G. Nadathur. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. J. of Func. and Logic Programming, 1999(2):1-62, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....second author supported by the CAPES Brazilian foundation. Email: fayala,flaviog mat.unb.br Email: fairouz cee.hw.ac.uk,fairouz macs.hw.ac.uk c 2002 Published by Elsevier Science B. V. automated deduction and theorem proving [24,25] to proof theory [31] to programming languages [8,20,23,26] and to higher order uni cation HOU [2,13] This paper concentrates on three di erent styles of substitutions: i) The style [1] which introduces two di erent sets of entities: one for terms and one for substitutions. ii) The suspension calculus [28,26] denoted susp , which introduces ....

....[31] to programming languages [8,20,23,26] and to higher order uni cation HOU [2,13] This paper concentrates on three di erent styles of substitutions: i) The style [1] which introduces two di erent sets of entities: one for terms and one for substitutions. ii) The suspension calculus [28,26], denoted susp , which introduces three di erent sets of entities: one for terms, one for environments and one for environment terms. iii) The s style [19] which uses a philosophy of de Bruijn s Automath [29] elaborated in the new item notation [18] The philosophy states that terms are built ....

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G. Nadathur. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. The Journal of Functional and Logic Programming, 1999(2):1-62, 1999.

....for research in mathematics, second author supported by the CAPES Brazilian foundation. Email: fayala,flaviog mat.unb.br Email: fairouz cee.hw.ac.uk c 2002 Published by Elsevier Science B. V. automated deduction and theorem proving [24,25] to proof theory [31] to programming languages [8,20,23,26] and to higher order uni cation HOU [2,13] This paper concentrates on three di erent styles of substitutions: i) The style [1] which introduces two di erent sets of entities: one for terms and one for substitutions. ii) The suspension calculus [28,26] denoted susp , which introduces ....

....[31] to programming languages [8,20,23,26] and to higher order uni cation HOU [2,13] This paper concentrates on three di erent styles of substitutions: i) The style [1] which introduces two di erent sets of entities: one for terms and one for substitutions. ii) The suspension calculus [28,26], denoted susp , which introduces three di erent sets of entities: one for terms, one for environments and one for environment terms. iii) The s style [19] which uses a philosophy of de Bruijn s Automath [29] elaborated in the new item notation [18] The philosophy states that terms are built ....

[Article contains additional citation context not shown here]

G. Nadathur. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. The Journal of Functional and Logic Programming, 1999(2):1-62, 1999.

....work remains to be done and in particular, to be conclusive, a prototype implementation of this method is necessary. Additionally, a formal distinction, from the practical point of view, between the s e calculus (and our procedure) and the suspension calculus developed by Nadathur and Wilson in [10, 9] (and used in the implementation of the higher order logical programming language Prolog) should be elaborated. This is meaningful, since the s e calculus and the calculus of [10, 9] have correlated nice properties. For instance the laziness in the substitution needed in implementations of ....

....between the s e calculus (and our procedure) and the suspension calculus developed by Nadathur and Wilson in [10, 9] and used in the implementation of the higher order logical programming language Prolog) should be elaborated. This is meaningful, since the s e calculus and the calculus of [10, 9] have correlated nice properties. For instance the laziness in the substitution needed in implementations of reduction, that arises naturally in the s e calculus, is provided as the informal but empirical concept of suspension of substitutions by the rewrite rules of Nadathur and Wilson. ....

G. Nadathur. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. The Journal of Functional and Logic Programming, 1999(2):1-62, 1999.

....work remains to be done and in particular, to be conclusive, a prototype implementation of this method is necessary. Additionally, a formal distinction, from the practical point of view, between the s e calculus (and our procedure) and the suspension calculus developed by Nadathur and Wilson in [NW98,NW99] (and used in the implementation of the higher order logical programming language Prolog) should be elaborated. This is meaningful, since the s e calculus and the calculus of [NW98,NW99] have correlated nice properties. For instance the laziness in the substitution needed in implementations of ....

....the s e calculus (and our procedure) and the suspension calculus developed by Nadathur and Wilson in [NW98,NW99] and used in the implementation of the higher order logical programming language Prolog) should be elaborated. This is meaningful, since the s e calculus and the calculus of [NW98,NW99] have correlated nice properties. For instance the laziness in the substitution needed in implementations of reduction, that arises naturally in the s e calculus, is Applying se Uni cation for Simply Typed HOU. Extended Version 25 provided as the informal but empirical concept of suspension ....

G. Nadathur and D. S. Wilson. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. The Journal of Functional and Logic Programming, 1999(2):1-62, 1999.

....replaces n by b and decrements by 1 all the free indices that refer to an element of the referential greater than n. So we do not define a general notion of simultaneous substitution fn=b; m=cg as the decrementing effect of such a substitution would be more technical (see [40] and the full papers [39, 41] for a complet development of this idea) The simultaneous substitution which can be easily defined is the substitution of an initial segment of the natural numbers f1=a 1 ; 2=a 2 ; n=ang. In this case all the other indices in the term have to be decremented by n. Notice that such a ....

G. Nadathur, A fine-grained notation for lambda terms and its use in intensional operations, Tech. Report TR-96-13, Department of Computer Science, University of Chicago, May 1996.

....provided as the informal but empirical concept of suspension of substitutions by Nadathur and Wilson rewrite rules, being their notion of substitution more general than the s e one. More recently their rewrite rules were published in the context of explicit substitution as the suspension calculus [49, 50]. Establishing formally the relations and di erences between the s e calculus and the suspension calculus remains as an important work to be done. 6 ( M N) M [N id] Beta) M N ) S] M [S] N [S] App) M ) S] M[1 (S ) Abs) M [S] T ] M [S T ] Clos) 1[M S] M (VarCons) ....

G. Nadathur and D. S. Wilson. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. The Journal of Functional and Logic Programming, 1999(2):1-62, 1999.

....in the s e calculus into a description of solutions of the corresponding HOU problems in the pure calculus. Additionally, a formal distinction, from the practical point of view, between the s e calculus (and our procedure) and the suspension calculus developed by Nadathur and Wilson in [NW98,NW99] (and used in the implementation of the higher order logical programming language Prolog) should be elaborated. This is meaningful, since the s e calculus and the calculus of [NW98,NW99] have correlated nice properties. For instance the laziness in the substitution needed in implementations of ....

....the s e calculus (and our procedure) and the suspension calculus developed by Nadathur and Wilson in [NW98,NW99] and used in the implementation of the higher order logical programming language Prolog) should be elaborated. This is meaningful, since the s e calculus and the calculus of [NW98,NW99] have correlated nice properties. For instance the laziness in the substitution needed in implementations of reduction, that arises naturally in the s e calculus, is provided as the informal but empirical concept of suspension of substitutions by Nadathur and Wilson rewrite rules. Establishing ....

G. Nadathur and D. S. Wilson. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. The Journal of Functional and Logic Programming, 1999(2):1-62, 1999.

....of an abstract machine [12] or the pruning of search space in unification algorithms [4, 5, 19] Also, this feature improves the substitution mechanism by allowing parallel substitutions of variables. An interesting discussion about composition of substitutions in calculus can be found in [29]. However, composition of substitutions and simultaneous substitutions are responsible of the following non left linear rule in oe: 1[S] Delta ( ffi S) SCons) S. Informally, if we interpret S as a list, 1 as the head function and as the tail function, then this rule corresponds to the ....

G. Nadathur. A fine-grained notation for lambda terms and its use in intensional operations. Technical Report TR-96-13, Department of Computer Science, University of Chicago, May 30 1996.

....may only need to be carried out up to the occurrence of the first disagreement between two unequal terms. Therefore, a good representation of terms should naturally allow the comparison to be interleaved with the fi reduction. An explicit substitution notation has been developed [9, 10] for the calculus to meet this requirement. The resulting notation, called the annotated suspension notation is used in the present implementation. Intuitively, a suspension consists of a term with a suspended substitution. A suspension term is a term of the form [t; ol; nl; env] where t is a ....

Gopalan Nadathur and Debra Sue Wilson. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. University of Chicago, Technical Report TR-93-13. (Accepted for publication in Journal of Functional and Logic Programming.

....of an abstract machine [40] or the pruning of search space in unification algorithms [25, 26, 55] Also, this feature improves the substitution mechanism by allowing parallel substitutions of variables. An interesting discussion about composition of substitutions in calculus can be found in [73]. Example 2.34 In the unification algorithm, the equation X[N= S] X[ S) N [S] where X is a meta variable, N is a ground term and S is a ground substitution, is a trivial equation when we add the composition operation: X[N [S] S] X[N [S] S] In i (and any calculi without ....

G. Nadathur. A fine-grained notation for lambda terms and its use in intensional operations. Technical Report TR-96-13, Department of Computer Science, University of Chicago, May 30 1996. Accepted for publication in Journal of Functional and Logic Programming.

....of such steps into a larger step that is easy to carry out and that has practical benefits such as providing for the combination of substitution walks. A compilation of this kind can be achieved through the identification of derived or admissible rules for our notation. This matter is discussed in [29]. 4.2 A modified syntax for terms At a formal level, the main addition to the syntax of de Bruijn terms that yields our notation is that of a suspension. In presenting this category of terms, it is necessary to also explain the structure of environments and environment terms. The syntax of these ....

....from the results of Sections 6 and 7 in showing properties of our system. This method of argument is similar in spirit to the one referred to as the interpretation method in [17] and used in [17] and [39] in proving confluence properties of a combinator calculus. We use this method again in [29]. 9 Conclusion We have described in this paper a notation for the terms in a lambda calculus and a system for rewriting expressions in this notation. Our notation is based on the de Bruijn representation of lambda terms but embellishes this so as to allow for the representation of a term with a ....

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Gopalan Nadathur. A fine-grained notation for lambda terms and its use in intensional operations. Technical Report TR-96-13, Department of Computer Science, University of Chicago, May 1996. To appear in Journal of Functional and Logic Programming.

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G. Nadathur. A Fine-Grained Notation for Lambda Terms and Its Use in Intensional Operations. J. of Func. and Logic Programming, 1999(2):1-62, 1999.

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