B. Nordstrom. Terminating general recursion. BIT, 28:605--619, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....calls involving an immediate component of the argument. This excludes functions that divide by repeated subtraction or that sort by recursively sorting shorter lists. Coding such functions using structural recursion requires ingenuity; consider Smith s treatment of Quicksort [26] Nordstrom [19] and I [21] have attempted to re introduce well founded relations to type theory, with limited success. In ZF set theory, well founded relations reclaim their role as the foundation of induction and recursion. They can express di#cult termination arguments, such as for unification and Quicksort; ....

Nordstrom, B., Terminating general recursion, BIT 28 (1988), 605--619

....to loops which terminate. This can be done by defining a well founded order on the data which the recursion is over, such that at each stage the computation on a value can only make use of computations on values lower in the order. This form of recursion is known as well founded recursion [Nor88] We will only use well founded recursion over the naturals with the usual less than ordering. We write # z x #[z]for# z:nat (z x) #[z] Rather than separate the proof of termination from well formedness, however, we build it in by defining a constant, natwfrec, which can only construct ....

Bengt Nordstrom. Terminating general recursion. Bit, 28:605--619, 1988.

....objects satisfying no condition that guarantees termination. As a consequence, there is no direct way of formalising them in type theory. The standard way of handling general recursion in type theory uses a wellfounded recursion principle derived from the accessibility predicate Acc (see [Acz77, Nor88] The predicate Acc captures the idea that an element a of type A is accessible by a relation OE if there exists no infinite decreasing sequence starting from a. A set A is said to be well founded with respect to OE if all its elements are accessible by OE. When using this predicate to write ....

....object is equal to itself. Although the following two predicates are not as general as the previous one, they play an important role in the following sections. Acc: Represents the standard accessibility predicate, which is the standard way to handle general recursion in type theory (see [Acz77, Nor88] 5 Given a set A, a binary relation OE on A and an element a in A, we can form the set Acc(A; OE; a) This set is inhabited if, given a i in A for 1 6 i, there exists no infinite descending sequence : OE a 2 OE a 1 OE a. If this is the case, we say that a is in the well founded part of OE ....

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B. Nordstrom. Terminating General Recursion. BIT, 28(3):605-- 619, October 1988.

....yield not only the intended results but also representations of the veracity of these results, that is, the reasons why they are correctly calculated. To ameliorate the latter problem certain notions of subtype were introduced into MLTT [Pet 84] For the W type Nordstrom introduced the acc type [Nor 87] In both cases these new types suppress the information which is deemed to be unnecessary from a computational point of view. Types such as these are known as types with information loss [Bac 89] Such additions undermine the desiderata of the theory (for example the principle of complete ....

Nordstrom, B., Terminating general recursion, Tech. rep., University of Gteborg, Programming methodology group, 1987.

....languages and then discuss some issues concerning the framework, which lead to another hierarchy of languages, LTC 0 ; LTC 1 ; LTC . The second part of the paper documents the implementation of this last hierarchy in the generic theorem prover, Isabelle, developed by Larry Paulson at Cambridge [4, 5]. We also describe work in progress on verifying, in Isabelle, the interpretations of the type theories TT 0 ; TT 1 and TT , that are presented in [6] in the corresponding languages of the new LTC hierarchy. Part One: The Framework 2 The LTC framework and the languages L i . The increasing ....

....for LTC. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science, pages 392 399. IEEE, 1988. 3] Peter Dybjer and Herbert Sander, A Functional Programming Approach to the Specification and Verification of Concurrent Systems. Formal Aspects of Computing (1989) 1:303 319 [4] L. C. Paulson, The Foundation of a Generic Theorem Prover. Technical report 130, University of Cambridge, 1988. 5] L. C. Paulson and T. Nipkow, Isabelle tutorial and user s manual. Technical report 189, University of Cambridge, 1990. 6] N. Mendler, A series of type theories and their ....

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B. Nordstrom. "Terminating General Recursion." In preparation (1987).

....scheme. However, we will see that this is not the case, indeed it is straightforward to optimise the resulting program such that the gain of efficiency is maintained. This can be exemplified by using the inductive presentation of general well founded recursion in Type Theory as proposed by [Nor88, Pau86]. We define the predicate Acc which defines the accessible subset of a relation 5 . We can define Acc in ALF inductively: Acc ( A; A)Set; A) Set acc (a A; b A; R(b, a) Acc(R, b) Acc(R, a) Le (Nat; Nat) Set From these it is possible to derive a typed fix point combinator which given ....

Bengt Nordstrom. Terminating General Recursion. BIT, 28(3):605--619, October 1988.

....calls involving an immediate component of the argument. This excludes functions that divide by repeated subtraction or that sort by recursively sorting shorter lists. Coding such functions using structural recursion requires ingenuity; consider Smith s treatment of Quicksort [26] Nordstrom [19] and I [21] have attempted to re introduce well founded relations to type theory, with limited success. In ZF set theory, well founded relations reclaim their role as the foundation of induction and recursion. They can express di#cult termination arguments, such as for unification and Quicksort; ....

Nordstrom, B., Terminating general recursion, BIT 28 (1988), 605--619

....[Bove99] shows that a standard Haskell implementation of the algorithm can be imported almost systematically into type theory. However the general recursion of the original is replaced by petrol powered recursion 4 over an induct 4 my phrase 10 ively defined accessibility predicate [Nor88] which can be expressed in ALF, but not its successors. My implementation, in chapter seven of this thesis, is dependently typed, and exploits the power of constraining constructors to represent substitutions as association lists in a way which captures the idea that each assignment gets rid of ....

Bengt Nordstrom. Terminating General Recursion. BIT, Vol. 28, pp605-- 619. 1988.

....it must not run into an infinite loop. Hence a side condition P # X is necessary which expresses that the program terminates (see Figure 2) Even though we recognize the importance of its definition, we rather concentrate in this work on the problem of partial correctness and refer the reader to [Nor88] for a discussion of techniques how to ensure termination. Under the assumption that proof terms indeed represent total functions they can be interpreted as programs calculating LF objects and types. The disastrous behavior resulting from dropping the totality property is demonstrated by the ....

Bengt Nordstrom. Terminating general recursion. BIT, 28:605--619, 1988.

....in second order logic (even with a meaning which is not clear in our framework) and corresponding tactics or strategies. To derive another program, we need inside the theory (to preserve properties) a general induction rule as in the extended propositional version of Martin Lof type theory [13] to derive more efficient algorithms (like, for instance, the quicksort program that uses general induction) by considering well order on the manipulated structures. Consequently, it is necessary to extend the theory by adapted rules (without destroying the essential correctness of programs) or to ....

Nordstrom, B., Terminating general recursion, Technical Report 46, Programming Methodology Group, Goteborg University, June 1988.

.... ourselves to structural recursion (which is a straightforward generalisation of primitive recursion) This is not such a serious restriction since we can still define general recursive programs by wellfounded recursion, which can be presented as an instance of structural recursion (e.g. see [Nor88]) 2.2 Incomplete covering ALF supports the construction of objects using pattern matching by generating complete sets of patterns. However, this mechanism is not foolproof as the following example shows: Unit Set void Unit lem (Unit) False lem(h) case h Unit of end absurd ....

Bengt Nordstrom. Terminating General Recursion. BIT, 28(3):605--619, October 1988.

....r normal forms, i.e. all x such that no R(y; x) exists are in Acc(R) because the precondition is vacuously true. We can now implement a refined version of fix which realizes general well founded recursion via structural recursion over the proof of Acc(R) 17 Due to Paulson [Pau86] and Nordstrom [Nor88] fix (A, B Set; R (A; A)Set; a A; b A;R(b, a) B) B; a A; Acc(R, a) B fix (A, B, R, h, a, acc( h 2 ) h(a, b, h 1 ]fix (A, B, R, h, b, h 2 (b, h 1 ) A source for well founded recursion are the internal versions of the structural orderings on inductively defined ....

Bengt Nordstrom. Terminating General Recursion. BIT, 28(3):605-- 619, October 1988.

....process. Another problem is program termination and, for some specification, we cannot find a solution in type theory because of a lack of information. A way to treat such situations in type theory, with propositions and subsets, is an extension of the theory, for example with general recursion [19]. 2.2 A logical framework We give here a presentation of intuitionistic type theory in a lambda calculus framework [7] like in [8] which has an important relationship with ELF [12] and Automath languages [2] This formulation shows a presentation of logical systems in typed theory. This ....

....the type concerned (here L(A) results in the elimination of the initial program in the extracted one. In type theory, we have the possibility to extracted direct efficient programs as the quicksort program (it needs extension of the kernel theory with subsets, propositions and general recursion [19] with new operators) It is not necessary to precise the form in our calculus to say that an occurence of append function appears in it, decreasing the efficiency. By application of the generalization strategy on this program, we can transform it by proof into an more efficient tail recursive ....

B. Nordstrom. Terminating general recursion. Technical Report 46, Programming Methodology Group, Goteborg University, June 1988.

....C(sup(b; u) c : WA1;A2 ) C(c) 5.2.2 The well founded part of a relation If A 1 is a set and A 2 is a binary relation on that set, then Acc A 1 ;A 2 (a) is true iff a is in the well founded part of A 2 . An application of this notion in the context of type theory can be found in Nordstrom [28]. He suggested to add general recursion along a well founded relation to a version of type theory in which propositions and sets are not identified and which is also extended with subset formation. Compare also the discussion in Dybjer [12] Formation rule. Acc : A 1 : set) A 2 : A 1 ) A 1 ....

B. Nordstrom. Terminating general recursion. BIT, 28:605--619, 1988.

....calls involving an immediate component of the argument. This excludes functions that divide by repeated subtraction or that sort by recursively sorting shorter lists. Coding such functions using structural recursion requires ingenuity; consider Smith s treatment of Quicksort [26] Nordstrom [19] and I [21] have attempted to re introduce well founded relations to type theory, with limited success. In ZF set theory, well founded relations reclaim their role as the foundation of induction and recursion. They can express difficult termination arguments, such as for unification and Quicksort; ....

Nordstrom, B., Terminating general recursion, BIT 28 (1988), 605--619

....simplicity is misleading; the side condition on variable names make it much more difficult to use than expected. 2.3. Strongly normalizing terms A term is strongly normalizing if and only if there is no infinite reduction path starting from it. The following definition is well known since [14] and expresses that, for a relation R, there is no infinite decreasing sequence starting from t: DEFINITION 9 (predicate Acc) The set AccR is the smallest set verifying: 8t: 8u: u R t ) u 2 AccR ) t 2 AccR : coqencoq.tex; 28 03 1997; 14:47; no v. p.5 6 B. Barras and B. Werner R2 R1 R1 y ....

B. Nordstrom. Terminating General Recursion. BIT, 28, 1988.

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B. Nordstrom. Terminating general recursion. BIT, 28:605--619, 1988.

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B. Nordstrom. Terminating general recursion. BIT, 28:605--619, 1988.

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B. Nordstrom. Terminating general recursion. BIT, 28, 1988.

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B. Nordstrom. Terminating general recursion. BIT, 28, 1988.

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B. Nordstrom. Terminating general recursion. BIT, 28:605--619, 1988.

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