a function of type modeatom (atom ftrue; falseg) such that applicable(p(t 1 ; t k ) q(u 1 ; u l ) j p = q k = l 8(1 i ....

Schonfinkel, M., Uber die Bausteine der mathematischen Logik, Math. Ann., 92:305--316 (1924).

....do in fact exist. The former approach, as adopted by the VDM # incorporates just such a proof, as the post condition has been constructed. Note 3. value The address (a) and data (v) arguments of these operators have been been separated from the memory arguments ( by the technique of currying (Schonfinkel 1924, Curry 1958) This allows use to interpret R(a) as an operation that reads from address a of any memory, and W (a, v) as an operator that writes v into address a of any memory. Given this model it is easy to show some key properties regarding the e#ects of multiple Writes to the same or di#erent ....

Schonfinkel, M.: Uber die bausteine der mathematischen logik. Mathematische Annalen, 92:305--16, 1924.

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Math. Annalen, 92:305--316, 1924. 10

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Math. Annalen, 92:305--316, 1924. 10

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. In Mathematische Annalen, volume 92, pages 305--316. 1924.

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. In Mathematische Annalen, volume 92, pages 305--316. 1924.

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Schonfinkel, M., Uber die Bausteine der mathematischen Logik. Math. Annalen 92, pp.305--316, 1924.

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. Mathematische Annalen, 92:305--316, 1924. English translation appears in [113].

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M. Schonfinkel. Uber die Bausteine der mathematischen Logik. In Mathematische Annalen, volume 92, pages 305--316. 1924.

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M. Schonfinkel. Uber die bausteine der mathematischen logik. In Mathematische Annalen, volume 92, pages 305--316. 1924.

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M. Schonfinkel. Uber die bausteine der mathematischen logik. Mathematische Annalen, 92:305--316, 1924.

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Moses Schonfinkel. Uber die Bausteine der mathematischen Logik. Mathematische Annalen, 92:305--316, 1924.

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Schonfinkel, M., Uber die Bausteine der mathematischen Logik. Matematische Annalen, 92 (1924), pp. 305-316.

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