Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-20/95, University of Pise, December 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....take into account a congruence on propositions representing computations. Viry [Vir98] had the idea that the classical rst order sequent calculus itself, turned into a rewrite system, can be viewed modulo and decomposed into computational and deductive parts using an oriented rewrite theory [Vir95]. In this paper we make precise what can be viewed as computation in the rst order sequent calculus. Doing that we obtain a system with a large computational part which is equivalent to the classical one. 2 Sequent calculus modulo Let us rst recall the notions from sequent calculus modulo ....

....application of the deduction rules, which has to be controlled undeterministically, and just do normalization 73 Sequent Calculus Viewed Modulo with the congruence. To do that we need to avoid that the congruence can interfere with the deduction. To ensure this, we use the techniques from [Vir95], so we turn the remaining deduction rules into rewrite rules to form an oriented rewrite theory. These rules have to be coherent with ER modulo E, which is achieved by coherence completion adding rules (see gure 7.5) to handle the case of a set of formulas being empty like 8C B C[a ....

Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-20/95, University of Pise, December 1995.

....guaranteeing this equivalence center around different variations on the notion of coherence, which is a form of relative confluence between equations and rules. Methods for checking coherence, or for achieving it by a process of relative completion, have been proposed by Viry in several papers [314,315,318]. Even when the rewrite theory is coherent and the language implementation supports rewriting modulo A, executing rewrite theories is nontrivial, because the rules R in general are neither Church Rosser nor terminating. Furthermore, some rules in R may have additional variables on their righthand ....

Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-9520, Dipartimento di Informatica, Universit`a di Pisa, December 1995. ftp: //ftp.di.unipi.it/pub/techreports/TR-95-20.ps.Z.

....There are di erent notions of coherence, which correspond to di erent correspondences between [t] A [R(E) A [t 0 ] A and [t] A[E [R]A[E [t 0 ] A[E . 4 We focus here on equational coherence, which states that the set of normal forms is preserved, except possibly for some cycles. See [17] for a discussion on the di erent types of coherence, where some interesting results related to coherence are given. Equational coherence is the most important of the coherence properties, since it can be checked by looking at critical pairs. Weak coherence and strong coherence may be more dicult ....

....pairs. Weak coherence and strong coherence may be more dicult to establish. De nition 1 [R(E) A is equational coherent if [E]A ### # # # # # # [R(E) A ## # # # # # # # [R(E) A ### # # # [R(E) A ## # # # # [E]A ## # # # # # # # [E]A ### # # # # # # Theorem 2 (From [17]) If [R(E) A is equational coherent, then if [t] E is a normal form for [R(E) A , then either [t] A[E is a normal form for [R]A[E , or there is a nonempty cycle [t] A[E [R]A[E [t] A[E . In the case of equational coherence the set of normal forms is not exactly preserved, since a cycle in ....

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Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-9520, Department of Computer Science, University of Pisa, December 1995. 29

....E 0 of equations that are Church Rosser, terminating, and sort decreasing modulo A; that is, the equational part must satisfy the same requirements as a functional module. Moreover, we require that the rules R in the module are coherent [67] or at least what might be called weakly coherent [44,68]) with the equations E 0 modulo A. This means that appropriate critical pairs between rules and equations are joinable, allowing us to intermix rewriting with rules and rewriting with equations without losing rewrite computations by failing to perform a rewrite that would have been possible ....

P. Viry. Rewriting modulo a rewrite system. Technical Report TR-95-20, Dipartimento di Informatica, Universit`a di Pisa, Dec. 1995. ftp://ftp.di. unipi.it/pub/techreports/TR-95-20.ps.Z. 65

.... and terminating modulo some axioms for which matching algorithms are available in the implementation, and that system modules are rewrite theories whose equational part satisfies the same requirements as a functional module, and where the equations and the rules are assumed to be weakly coherent [61, 60] modulo the axioms (see Section 4.3) In META LEVEL: ffl Maude terms are reified as elements of a data type Term of terms; ffl Maude modules are reified as terms in a data type Module of modules; ffl the processes of reducing a term to normal form in a functional module and of finding whether ....

....8 and a set E of equations that are Church Rosser, terminating and sort decreasing modulo A; that is, the equational part must be equivalent to a functional module. ffl The rules R in the module are coherent [61] or at least what might be called weakly coherent [38, Section 5.2. 1][60]) with the equations E modulo A. This means that appropriate critical pairs exist between rules and equations, allowing us to intermix rewriting with rules and rewriting with equations without loosing rewrite computations by failing to perform a rewrite that would have been possible before an ....

P. Viry. Rewriting modulo a rewrite system. TR-95-20, C.S. Department, University of Pisa, 1996.

....gives very natural specifications of CCS (see [18] and of the # calculus ( 20] However, since (conditional) term rewriting modulo arbitrary equational theories is generally too complex or even undecidable, it is hard to implement this approach directly. Instead, following the ideas of Viry in [18, 19], we propose to decompose E into a set of directed equations ER and into a set AC expressing associativity and commutativity of certain binary operators in #. If ER is terminating modulo AC , then rewriting by R modulo E can be implemented by a combination of normalizing by ER and rewriting by R, ....

....unique normal forms. In contrast, rules in R describe transitions between states of the system under consideration. Here, the conditions accommodate for the fact that the behavior of a complex system may depend on the behavior of its components. Our definition of an ORT di#ers from the one in [19] with regard to the following aspects: Viry does not take conditions in the transition rules into account, which is crucial for many applications. We do not necessarily assume the transition relation induced by an ORT to be congruent. This will be justified later. In the following example ....

P. Viry. Rewriting modulo a rewrite system. Technical Report TR--95--20, Universit a di Pisa, Dipartimento di Informatica, December 1995.

....the axiom 8y 0 y = y is congruent to the equality axiom 8y y = y. Hence, it can be dropped. Using the terminology introduced by Plotkin, these axioms have been builtin (Plotkin, 1972; Andrews, 1971; Peterson and Stickel, 1981; Stickel, 1985; Jouannaud and Kirchner, 1986; March e, 1994; Viry, 1995; Viry, 1998) In the example above, the congruence is just the congruent closure of the relation induced on terms by the term rewriting system. In many situations, it is also natural to consider congruences de ned directly at the proposition level. For instance, we may add to the previous system ....

Viry, P. (1995). Rewriting modulo a rewrite system. Technical report TR-20/95, Dipartimento di informatica, Universita di Pisa.

.... modulo, whose main landmarks are the study of associative commutative completion (Peterson and Stickel, 1981) the general study of coherence of an equational theory with respect to a rewrite system (Jouannaud and Kirchner, 1986) its uni ed presentation in (Bachmair, 1987) and its extension in (Viry, 1995; March e, 1994) A di erent and complementary way to deal with equality consists in orienting equality, i.e. determining which object is greater than the other. The integration of rewrite based techniques and ordering in rstorder theorem proving has lead to very powerful results and systems. ....

Viry, P.: 1995, `Rewriting modulo a rewrite system'. Technical report TR-20/95, Dipartimento di informatica, Universita di Pisa.

....language Maude [CELM96] which builds up directly on rewriting logic, for this purpose. Regarding e#ciency, it is desirable to reduce the generally large and complex equational theory to rewriting modulo associativity and commutativity, involving coherence techniques as investigated in [Vir95] see also [Vir96] for an application to the # calculus) This would enable us to employ rewriting tools such as ELAN [BKK 96] for our implementation. It should be noted that the overall structure of a system specified in rewriting logic is that of a transition system. Though, as stated ....

Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-95-20, Dipartimento di Informatica, December 01 1995. URL: ftp://ftp.di.unipi.it/pub/ techreports/TR-95-20.ps.Z.

....having 2 as trigger and 1 : as initial configuration. In a non conditional rewriting system, this is not necessarily true, because rewriting steps can be freely contextualized (and instantiated) This problem is well known, and some ad hoc solutions have been already proposed in the literature [23,28]. Our methodology offers a unifying view for many analogous situations. The basic idea is to stretch tiles into ordinary rewriting rules, preserving the capacity to distinguish between configurations and effects: s a Gamma b s 0 7 ffi F NaN F NaN ( s;b F NaN F NaN 66 a;s 0 ....

P. Viry. Rewriting Modulo a Rewrite System. Technical Report TR-95-20. Department of Computer Science, University of Pisa (1995).

.... NaN 1 id F NaN F NaN 1 In unconditional rewriting systems, this is not necessarily true, because rewriting steps can be freely contextualized (and instantiated) This problem is well known in rewriting logic, and some partial solutions have been already proposed in the literature [25, 36]. However, our methodology seems to offer a unifying view for a wide class of related problems. The basic idea is to stretch tiles into ordinary rewriting cells as pictured below, maintaining the capability to distinguish between configurations and observations. ffi s F NaN F NaN a F ....

....v Gamma u g. 4 Rewriting CCS Processes via Tiles Milner s Calculus for Communicating Systems (CCS) 32] is among the better well known and studied concurrency models. In the recent literature, several ways in which CCS can be conservatively represented in rewriting logic have been proposed [25, 36]. We present the executable implementation defined through the translation into Maude of the tile system for full CCS. This work extends the translation given in [19, 30] for a finitary version of CCS (i.e. without replicator) 4.1 CCS and its Operational Semantics Let Delta (ranged over by ff) ....

P. Viry, Rewriting Modulo a Rewrite System, Technical Report TR-95-20, Department of Computer Science, University of Pisa (1995).

....is by using R modulo E. However, ER and R have to match certain (strong) coherence properties to make this approach also complete, i.e. to ensure that every term derivable via R modulo E can be obtained by rewriting wrt. R and ER modulo AC . A detailed account on this topic can be found in [21]. It should be noted that the overall structure of a system specified in rewriting logic is that of an LTS. Though, as stated above, a single transition may represent concurrent activities in different subcomponents. Hence, we are able to reuse the e#cient implementation of LTSs in Truth even ....

Patrick Viry, "Rewriting modulo a rewrite system," Tech. Rep. TR-95-20, Dipartimento di Informatica, Dec. 01 1995.

....transitions, while equations can be applied in both directions and are interpreted as equalities. Typically, rules specify transitions between states or logical deduction steps, and are applied to objects whose internal structure is specified by the equations. In an oriented rewrite theory [Vir95], the operational and semantic interpretation of a rule need not be be same. This is achieved by partitioning the set of rules into R and ER (equational rules) rules in both sets can be applied only in one direction, but rules of ER have an equational interpretation Preprint submitted to ....

....i.e. specify the internal structure of the objects on which rules of R are applied. This is usually referred to as building in equality . Obviously, when moving around rules and equations, one must make sure that (some notion of) derivability is preserved. Sufficient conditions are given in [Vir95], based on the notion of coherence. Case studies showing how complex rewrite theories can be made executable in the above sense have been presented in [Vir95] Our motivation here is to investigate a case study showing how the notion of an oriented rewrite theory and the associated coherence ....

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Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-20/95, Universit`a di Pisa, 1995. available from http://www.kurims.kyoto-u.ac.jp/~viry.

....rewrite rules. It also provides a semi decision procedure for solving arbitrary equations (unfailing completion [2] In a programming environment, it can be seen as a program transformation technique for constructing programs with desired properties (usually convergence, confluence or coherence [10, 16]) Equational programming, or its generalization rewriting logic, is a very high level specification language with a direct operational counterpart, i.e. specifications are immediately executable as long as the rules and equations verify some of those properties. Ongoing research on completion has ....

P. Viry. Rewriting modulo a rewrite system. Technical Report TR-95-20, Universit `a di Pisa, 1995. Available from http://www.kurims.kyoto-u.ac.jp/~viry/.

.... these calculi can be mapped to rewriting logic in a natural way, because rewriting logic does not make any assumption about the computational model (everything must be explicited) and their interaction can be studied in a simple and unified framework, using for instance the methods proposed in [Vir95] The paper is structured as follows. In a first part, we argue the choice of calculus as our basis for describing I O, then give an executable specification of calculus in terms of rewrite rules and prove its correctness. In a second part, we show how this specification can be used as well ....

....may itself be presented by a smaller equational theory E together with a rewrite system R convergent modulo E, and rewriting by Choice modulo SISE can be replaced by rewriting by Choice and R modulo E. This transformation and sufficient conditions guaranteeing its correctness are presented in [Vir95] This was the idea behind the fact of writing some equations with = and other with = Gamma . From now on, let E be the set of non oriented equations in SISE, namely AC( and AC( j ) and R be the set of equational rules in SISE written with = Gamma or Gamma . including the ....

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P. Viry. Rewriting modulo a rewrite system. Research Report RR95 -20, Dipt. di Informatica, Universit`a di Pisa, 1995. Available from http://www.di.unipi.it/~viry/.

....This is fine on a theoretical point of view, but a straightforward implementation is not realistic, since it would imply matching modulo an arbitrary 1 Supported by an HCM fellowship, EuroFOCS Network Preprint submitted to Elsevier Science 1 August big equational theory. We have shown in [Vir95] that an effective implementation is possible in many common cases, by orienting equations of the equational theory into rewrite rules, thus ending up with two sets of rules : the reduction rules (denoted with Gamma ) describe transitions between states and the equational rules (denoted with = ....

....middle, weak coherence implies preservation of derivations. The coherence properties can be verified by checking critical pairs between rewrite rules, and in the weak and strong cases ensuring that a non linear equational rule can never be applied above a reduction rule. The reader is referred to [Vir95] for more details. In this paper, We take advantage of this result to design and implement an I O model for rewrite interpreters that will be totally explicit in the same framework as any other programs. The model we propose is based on calculus [Mil91] a well studied calculus exhibiting ....

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P. Viry. Rewriting modulo a rewrite system. Technical Report TR-9520, Dipartimento di Informatica, Universit`a di Pisa, 1995.

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Patrick Viry. Rewriting modulo a rewrite system. Technical Report TR-95-20, Dipartimento di Informatica, December 01 1995.

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