Huet, G. and Lang, B. Proving and Applying Program Transformations Expressed with Second-Order Logic. Acta Informatica 11 (1978) 31--55.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....used as the syntactic representation of binding operators. Theorem proving is undecidable, but it is unfortunate that each resolution step is undecidable. We can recover decidability by restricting unification. Limiting the search gives unpredictable results. Second order matching is decidable [18], though second order unification is not [11] Perhaps first order unification plus second order matching is a practical compromise. Ketonen s EKL proves theorems using first order unification plus higher order matching. Ketonen claims that higher order matching is decidable, without proof [20] ....

G. P. Huet, B. Lang, Proving and applying program transformations expressed with second-order patterns, Acta Informatica 11 (1978), pages 31--55.

.... known since the invention of typed # calculi (see for the special case of the quantifiers in [25] it was not before the late seventies that the overall importance of higherorder abstract syntax (a term coined by [28] for the representation of binding in logical rules and program transformations [29] and for implementations [28] was recognized. The term shallow embedding (invented in [30] extends higherorder abstract syntax (HOAS) to a semantic definition and is contrasted to deep embeddings . Moreover, throughout this paper, we will distinguish typed and untyped shallow embeddings. ....

G. Huet, B.L.: Proving and applying program transformations expressed with second order patterns. (Acta Informatica)

.... known since the invention of typed # calculi (see for the special case of the quantifiers in [25] it was not before the late seventies that the overall importance of higherorder abstract syntax (a term coined by [28] for the representation of binding in logical rules and program transformations [29] and for implementations [28] was recognized. The term shallow embedding (invented in [30] extends higherorder abstract syntax (HOAS) to a semantic definition and is contrasted to deep embeddings . Moreover, throughout this paper, we will distinguish typed and untyped shallow embeddings. ....

G. Huet, B.L.: Proving and applying program transformations expressed with second order patterns. (Acta Informatica)

....terms, the application of a substitution to a term and the computation of a matching substitution are rather simple operations. Thus, a rewriting step itself is a rather simple operation. This is not longer true in the second order world, i.e. if the terms contain second order variables; see [4] where Huet and Lang present an algorithm to compute all principal matches for the second order case. For second order variables, substitution application incorporates fi reduction, and consequently matching may involve fi expansion. The research reported here was partially supported by SERC ....

G'erard Huet and Bernard Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31--55, 1978.

....x of P , we can treat x as an arbitrary constant from now on. The subgoal may now be rewritten as #= Find : i(name x, id x, age x) name x, if age x 30 then id x else 0) 8 To make further progress, we need the concepts of imitation and projection from higher order matching theory [12], embodied in the following rules: Imitation Rule Given g : B 1 . B n # C To Find: f u = g(v 1 , v n ) Solution: f = # x g(h 1 x, h 2 x, h n x) #= Find: h i u = v i for 1 # i # n Projection Rule To Find: f(a 1 , a k ) a i Solution: f = # k i = # ....

G. Huet and B. Lang. Proving and applying program transformations expressed with second-order patterns. Acta informatica , 11:31--55, 1978.

....of their bound variable exactly once, are considered. A context may be viewed as a linear second order function with one argument, where the binder is left implicit and the hole is the single occurrence of the bound variable. An algorithm for general second order matching is given by Huet and Lang [15]. Curien, Qian and Shi [8] give an algorithm for second order matching that improves eciency for the case of right hand sides with many bound variables and few constants. Second order and third order matching are NP complete [7] while fourth order matching is decidable but NEXPTIME hard [26] For ....

....and want to avoid exponential behavior we may remove equations x t and X [ t[ where x or X occurs only once in the problem. It is interesting to compare the working of our rules to the rules Simpli cation, Imitation and Projection that are used in the more general higher order cases [9, 15]. Simpli cation is essentially a Decompose with respect to a function symbol, while Imitate and Project together correspond to Variable Split. Applied to context matching they would construct a context step by step, Imitation would add a function symbol at the bottom and Project would put in the ....

Gerard Huet and Bernard Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31-55, 1978.

....and verify that P satis es suitable conditions. The schemata approach determines much more concise program derivations than the rules strategies approach, but the schemata approach has its drawbacks: i) the matching task is not always easy and in some cases matching may even be undecidable [HuL78], and (ii) the choice of the right schema transformation one should apply, often requires a deep insight on the program behaviour and may be dicult to mechanize. We have not described our list introduction strategy in full detail. Nevertheless, the reader may realize that each individual ....

G. Huet and B. Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31-55, 1978.

....higher and not just second order means the code is complicated and not easy to understand. It proved difficult to work with and so I decided to write my own special purpose second order mapping. If more time had been available I would have modified the second order algorithm of Huet and Lang [Huet Lang 78] but given the time constraints, I implemented a restricted algorithm of my own. The algorithm considers various cases outlined in Figure 5 1. 1 The algorithm does not allow any variable (first or second order) to map to two different images. The main restriction is that first order ....

G Huet and B Lang. Proving and applying program transformations expressed with second-order patterns. Acta Information, 11:31-55, 1978.

....representing and reasoning about software development steps in QED: ffl by higher order functions, ffl by meta functions. 4.1 Representation of Steps by Higher Order Functions The formalization of transformations using higher order patterns has been considered by several researchers. In [19], for example, program transformations for recursion removal are expressed as second order patterns defined in the simply typed calculus [5] As opposed to this treatment we use the powerful framework of QED and demonstrate that it is possible to formalize and verify a large development step ....

G. Huet and B. Lang. Proving and applying program transformations expressed with second-order-patterns. Acta Informatica, 11:31--55, 1978.

....In this respect, the content language OpenProof (and for the same reason already OpenMath) is more problematic than traditional agent content languages. However, since we can restrict OpenProof to second order expressions, we only need second order matching, which is known to be decidable [ Huet and Lang, 1978 ] We are currently investigating whether more restrictive policies for addressing services via OpenProof can be captured with comparably more lightweight mechanisms. 5.3 A Categorisation of Mathematical Services In this section we brie y categorise mathematical services by their behaviour and ....

G. Huet and B. Lang. Proving and applying Program Transformations expressed with Second Order Logic. Acta Informatica, 11:31-55, 1978.

....etc. Transformations need to be aware of variables by means of extra conditions to avoid problems such as free variable capture during substitution and lifting variable occurrences out of bindings. Transparent handling of variable bindings is desirable. Higher order abstract syntax (hoas) [17, 29, 43] gives a solution to such problems by encoding variable bindings as lambda abstractions. In addition to dealing with the problem of variable capture, hoas provides higher order matching which synthesizes new functions for higher order variables. One of the problems of higher order matching is that ....

G. Huet and B. Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31-55, 1978.

....a rule s left hand side into a given application graph, limited to the case where the images of the rule tips are given, is decidable and finitary on finite graph, i.e. it has mostly finitely many solutions. 15 In several contexts, such as automatical reduction, one prefers a 15 On finite DAGs [11] gives this result. unitary matching problem, i.e. a matching problem with at most one solution. Under certain circumstances the matching is determined by the mapping of rule tips into the application graph. An example is the condition imposed on left hand rule sides in CRSs (see Def. 2.2.8) ....

G. P. Huet and B. Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31--55, 1978.

....= H also tries to imitate the rigid term but by building a term with which to instantiate the free variable in a single step. Beyond this the two algorithms differ substantially. The algorithm presented here is significantly restricted and is for matching, not unification. Huet and Lang [5] adapt Huet s original algorithm to the process of matching second order terms, which is known to be decidable. Their application area is program transformation which is close to the intended application of the algorithm presented here. Simon [12] also presents an algorithm for second order ....

G. Huet and B. Lang. Proving and applying program transformations expressed with 2nd order patterns. Acta Informatica, 11:31--55, 1978.

....Higher order matching was first studied by Lewis D. Baxter and Gerard Huet in the midseventies in their PhD theses [2, 11] Baxter [2] showed that second order matching is NP complete by reduction to the one in three SAT problem. At the same time this case was independently investigated by Huet [11, 12]. In 1975 Huet devised a semi decision algorithm for (undecidable even in the second order case) unification problem [10] Later an analysis of behavior (termination, running time, etc. of this algorithm and its modifications was a source of a couple of results about decidability and complexity ....

....e.g. the one introduced by Dale Miller [15] we would rather avoid it when speaking about unification of an arbitrary term with a closed (or ground) one. 1 ToMasz Wierzbicki: Complexity of the higher order matching order decidable lower bound upper bound 1 yes [18] NLOGSPACE [8] 2 yes [12] NP complete [2, 12, 5] 3 yes [6] NP complete [25, 5] 4 yes [16] NEXPTIME hard 2 NEXPTIME ( 5] # not elementary recursive [24] Table 1: Decidability and complexity of higher order matching also presented an algorithm for the fourth order case (the case shown to be decidable a year before ....

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Gerard P. Huet, Bernard Lang, Proving and applying program transformations expressed with second order patterns, Acta Informatica, 11 (1978) 31--55.

....patterns [Mil90] Although we use a language with full higher order uni cation many of its uses are for higher order matching of a pattern to a sub term of a speci cation. Higher order matching appears in existing work on functional program transformation and synthesis, for example in [HL78] 3 Proof Planning A proof of a theorem can be attained by applying sound derivation rules to certain axioms until the theorem is reached. Alternatively, one can start with a theorem and back chain through the rules until axioms are reached. The search space of these rules, however, is too ....

G. Huet and B. Lang. Proving and applying program transformation expressed with second order patterns. Acta Informatica, 11:31-55, 1978. Logic Program Synthesis in a Higher-Order Setting 20th January 13

....higher order patterns [18] Although we use a language with full higher order uni cation many of its uses are for higher order matching of a pattern to a sub term of a speci cation. Higher order matching appears in existing work on functional program transformation and synthesis, for example in [11]. 3 Proof Planning A proof of a theorem can be attained by applying sound derivation rules to certain axioms until the theorem is reached. Alternatively, one can start with a theorem and back chain through the rules until axioms are reached. The search space de ned by these rules, however, is ....

G. Huet and B. Lang. Proving and applying program transformation expressed with second order patterns. Acta Informatica, 11:31-55, 1978.

....here in the decidable cases. Among them we can mention higher order pattern matching that is decidable and unitary as a consequence of the decidability of pattern uni cation [Mil91, DHKP96] higher order matching which is known to be decidable up to the fourth order [Pad96, Pad00, Dow94, HL78] the 9 decidability of the general case being still open) many rst order equational theories including associativity, commutativity, distributivity and most of their combinations [Nip89, Rin96] For example when T is empty, the syntactic matching substitution from t to t 0 , when it ....

G. Huet and B. Lang. Proving and applying program transformations expressed with secondorder patterns. Acta Informatica, 11:31-55, 1978.

....can not be used to define this notion of instance since that would produce a substitution replacing both p and q with the same variable. Matching must be used to solve the constraint A v = B. Since the object logic is first order, at most second order matching is needed, which remains decidable [9]. In case second order matching yields more than one matching substitution, the rule schema above remains valid for any substitution. WLG R can be thought of as a tactic that is applied backward (or goal directed) to the conclusion of the rule, yielding the premises as subgoals. The intuitive ....

G'erard Huet and Bernard Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31--55, 1978.

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Huet, G. and Lang, B. Proving and Applying Program Transformations Expressed with Second-Order Logic. Acta Informatica 11 (1978) 31--55.

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G. P. Huet, B. Lang, Proving and applying program transformations expressed with second-order patterns, Acta Informatica 11 (1978) 31-55.

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Huet, G.P., Lang, B.: Proving and applying program transformations expressed with second-order patterns. Acta Informatica 11 (1978) 31-55

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Huet, G., Lang, B.: Proving and applying program transformations expressed with second-order patterns. Acta Informatica 11 (1978) 31--55

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Huet, G., and Lang, B. Proving and applying program transformations expressed with second-order patterns. Acta informatica 11 (1978), 31--55.

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G. P. Huet, B. Lang, Proving and applying program transformations expressed with second-order patterns, Acta Informatica 11 (1978), pages 31--55.

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G. P. Huet, B. Lang, Proving and applying program transformations expressed with second-order patterns, Acta Informatica 11 (1978), pages 31--55.

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