J.L. Bates and R.L. Constable, Proofs as programs, ACM Transactions on Programming Languages and Systems 7(1) (1985) 113--136.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is provable under the set of conditional substitutions [ x 1 (yh 1) Yc yh 1 ] x 1 (yh 1) Yc yh ] In a final step the conditional substitutions and rippling sequences that prove the induction step are translated into a sequent proof. Following the proofs asprograms principle [3] one can then extract a proof expression that describes an algorithm for the given specification. The extended rippling technique, which is described in detail in [24] has been implemented as NuPRL tactic and applied to several program specifications such as quotient remainder, append, last, log ....

J. Bates & R. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, 1985.

....of the parse state after an error is found. There is only one assumption made here that is not trivially true for most parsers used today: the measure of distance assumes that all handles, even after the point of error, are reduced, if possible. This feature is available in some modern parsers[5] that can perform non corrective syntax analysis[13] Further, 16] presents a suitable parser generator for continuous LR(k) grammars which produces parsers as fast as traditional parsers generated by lex and yacc. It proves that the function f can be computed efficiently. Continuous grammars ....

Joseph Bates and Alon Lavie. Recognizing substrings of LR(k) languages in lineax time. ACM Transactions on Programming Languages and Systems, 16(3):1051- 1077, May 1994.

....The NuPRL system [3] was designed for interactive writing of machine checked constructive proofs and for extracting algorithms from the proofs. The extracted algorithms are guaranteed to be correct which makes it possible to use NuPRL as a programming language with built in veri cation[1,5,7,8,9,10]. However it turned out that proofs written without algorithmic eciency in mind often produce very inecient algorithms exponential and double exponential ones for problems that can be solved in polynomial time. In this paper we present some general principles of ecient programming in ....

Bates, J. L. and R. L. Constable, Proofs as programs, ACM Transactions on Programming Languages and Systems 7 (1985), pp. 53-71.

....nondeterminism, computations on continuous and higher type data, and the operational semantics and expressive power of functional languages [51, 48, 53, 28, 1] On the other hand proof theory provides formal methods for program development. For example, the so called proofs as programs paradigm [2, 50, 59], according to which a constructive proof of a speci cation corresponds to a satisfying program, has led to a method of program synthesis from proofs which has been implemented in a number of interactive proof systems [21, 39, 42, 30, 47, 3] It is the objective of this project to pursue current ....

J. Bates, R. Constable, Proofs as Programs. ACM Transactions on Programming Languages and Systems 7(1), 113-136, 1985.

....by the German Academic Exchange Service DAAD with a fellowship while the second author was a visiting scientist at Cornell University, Ithaca, NY, USA. 2 Uwe Egly, Stephan Schmitt On Intuitionistic Proof Transformations, their Complexity, according to the proofs as programs paradigm [1]. Hence, f forms a correctly verified program term with respect to the given specification. Since the interactive nature of the proof process stands in contrast to an efficient development of programs, every effort has been made in order to support automated proof search in fragments of ITT. This ....

J. L. Bates and R. L. Constable. Proofs as Programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, January 1985.

....the NuPRL proof development system [8] since its underlying logical calculus can deal with a rich variety of problems from mathematics and programming and allows to formalize even high level strategies in a natural way. Since it is based on the proofs as programs paradigm to program synthesis [2] it allows to treat algorithm design strategies as proof procedures and to integrate a great variety reasoning techniques on all levels of abstraction. Finally it supports interaction with a human expert (programmer) whenever the automated strategies turn out to be too weak. 2 All our automated ....

....allows to use the above techniques as trusted external refiners. However, whenever a piece of code shall be extracted from the proof, it is necessary to convert the proofs generated by a search procedure back into a constructive sequent proof which, according to the program as program paradigm [2], can be turned into a program. In [34, 20] we have developed an embedding of connection based proof methods into NuPRL based on such conversions. The proof method described in [20] constructs a matrix proof closely related to a cut free sequent proof in LJmc , the multiple conclusioned sequent ....

J. L. Bates and R. L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, January 1985.

....that eliminates detours in proofs, and hence presents the proof in a pure, expanded form suitable for further analysis. This can e.g. be used to carry out the proofs as programs paradigm in our setting. 1. 2 Proofs as programs The proofs as programs paradigm has been stressed by Constable [2] as providing a suitable interface between mathematics and computer science. This is usually carried out in a setting 4 where the underlying logic is intuitionistic. However, using this logic for mathematical proofs is still somewhat unusual, since some familiar logical laws do not hold any more ....

Joseph L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, January 1985.

....of tactics, and record derivation histories as an execution trace of the application of tactics. One key idea is that tactics are compositional: higher level tactics are constructed from more rudimentary using defined control primitives. This is similar to the approach used in LCF[12] and NuPRL[1, 8]. Given such a derivation history and a modified specification, the correspondence problem [21, 20] addresses how during replay a correspondence between program parts of the original and modified program is established. Our approach uses a combination of name association, structural properties, ....

....of tactics, and record derivation histories as an execution trace of the application of tactics. One key idea is that tactics are compositional: higher level tactics are constructed from more rudimentary using defined control primitives. This is similar to the approach used in LCF[12, 18] and NuPRL[1, 8]. ffl An approach to the correspondence problem is described [21, 20] The correspondence problem addresses how during replay a correspondence between program parts of the original and modified program is established Our approach uses a combination of name association, structural properties, and ....

Bates, J. L., and Constable, R. L. Proofs as programs. ACM Transactions on Programming Languages and Systems 7, 1 (January 1985), 113--136.

....release 5 of the NuPRL proof development system [9] a matrix proof can only be used to establish the truth of a given formula. If, however, the formula is a part of a program derivation, one must be able to extract a piece of code from the proof, which according to the proofs as program paradigm [2], is essentially the same as providing a constructive sequent proof. In order to integrate matrix methods into proof assistants it is therefore necessary to reconstruct a sequent proof from a given matrix proof. As matrix proofs are compact representations of sequent proofs, converting matrix ....

J. Bates and R. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, 1985.

....is 55 LECTURE 12. MAXIMUM SUBSEQUENCE PROBLEM 56 not interesting, since the maximum subsequence will always be X 1 ; X 2 ; X n , so the interesting case is when some of the scores are negative. The following algorithm for finding a maximum subsequence was given by Bates and Constable [5] and Bentley [7, Column 7] Suppose we already knew that the maximum subsequence B of X 1 ; X 2 ; X k has score b. How can we find the maximum subsequence of X 1 ; X 2 ; X k ; X k 1 If X k is included in B, then it is easy: if X k 1 0, we will add X k to B, and if not, we will ....

Joseph L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, January 1985.

....specific algorithms and give their derivations. Finally we comment about higher order properties of the system and some possible extensions to higher order reasoning. We now make some comments about other approaches to program generation. The constructive type theory approaches of [24] Nuprl [5], or the calculus of con3 structions [12] identify the proof that a term has a type with the proof that a term satisfies a specification. From such a proof, a term (program) in the typed or untyped lambda calculus satisfying the specification can be extracted. Such systems involve constructive ....

Joseph L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, January 1985.

....Studies to relax such hypotheses have been carried out but they generalize one hypothesis at a time, and no uni ed approach has been proposed. To overcome limitations of scheduling theory, it is important to study its connections to speci cation theory and take advantage of their complementarity [8, 13]. The speci cation based approach consists in building a timed model of the scheduled system or of an abstraction of it. Then, timed analysis tools are used either to check that the exact model meets scheduling requirements or to extract from the abstraction a scheduler [6, 10] A major diculty in ....

Z. Liu and M. Joseph. Specication and verication of fault-tolerance, timing, and scheduling. ACM Transactions on Programming Languages and Systems, 21(1):46-89, January 1999.

....it will also be able to synthesize programs. 1 Introduction The aim of the work presented here is to automate the synthesis of logic programs. This is done by adapting techniques from areas such as middle out reasoning in explicit proof plans [Bundy 88, Bundy et al. 90a] proofs as programs [Bates Constable 85] and deductive synthesis [Bibel 80] We synthesize pure logic programs [Bundy et al. 90b] from specifications in sorted, first order theories. The approach encompasses two levels of reasoning: An object level, which is a sorted, first order predicate logic with equality, and a meta level, which ....

....an SERC Senior Fellowship. 1 synthesized by leaving the program unspecified when planning a verification proof. Section 7 contains a summary and suggestions for future work. 2 Related Work In program synthesis from specifications 1 , there are two main approachs, i.e. proofs as programs [Bates Constable 85] and deductive synthesis [Bibel 80, Biundo 88] Proofs as programs is based on what is known as the Curry Howard isomorphism [Howard 80] whereby a proposition is identified with a type of terms in the calculus that represent evidence for its truth. Under this isomorphism, a proposition is true ....

Joseph L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113-- 136, January 1985. 10

....inductive proofs [Bundy et al. 93] and has been rather successfully used in the automatic planning [Bundy 88] of inductive proofs in a functional framework. Conventional rippling 2 has also been applied in logic program synthesis and transformation by means of the proofs as programs approach [Bates Constable 85, Bundy et al. 90, Wiggins 92b] However, the relational nature of logic programming pinpoints the weakness of conventional rippling for dealing with relational expressions. Conventional rippling makes an extensive use of information contained in the syntactic structure of a functional expression. ....

Joseph L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, January 1985.

.... the subarea of program specification and program verification, the term verification refers to the more formal attempt to prove programs correct, which leads to the particular notion of deductive programming (cf. Bie85 ] BB93 ] e.g. and, in particular, to proofs as programs (cf. BC85 ] An essential flaw in the area focusing on human factor issues is to completely ignore the formal approaches developed so far. This is nicely illustrated by the claim: The paradox is the potential production of conclusions and recommendations about verification and validation which ....

Joseph L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):113-- 136, 1985.

.... in a way that they conform to human reading conventions [5] or they are translated into natural language to obtain proofs similar to those found in mathematical textbooks [10] Furthermore proofs can be worked up for planning or synthesis tasks if plans or programs should be extracted form proofs [1, 14]. All these applications require a specific proof, i.e. it is not enough to know that some proof exists. As long as we instantiate function variables with function symbols, the proof ae( P 0 ) of has the same structure as the schematic proof P 0 . However, the success of the method is ....

....it actually consists of two replacements which have to be performed separately at different positions. Finally, the replace( 2 (5) step is redundant and should be omitted. Thus 2 (P 0 ) has to be patched for obtaining a proof of 2s which can be processed subsequently in applications like [5, 10, 1, 14]. This example reveals the need for patching proofs and we show that the necessary modifications can always be performed in a uniform way for an arbitrary second order substitution. For that purpose we construct two algorithms for patching a single proof step (Section 3) and based on that for ....

J. L. Bates and R. L. Constable. Proofs as Programs. ACM Transactions on Programming Languages and Systems, 7(1):113--136, 1985.

....operation and for relational operation. Types. Although there is some argument for just accepting a Hindley Milner type system and be done with it, this seems counter to the strong constructivist basis of the system. There is plenty to investigate. There are several systems, such as Nuprl[4, 13] and Larch[21] that include intentional information concerning types. Larch is not constructive but Nuprl is. Again, the dualistic nature of constructionism causes us to pause and determine where we should go from here. But even to bootstrap, some initial decisions must be made. Elementary Data ....

J. Bates and R. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1), 1985.

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J. Bates, A. Lavie: Recognizing Substrings of LR(k) Languages in Linear Time. ACM Transactions on Programming Languages and Systems 16(3) (1994), 1051--1077.

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J. L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):53--71, 1985.

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J. L. Bates and Robert L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):53-71, 1985.

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J. L. Bates and R. L. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7(1):53--71, 1985.

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J.L. Bates and R.L. Constable, Proofs as programs, ACM Transactions on Programming Languages and Systems 7(1) (1985) 113--136.

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J. Bates and R. Constable. Proofs as programs. ACM Transactions on Programming Languages and Systems, 7:113--136, 1985.

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J.L. Bates and R.L. Constable. Proofs as Programs. ACM transactions on Programming Languages and Systems, 7, 1985.

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