Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science. To appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....on arbitrary higher order terms, therefore applies on higher order rewriting based on plain pattern matching. On the other hand, because our ordering includes reduction, it can be adapted to operate on terms in long normal form, hence applying to the higher order rewriting a la Nipkow [10], and yielding an ordering much stronger than the already existing ones [9, 7] to the exception of [3] which needs an important user interaction. To hint at the strength of the ordering described in the present paper, let us mention that almost all examples that we have found in the literature ....

Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3--29, February 1998. 15

....reduce the don t know nondeterminism due to the selection of the inference rule to be applied next. Finally, in Section 5 we draw some conclusions and directions of future work. 2 Preliminaries We first describe the meta language of simply typed # calculus. The notation is roughly consistent with [2, 9, 15]. 2.1 The Simply Typed # Calculus Starting with a fixed set of base types B, the set of all types is the closure of B under the function space constructor #. The letter # ranges over types. Function types associate to the right, i.e. we parse # 1 # 2 # 3 as # 1 (# 2 # 3 ) We ....

....=R t t =R u s =R u s =R t #x.s =R #x.t s =R s # t =R t # (s t) R (s # t # ) l l =R r # ## t It has been shown [18] that =R coincides with the model theoretical semantics for higher order equational logic. Moreover, we have the following relationship between rewriting and equational logic [9]: # . 1) 7 An equation is a pair (s, t) of terms of the same type. We distinguish between oriented equations, written as s # t, and unoriented equations, written as s t. We denote by Eq(F , the set of equations over variables and function symbols . A flex flex equation is an ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Technical report, Institut fur Informatik, TU Munchen, 1994.

....if u is an instance of l by some substitution . Matching here is syntactic, that is, u is ff convertible to the instance of l. In contrast, the more sophisticated notions of higher order rewriting defined by Klop (Combinatory Reduction Systems [30,31] Nipkow (Higher order Rewrite Systems [39,34]) and van Raamsdonk and van Oostrom (Higher Order Rewriting Systems [49,50] generalizing both) are based on higher order pattern matching, that is, u must be fijff convertible to the instance of l. Definition 6 (Rewrite rules and rewriting) A rewrite rule is a pair l r of terms such that: 1) ....

....this case, but this would be an endless game. The use of higher order matching, on the other hand, chooses the appropriate value for the higherorder free variables so as to cover all cases. The local confluence of these rules can be checked on higher order critical 27 pairs, as shown by Nipkow [39,34]. The computation of these critical pairs can be done in linear time [43] thanks to the hypothesis that the left hand sides are patterns. We now show that this example follows the General Schema, by showing first that the free variables of the right hand sides are accessible in their respective ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192, 1998.

....to generalize both first order term rewriting and rewrite systems with bound variables like Church s calculus. In 1991, after Miller s decidability result of the pattern unification problem [20] Nipkow introduced Higher order Rewrite Systems (HRSs) 23] called Pattern Rewrite Systems (PRSs) in [18]) to investigate the metatheory of logic programming languages and theorem provers like Prolog [21] or Isabelle [25] In particular, he extended to the higher order case the decidability result of Knuth and Bendix about local confluence of first order term rewrite systems. 1 At the same time, ....

....only built from variables, function symbols and the abstraction operator like terms, but also from metavariables Z; Z 0 ; of fixed arity. In the left hand sides of rules, the metavariables must be applied to distinct bound variables (a condition similar to the one for patterns a la Miller [18]) By convention, a term Z(x i 1 ; x i k ) headed by [x 1 ] x n ] can be replaced only by a term u such that FV (u) fx 1 ; xng fx i 1 ; x i k g. For example, in a left hand side of the form f( x] y]Z(x) the metaterm Z(x) stands for a term in which y cannot ....

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R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192, 1998.

....a typing discipline including ML like polymorphism. This ordering is powerful enough to deal with many non trivial examples and can be automated. Besides, all aforementioned previous methods operate on terms in j long fi normal form, hence apply only to the higher order rewriting a la Nipkow [MN98], based on higher order pattern matching modulo fij. HORPO is the first method which operates on arbitrary higher order terms, therefore applying to the other kind of rewriting, based on plain pattern matching, where fi reduction is considered as any other rewrite rule. Furthermore, HORPO can ....

....are using the same kind of interpretations as in the first order case) and on the other hand to adapt the notion of dependency graph for the higher order case. 2. We can adapt, in the same way as it can be done for HORPO [JR01] the method to be applicable to higher order rewriting a la Nipkow [MN98], i.e. rewriting on terms in j long fi normal form. 3. We will study other possible interpretations to build I using functionals in a similar way as in [dP96] but with two relevant differences. First due to the fact that we are building a quasi ordering we can use weakly monotonic functionals ....

Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3--29, 1998.

....a typing discipline including ML like polymorphism. This ordering is powerful enough to deal with many non trivial examples and can be automated. Besides, all aforementioned previous methods operate on terms in j long fi normal form, hence apply only to the higher order rewriting a la Nipkow [MN98], based on higher order pattern matching modulo fij. HORPO is the first method which operates on arbitrary higher order terms, therefore applying to the other kind of rewriting, based on plain pattern matching, where fi reduction is considered as any other rewrite rule. Furthermore, HORPO can ....

....several examples which could not be handled by the previous methods. Finally let us mention some work already in progress and some future work we plan to do. 1. We can adapt, in the same way as it can be done for HORPO [JR01] the method to be applicable to higher order rewriting a la Nipkow [MN98], i.e. rewriting on terms in j long fi normal form. 2. We will study other possible interpretations to build I using functionals in a similar way as in [dP96] but with two relevant differences. First due to the fact that we are building a quasi ordering we can use weakly monotonic functionals ....

Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3--29, 1998.

....of polymorphic higher order function symbols. This ordering is powerful enough to deal with many non trivial examples and can be automated. Besides, all aforementioned previous methods operate on terms in j long fi normal form, hence apply only to the higher order rewriting a la Nipkow [MN98], based on higher order pattern matching modulo fij. HORPO is the first method which operates on arbitrary higher order terms, therefore applying to the other kind of rewriting, based on plain pattern matching, where fi reduction is considered as any other rewrite rule. Furthermore, HORPO can ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3--29, 1998. 28

....combinatory reduction systems (CRSs) Khasidashvili s expression reduction systems (ERSs) and Nipkow s higher order pattern rewriting systems (PRSs) 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It s an extension of some confluence results in [CR36, Hue80, Toy88, Klo80, Kha92, Raa93, Tak, MN94, Oos94, ORb] and we refer the reader to these papers and to the handbook chapters [DJ, Klo] for motivation and for standard definitions as well. Here we will mainly be concerned with proving our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system ....

....our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system for which the rewrite rules do not depend on one another is called orthogonal. Formalising this notion can be quite involved depending on the rewrite formalism it is applied to ([Hue80, Klo80, HL, GLM, MN94, Oos94]) but the intuition to be captured is always the same: an application of a rule replaces some substructure by another one, and in orthogonal systems we moreover have that if two distinct substructures can be replaced then these substructures are independent. Some (non)examples are: 1. The rules F ....

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Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. On ftp.informatik.tu-muenchen.de as local/lehrstuhl/nipkow/hrs.dvi.gz, November 1994. To appear in TCS.

....framework for reductions with substitutions (also referred to as higher order rewriting) as in the # calculus [5] and its extensions. Restricted rewriting systems with substitutions were first studied in Pkhakadze [42] and Aczel [1] Several interesting formalisms have been introduced later [24, 51, 36, 48]. We refer to van Raamsdonk [49] for a survey. Expression Reduction Systems Here we use Expression Reduction Systems (ERSs) defined in [24] under the name of CRSs) The present formulation follows [27] and is simpler. Definition 1 Let # be an alphabet comprising variables x, y, z, ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192:3--29, 1998.

....more complicated. Also unification of higher order patterns is technically more complicated than that of first order terms; it still decidable [21] and moreover has linear complexity [31] Here we do not give a completely formal definition of critical pair, which can be found for instance in [22, 18]. Definition 1. 1. A rewrite rule x:l x:r of a higher order rewriting system is left linear if every variable x 2 x occurs exactly once in l. 2. Let C[ls] gt indicate a most general overlap between two redexes, with l r and g d rewrite rules. Then (C[rs] dt) is a critical pair. Note ....

....case there must be a critical pair (r; r 0 ) such that t = C[roe] and t 0 = C[r 0 oe] Since critical pairs are confluent, r and r 0 have a common reduct. As a consequence, t and t 0 have a common reduct. The result of [11] is generalised to higher order term rewriting systems by Nipkow [22, 18]. The proof proceeds basically as in the first order case, but it is technically significantly more difficult to show the key auxiliary result. Theorem 3. A higher order rewriting system is locally confluent if all its critical pairs are confluent. Consider for example the rewrite rules for ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192:3--29, 1998.

....for the class of almost orthogonal and fully extended higher order rewriting systems. The present paper is rather concise in nature; for a detailed account the interested reader is referred to [Raa96] 2 Preliminaries In this section we recall the definition of higher order rewriting systems [Nip91, MN94], following the presentation in [Oos94, Raa96] We further give the definitions of almost orthogonality and full extendedness. The reader is supposed to be familiar with simply typed calculus with fi reduction (denoted by fi ) and restricted j expansion (denoted by j ) see for instance ....

Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Technical Report TUM-I9433, Technische Universitat Munchen, August 1994.

....not affected. Now Finite Family Developments (FFD) states: If a transformation sequence is unbounded in length, then the generation the parts occurring in the sequence belong to, is unbounded. 1 In this paper it is shown that FFD holds for higher order pattern rewriting systems (PRSs, Nip, MN94] a class of term rewriting systems comprising both first order term rewriting systems (TRSs, cf. Klo] and the lambda calculus (cf. Bar84] The outline of the paper is as follows. In Section 2 higher order pattern rewriting systems are recapitulated. In Section 3 a simpler version of FFD ....

.... P (X) for all (exactly one) variables in the common domain (and the identity on the other variables) Lemma 2.6 (Substitution) Let P be a PRS. 1. If M iP N and oe iP , then M oe iP N . 2. If P is a linear pattern and oe P , then P oe P P . Proof 1. This is (a special case of) MN94, Thm. 3.9] 2. By induction on the pattern P . 2 Note that the second item fails in general for non linear patterns or for terms which are not patterns. In the sequel we assume all patterns to be linear, hence all rewrite systems to be leftlinear. This entails no loss in generality, since a ....

Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Technical Report I9433, Institut fur Informatik, TU Munchen, November 1994. To appear in TCS.

....rewriting systems, we have: Corollary 5.2. Let H be a left linear patterm rewriting system, such that for every critical pair (s;t) s ffi Gamma t, then H is confluent. The corollary extends Huet s result in the first order case and is a generalisation to the higher order case as well. cf. [MN94] for an extension to first order like HRSs) Let us remark that the result does extend the confluence by weak orthogonality result of Section 4, but only in the case of patterm rewriting systems, since the proof here uses the term structure in an essential way. 8 Toyama In [Toy88] Toyama ....

Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. On ftp.informatik.tu-muenchen.de as local/lehrstuhl/nipkow/hrs.dvi.gz, November 1994.

....[28] Examples of higher order patterns are x; y:F (x; y) and x:f(G(z:x(z) where the latter is at least third order. Non patterns are for instance x; y:F (a; y) and x:G(H(x) 2. 1 Higher Order Rewriting The following definitions for higher order rewriting are in the lines of [24, 19]. Definition1. A rewrite rule is a pair l r such that l is a pattern but not j equivalent to a free variable, l and r are long fij normal forms of the same base type, and FV(l) FV(r) A Higher Order Rewrite System (HRS) is a set of rewrite rules. The letter R always denotes an HRS. Assuming a ....

....and a substitution is R normalized if if all terms in the image of are in R normal form. In contrast to the first order notion of term rewriting, Gamma is not stable under substitution: reducibility of s does not imply reducibility of s. Its transitive reflexive closure is however stable [19]: Lemma 2. Assume an GHRS R. If s Gamma R t, then s Gamma R t. A reduction is called confluent, if any two reductions from a term t are joinable, i.e. if t Gamma u and t Gamma v then there exists w with u Gamma w and v Gamma w. For results on confluence of higher order ....

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Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Technical report, Institut fur Informatik, TU Munchen, 1994.

....the case s = t 2 C is contained in the case s ffl Gamma u. 9 Note that the Parallel Moves Lemma does not hold for orthogonal higherorder rewrite systems with bound variables as observed in the literature, see [vO97] and [vR99] A proof of confluence in such rewrite systems can be found in [MN98]. Now we conclude this section by the main result of this section and an example of its application. Theorem 25 (confluence by orthogonality) Every orthogonal TRS is confluent. Proof. By Lemma 16 and the Parallel Moves Lemma (Lemma 24) Example 26 (confluence by orthogonality) The example TRS ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192:3--29, 1998.

....lemma to this case as done in [Vir96] had to be employed. 2 1 Motivation Higher order rewrite systems (HRS) as an extension of term rewriting to simply typed terms were introduced by Nipkow [Nip91] Several confluence and local confluence results could be generalized to the higher order case [MN94]. In particular the convergence of all critical pairs implies local confluence and (weak) orthogonality implies confluence. Since the logic programming language Elf [Pfe94] provides a higher order setting it can be used in a similar way as Prolog [Mil91] as done by Felty in [Fel92] to implement ....

R. Mayr and T. Nipkow. Higher-Order Rewrite Systems and their Confluence. Technical Report TUMI9433, Technical University, Munich, August 1994.

....step which is not perpetual. Example 7.1. Consider the higher order rewrite system with rules: f(yz:F (x:y(x) z) f F (x:c; Omega Gamma app(abs(x:F (x) S) beta F (S) where the first rule contains a function variable (y) as argument to a free variable (F ) the second rule is the usual [46] higher order rendering of the fi rule from calculus, and Omega = app(abs(x:app(x; x) abs(x:app(x; x) Then f(yz:app(abs(x:y(x) z) beta f(yz:y(z) is non erasing but critical. This can be seen from the following diagram, of which the bottom part is the only reduction starting from ....

Mayr, R. and Nipkow, T. (1998), Higher-order rewrite systems and their confluence, Theoret. Comp. Sci. 192, 3--29.

....proving strong normalization of higherorder rewrite rules based on ordering comparisons. These orderings are either quite weak [13, 11] or need an important user interaction [5] Besides, they operate on terms in j long fi normal form, hence apply only to the higher order rewriting a la Nipkow [15], based on higher order pattern matching modulo fij. To our knowledge, our ordering is the first to operate on arbitrary higher order terms, therefore applying to the other kind of rewriting, based on plain pattern matching. And indeed we want to stress four important features of our approach. ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3-29, 1998.

....proving strong normalization of higherorder rewrite rules based on ordering comparisons. These orderings are either quite weak [13, 11] or need an important user interaction [5] Besides, they operate on terms in j long fi normal form, hence apply only to the higher order rewriting a la Nipkow [15], based on higher order pattern matching modulo fij. To our knowledge, our ordering is the first to operate on arbitrary higher order terms, therefore applying to the other kind of rewriting, based on plain pattern matching. And indeed we want to stress four important features of our approach. ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3-29, 1998.

....if u is an instance of l. Matching here is syntactic, that is, u is ff convertible to the instance of l by some substitution . In contrast, the more sophisticated notions of higher order rewriting defined by Klop (Combinatory Reduction Systems [29, 30] Nipkow (Higherorder Rewrite Systems [38, 33]) and Raamsdonk and Oostrom (Higher Order Rewriting [45, 46] generalizing both) are based on higher order pattern matching, that is, u must be fijff convertible to the instance of l by some substitution . We will later see in Section 4 that our results apply to these more sophisticated notions ....

....for this case, but this would be an endless game. The use of higher order matching, on the other hand, chooses the appropriate value for the higher order free variables so as to cover all cases. The local confluence of these rules can be checked on higher order critical pairs, as shown by Nipkow [38, 33]. The computation of these critical pairs can be done in linear time [41] thanks to the hypothesis that the lefthand sides are patterns. We now show that this example follows the General Schema, by showing first that the free variables of the righthand sides are accessible in their respective ....

R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. TCS, 192, 1998.

....when the hole is replaced by an expression of compatible type, environments always produce well typed expressions: Lemma 9. If Gamma OE Sigma E[ Gamma ffi ffi : A ffi ] A, and Delta OE Sigma M : A ffi with Delta Gamma ffi , then Gamma OE Sigma E[ M ] A. In [9], the definition of substitution makes use of the existence and uniqueness of long fij normal forms. In the LF calculus, these find an analogue in the concept of canonical form: Definition10. We define canonical forms for terms and type families by the judgements Gamma Sigma M A M is ....

.... Delta OE Sigma N [M=x]A fx7 M;y7 Ng: Gamma;y:A) OE Delta Definition13. Given any well typed term Gamma Sigma M : A and substitution = fx 7 Ng : Gamma Delta, define M to be the (unique) canonical form of Delta Sigma (x : B:M ) N : N=x]A Note that here, in analogy to [9], we define the result of a substitution application to be a canonical term. Definition14. 1. Given two substitutions 1 = fx 7 Mg : Gamma 1 Gamma 2 and 2 : Gamma 2 Gamma 3 , the composition 2 ffi 1 is the substitution fx 7 2 Mg : Gamma 1 Gamma 3 . 2. A substitution = ....

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Mayr, R., Nipkow, T. Higher-Order Rewrite Systems and their Confluence. Technical Report TUM-I9433, Technische Universitat Munchen, 1994

....advance in order to overcome these difficulties is due to D. Miller [8] who identified a subclass of higher order terms, called higher order patterns for which the unification problem is decidable, and moreover uniqueness of most general unifiers hold. Making use of this result, T. Nipkow [7, 9] was able to state and prove an analogous of the Critical Pair Lemma for the case of higher order, simply typed TRSes. Nipkow s Higher Order Term Rewriting Systems (HTRS) are similar to Klop s Combinatory Reduction Systems (CRS) For a detailed analysis of the relation between these two, see [16] ....

Mayr, R. , Nipkow, T. Higher-Order Rewrite Systems and their Confluence. Tech. Report, Technische Universitat Munchen, 1994

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R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 1997. C. Prehofer, 6/97 Higher-Order Equational Logic 100

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Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science. To appear.

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Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Technical report, Institut fur Informatik, TU Munchen, 1994.

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