V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In A. Nerode, editor, Logical Foundations of Computer Science, pages 379--392. Springer LNCS 813, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the right hand side of the Beta rule depends on the substitution calculus W , we should denote this rule as BetaW , but we prefer to omit the index W to make easier the notation. 2 The same remark done for the Beta rule applies here. 3 Another notion of substitution calculus can be found in [38]. 1. W is strongly normalizing 4 . 2. W is confluent. 3. W normal forms of terms are pure terms. 4. W (a b) W (a) W (b) and W (#a) #W (a) 5. For any substitution s in W , 1[lif t W (s) W 1 and for any m # 1, m 1[lif t W (s) W m[s] #] 6. For any term k in W , 1[cons W ....

V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In LICS, 1994 This article was processed using the L A T E X macro package with LLNCS style

....= Q=u) t N ) redexes in W= Q=P ) are created by P=Q. t Q e W u 2 o u Q=u o 0 3 W u = Q=u) u=Q W 2 s N Q=P = Q=u) N s 0 3 W N= Q=u) One can verify that the calculus [L ev78] orthogonal TRSs [HuL e91] orthogonal Higher Order Rewriting Systems (HORSs) [OR94, Oos94], and hence other orthogonal higher order rewrite systems, such as CRSs [Klo80] ERSs [Kha92] and HRSs [Nip93] and Orthogonal Term Graph Rewriting Systems [KKSV93] form DRSs; the latter are equivalent to Maranget s orthogonal DAG (Directed Acyclic Graphs) rewriting systems, defined via labelled ....

Van Oostrom V., van Raamsdonk F. Weak orthogonality implies confluence: the higher-order case. In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, St. Petersburg, Nerode A., Matiyasevich Yu. V. eds., Springer LNCS, vol. 813, pp. 379-392, 1994.

....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, ....

....term. The Strong Church Rosser (confluence) property is established for OERSs in [24, 27] the Finite Developments Theorem [5, 29] is proved first, from which strong confluence follows by a standard argument. Strong confluence for other higher order rewriting formats are obtained, among others, in [29, 48, 44, 37]. 5 or strongly equivalent, or permutation equivalent 7 Theorem 4 (Finite Developments) All complete developments of a set of redexes in a term t, in an OERS, end at the same term s, and the residuals in s of any redex in t along any complete development are the same. Theorem 5 (Strong ....

V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: The higher-order case. In A. Narode and Yu. V. Matiyasevich, editors, 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, volume 813 of LNCS, pages 379--392. Springer-Verlag, 1994.

....for reductions with substitutions (also referred to as higher order rewriting) as in the calculus [Bar84] and its extensions. Restricted rewriting systems with substitutions were first studied in Pkhakadze [Pkh77] and Aczel [Acz78] Several interesting formalisms have been introduced later [Kha90, Nip93, Wol93, OR94]. We refer to Klop et al. KOR93] and van Oostrom [Oos94] for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs) defined in Khasidashvili [Kha90, Kha92] ERSs are called CRSs in [Kha92] the present formulation is simpler. Definition 2.1 Let Sigma be ....

Van Oostrom V., van Raamsdonk F. Weak orthogonality implies confluence: the higherorder case. In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, `Logic at St. Petersburg', LFCS'94, Springer LNCS, vol. 813, Narode A., Matiyasevich Yu. V. eds. St. Petersburg, 1994. p. 379-392.

....Reduction Systems (CRSs) 38] to provide a uniform framework for reductions with substitutions like that in the calculus [4] and its extensions. Restricted rewriting systems with substitutions were first studied by Pkhakadze [52] and Aczel [1] Several interesting formalisms were introduced later [24,47,68,51]. See [57] for a survey. We will refer to such systems using a collective name higher order rewriting . Here we use Expression Reduction Systems (ERSs) 24,26] 1 ERSs are based on the syntax of Pkhakadze [52] and were introduced by the present author independently from other formats of ....

.... (implying conservation) in OTRSs and the calculus, respectively, and similar perpetuality criteria for fully extended orthogonal CCERSs were obtained by Khasidashvili et al. [36,37] where it is also demonstrated that in formats of higher order rewriting where function variables can be bound [47,68,51], even non erasing redexes may not be perpetual. A strictly stronger criterion of perpetuality of fi redexes is obtained by Honsell and Lenisa [18] by using semantical methods. Bohm and Intrigila [7] showed that any step is perpetual in the ffi k 34 calculus, which is obtained from the I ....

V. van Oostrom and F. van Raamsdonk, Weak orthogonality implies confluence: the higher-order case, in: A. Nerode and Yu.V. Matiyasevich, eds., Proc. Third International Conference on Logical Foundations of Computer Science, LFCS'94, Lecture Notes in Computer Science, Vol. 813 (Springer, Berlin, 1994) 379-392. 39

.... valid for orthogonal CRSs as well (see [Raa96] for a detailed comparison of various forms of higher order rewriting) However, using an example due to van Oostrom [Oos97] we will demonstrate that our results cannot be extended to higherorder rewriting systems where function variables can be bound [Wol93, Nip93, OR94], as they can exhibit pretty strange behaviour not characteristic of the calculi. In order to prove our perpetuality criteria, we first generalize the constricting or zoom in perpetual strategy, independently discovered by Plaisted [Pla93] Srensen [Sr95] Gramlich [Gra96] and Melli es [Mel96] ....

.... from [Kha94c] CERSs extend Expression Reduction Systems [Kha92] a formalism of higher order (rather, second order) rewriting close to Combinatory Reduction Systems [Klo80] We refer to [Raa96] for an extensive survey of the relationship among various formats of higher order rewriting (such as [Klo80, Kha92, Wol93, Nip93, OR94, Oos94]) and refer to [Klo92] for a survey of results on conditional TRSs. Terms in CERSs are built from the alphabet like in the first order case, except some symbols may have binding power. For example, a fi redex in the calculus appears as Ap(x t; s) where Ap is a function symbol of arity 2, and ....

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Van Oostrom V., van Raamsdonk F. Weak orthogonality implies confluence: the higher-order case. Proc. of LFCS'94, LNCS 813, pp. 379-392, 1994.

....confluent. 8 Related Work As already indicated in the introduction, HRS are closely related to CRS [11, 13] Confluence of CRS has been investigated for orthogonal systems. This has lead to a notion of 25 higher order rewriting system (HORS) which is parameterized by a substitution calculus [25, 23]. HORS generalize both CRS and HRS. In the case of HRS the substitution calculus is the simply typed calculus. It has been shown that all weakly orthogonal HORS are confluent [25, 23] Although no notion of critical pair has been defined for HORS, weak orthogonality for HRS can be translated as ....

....lead to a notion of 25 higher order rewriting system (HORS) which is parameterized by a substitution calculus [25, 23] HORS generalize both CRS and HRS. In the case of HRS the substitution calculus is the simply typed calculus. It has been shown that all weakly orthogonal HORS are confluent [25, 23]. Although no notion of critical pair has been defined for HORS, weak orthogonality for HRS can be translated as follows: Definition 8.1 An HRS R is called weakly orthogonal iff it is left linear and all of its critical pairs are of the form hu; ui. By the above result it follows that all weakly ....

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In A. Nerode, editor, Logical Foundations of Computer Science, volume 813 of Lect. Notes in Comp. Sci., pages 379--392. Springer-Verlag, 1994.

....some redex in M 0 . Then R is in fact a finite reduction; it is called a development of M 0 . If in the final term of R no descendant of a redex in M 0 is left, R is a complete development. There are many proofs of FD. We refer to Barendregt [Bar84] Krivine [Kri93] van Oostrom van Raamsdonk [OR94], de Vrijer [Vri85] So, a development arises by forbidding contraction of redexes created as in I III. It turns out that types II and III of redex creation are a somewhat more innocent way of creation. If we forbid only contraction of type I redexes in a reduction, we have by definition a ....

....the fij calculus are trivial; in other words, that fij calculus is weakly orthogonal. From this it follows by a recent general theorem for higher order rewrite systems (HRSs or CRSs) of which fij calculus is a typical example, that the fij calculus is confluent (cf. van Oostrom van Raamsdonk [OR94]) By contrast the CR proof of Curry Feys is ad hoc and uses postponement of j reductions. A simpler proof using commutation of fi and j reductions was given in [Klo80] However, standard proofs of confluence for fi that rely on keeping track of residuals do not carry over easily to fij. This ....

V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: The higher-order case. In A. Nerode and Yu.V. Matiyasevich, editors, Proceedings of the third International Symposium on Logical Foundations of Computer Science, volume 813 of Lecture Notes in Computer Science, pages 379--392. Springer, 1994.

.... 10 of paralleloutermost reduction may also be used as essential property (this is exactly what is needed in the construction) And it seems plausible ( Mid97b] that hyper normalization of parallel outermost reduction for weakly orthogonal TRSs can be proved by using ideas of [SR93] [OR94], Mid97a] Yet, this remains to be checked in detail. Concerning normalization of outermost fair reduction we are not aware of any further positive results (besides the one mentioned above) For instance, it seems to be open whether outermost fair reduction is also normalizing for weakly ....

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: The higher-order case. In A. Nerode and Y.V. Matiyasevich, editors, Proc. 3rd Int. Symp. on Logical Foundations of Computer Science, volume 813 of Lecture Notes in Computer Science, pages 379--392, St. Petersburg, Russia, July 1994. Springer-Verlag.

.... claim that all our results are valid for orthogonal fully extended CRSs as well (see [61] for a detailed comparison of various forms of higher order rewriting) We will demonstrate, however, that our results cannot be extended to higher order rewriting systems where function variables can be bound [71, 52, 57], since already the Conservation Theorem fails for these systems. The paper is organized as follows: In Section 2, we introduce CCERSs and show how several rewrite and transition systems can be encoded as CCERSs. In Section 3, we prove some standard results for orthogonal CCERSs. In Section 4, we ....

....case there are several conceptually similar, but notationally often quite different, proposals. The first general higher order format was introduced long ago by Klop [37] under the name of Combinatory Reduction Systems (CRSs) Since then, several other interesting formalisms have been introduced [27, 71, 52, 45, 57]. Restricted rewriting systems PERPETUALITY AND UNIFORM NORMALIZATION 5 with substitutions were first studied by Pkhakadze [58] and Aczel [2] See van Raamsdonk [61] for a detailed comparison of various forms of higher order rewriting. It is often of interest to have the possibility of putting ....

[Article contains additional citation context not shown here]

Van Oostrom, V. and van Raamsdonk, F. (1994), Weak orthogonality implies confluence: the higher-order case, in [51, pp. 379--392].

....in our last proof where the uncurrying is type directed: it would be nice if the underlying formalism provided support for system transformations involving type changes. An even more drastic approach would be to use a formalism where the substitution calculus is a plug in such as HORS [26]: this could provide for more advanced notions of static reduction, for example including arithmetic as needed by the first order power example. One worry remains, however: the typed systems (including HRSs) work on fij long normal forms. It is not clear to which extent this interferes with the ....

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. Technical Report CS-R9501, CWI, 1995.

....This requires neither left linearity nor termination of the system. 1 Introduction Several frameworks of rewriting systems for higher order expressions have been proposed [Klo80, Nip91, MN94, LS93, KvO95] Van Oostrom and van Raamdonk proposed a framework of Higher Order Rewriting Systems (HORSs) [vO94, vOvR94, vR96], capable of unifying the existing theory of rewriting, e.g. Combinatory Reduction Systems (CRSs) Klo80] another variation of) Higher order Rewriting Systems (HRSs) by Nipkow [Nip91] and Term Rewriting Systems (TRSs) They also presented a sufficient condition for the Church Rosser property ....

....some abstract conditions are first presented on rewrite rules and substitution calculi, and then properties such as the Church Rosser property are derived for HORSs satisfying the conditions. In particular, a non overlapping left linear pattern HORS is shown to satisfy the conditions (see also [vOvR94], vR96] It is well known that a non overlapping left linear TRS has the Church Rosser property. Without left linearity and with a slight modification of the non overlap requirement, some results [Che81, dV90, TO94, MO95] have concluded the unique normal form property of TRSs. The unique normal ....

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V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: The higher-order case. In 3rd Int. sympo. on Logical Foundations of Computer Science, pp. 379--392, 1994.

.... Expression Reduction Systems Klop introduced Combinatory Reduction Systems (CRSs) in [Klo80] to provide a uniform framework for reductions with substitutions (also referred to as higher order rewriting) as in the calculus [Bar84] Several interesting formalisms have been introduced later [Kha92, Nip93, OR94]. We refer to van Raamsdonk [Raa96] for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs) defined in [Kha92] under the name of CRSs) the present formulation follows [GlKh94] and is simpler. Definition 2.1 Let Sigma be an alphabet, comprising ....

....Q are co initial reductions in an OERS, then (P P 0 ) Q L P=Q P 0 = Q=P ) and P= Q Q 0 ) L (P=Q) Q 0 . The following strong Church Rosser (confluence) property is proved for ERSs in [Kha92] the same result for other higher order rewriting formats are obtained, among others, in [Klo80, Nip93, KOR93, OR94, Oos94, KvO95, Raa96]. Theorem 2.1 (Strong Church Rosser) For any co initial reductions P and Q in an OERS, P Q=P L Q P=Q. 3 Relative Normalization In this section, we review some notions and results concerning relative normalization from [GlKh94] Definition 3.1 Let S be a set of terms in an OERS R. We call a ....

Van Oostrom V., van Raamsdonk F. Weak orthogonality implies confluence: the higher-order case. In: Proc. of LFCS'94, Springer LNCS, vol. 813, Narode A., Matiyasevich Yu. V. eds. St. Petersburg, 1994. p. 379-392.

.... 6 of parallel outermost reduction may also be used as essential property (this is exactly what is needed in the construction) And it seems plausible ( Mid97b] that hyper normalization of parallel outermost reduction for weakly orthogonal TRSs can be proved by using ideas of [SR93] [OR94], Mid97a] Yet, this remains to be checked in detail. 4 Discussion Right normalization of semi complete TRSs is a transformation which is incompatible with the usual (position selection) reduction strategies like (leftmost, parallel) innermost or outermost, namely in the following sense. ....

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: The higher-order case. In A. Nerode and Y.V. Matiyasevich, editors, Proc. 3rd Int. Symp. on Logical Foundations of Computer Science, volume 813 of Lecture Notes in Computer Science, pages 379--392, St. Petersburg, Russia, July 1994. Springer-Verlag.

....18] Confluence of CRSs has been investigated for orthogonal systems. HRSs are the same as Wolfram s higher order term rewriting systems. This abundance of slightly different frameworks has lead to a notion of higher order rewriting system (HORS) which is parameterized by a substitution calculus [31, 29] and generalizes all of the aforementioned frameworks. In the case of HRSs PRSs the substitution calculus is the simply typed calculus. It has been shown that all weakly orthogonal HORSs are confluent [31, 29] Although the notion of weak orthogonality for HORSs is defined directly rather than in ....

....higher order rewriting system (HORS) which is parameterized by a substitution calculus [31, 29] and generalizes all of the aforementioned frameworks. In the case of HRSs PRSs the substitution calculus is the simply typed calculus. It has been shown that all weakly orthogonal HORSs are confluent [31, 29]. Although the notion of weak orthogonality for HORSs is defined directly rather than in terms of critical pairs, it can be translated as follows: Definition 7.1 A PRS R is called weakly orthogonal iff it is left linear and all of its critical pairs are of the form hu; ui. A direct proof of ....

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In A. Nerode, editor, Logical Foundations of Computer Science, volume 813 of Lect. Notes in Comp. Sci., pages 379--392. Springer-Verlag, 1994.

....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 ....

....the dependent steps are both inner and outer. 2. The rewrite rules for parallel or : por(x; T ) T por(T; x) T are development closed since steps only depend on each other in the case of por(T; T ) The result then is T for either rewrite rule (so in fact the system is weakly orthogonal [ORb]) and since the empty step is a special case of a development step, the system is confluent. Remark. Throughout the text there are many references to [Oos94] This is only meant for easy reference. Many of the results can be found at many other places. 2 Rewriting We fix the no(ta)tions for the ....

[Article contains additional citation context not shown here]

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. pp. 379--392 in LFCS'94, LNCS 813.

....rewriting rule 2 Theta x x x to the term 2 Theta 3 gives rise to the duplication of the term 3 in the result 3 3. How this duplication is actually performed (for example, using sharing) depends on the designer s implementation of substitution. This decomposition has been shown useful in [OR94, Oos94] in the case of first order term rewriting systems (TRSs, DJ90, Klo92] and higher order term rewriting systems (Klop s combinatory reduction systems (CRSs) Klo80] Nipkow s higher order rewrite systems (HRSs) Nip93] We will indicate how, using this decomposition, results can be proved ....

....case that all the non determinism of the rewriting process is due to parallelism. That is, any two distinct steps that can be performed take place in disjoint parallel parts of the structure. In the case of term rewriting such orthogonal systems are known to be confluent ( Klo80, Nip93] In [OR94, Oos94] it was shown that confluence of orthogonal term rewriting systems can be viewed as depending on the underlying calculus for substitutions. In particular, it was shown that the Finite Developments theorem ( Klo80, Bar84] can be reduced to normalisation of the substitution calculus. The Finite ....

[Article contains additional citation context not shown here]

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In [NM94, pp. 379--392], 1994.

.... that all our results are valid for orthogonal fully extended CRSs as well (see [Raa96] for a detailed comparison of various forms of higher order rewriting) We will demonstrate, however, that our results cannot be extended to higher order rewriting systems where function variables can be bound [Wol93, Nip93, OR94], since already the Conservation Theorem fails for these systems. The paper is organized as follows: In Section 2, we introduce CCERSs and show how several rewrite and transition systems can be encoded as CCERSs. In Section 3, we prove some standard results for orthogonal CCERSs. In Section 4, we ....

....there are several conceptually similar, but notationally often quite different, proposals. The first general higher order format was introduced long ago by Klop [Klo80] under the name of Combinatory Reduction Systems (CRSs) Since then, several other interesting formalisms have been introduced [Kha92, Wol93, Nip93, Lor93, OR94]. Restricted rewriting systems with substitutions were first studied by Pkhakadze [Pkh77] and Aczel [Acz78] See van Raamsdonk [Raa96] for a detailed comparison of various forms of higher order rewriting. It is often of interest to have the possibility of putting restrictions on the generation of ....

[Article contains additional citation context not shown here]

Van Oostrom V. and van Raamsdonk F., Weak orthogonality implies confluence: the higher-order case. In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS, vol. 813, 1994. pp. 379--392.

.... and we claim that all our results are valid for orthogonal CRSs as well (see [Raa96] for a detailed comparison of various forms of higher order rewriting) We will demonstrate, however, that our results cannot be extended to higher order rewriting systems where function variables can be bound [Wol93, Nip93, OR94], since already the Conservation Theorem fails for these systems. The paper is organized as follows: In Section 2 we introduce CCERSs and show how several rewrite and transition systems can be encoded as CCERSs. In Section 3 we prove the standard results for orthogonal CCERSs. In Section 4 we ....

....there are several conceptually similar, but notationally often quite different, proposals. The first general higher order format was introduced long ago by Klop [Klo80] under the name of Combinatory Reduction Systems (CRSs) Since then, several other interesting formalisms have been introduced [Kha92, Wol93, Nip93, Lor93, OR94]. Restricted rewriting systems with substitutions were first studied by Pkhakadze [Pkh77] and Aczel [Acz78] See van Raamsdonk [Raa96] for a detailed comparison of various forms of higher order rewriting. It is often of interest to have the possibility of putting restrictions on the generation of ....

[Article contains additional citation context not shown here]

Van Oostrom V. and van Raamsdonk F., Weak orthogonality implies confluence: the higher-order case. In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS, vol. 813, 1994. pp. 379-392.

....x x to the term 2 Theta 3 gives rise to the duplication of the term 3 in the result 3 3. How this duplication is actually performed (for example, using sharing) depends on the designer s implementation of substitution. This decomposition has been shown useful for the study of orthogonality [ORb, Oos94a] in the case of first order term rewriting systems (TRSs, DJ, Klo] and higher order term rewriting systems (e.g. Klop s combinatory reduction systems (CRSs, Klo80] Nipkow s higher order rewrite systems (HRSs, Nipa] Using this decomposition, results can be proved uniformly for structured ....

....all the non determinism of the rewriting process is due to rewrite steps which are simultaneous . That is, any two distinct steps that can be performed take place in disjoint (independent) parts of the structure. In the case of term 1 rewriting such orthogonal systems are known to be confluent ([Klo80, Nipb, ORb]) In [ORb, Oos94a] it was shown that confluence of orthogonal term rewriting systems can be viewed as depending on the underlying calculus for substitutions. In particular, it was shown that the Finite Developments theorem ( Klo80, Bar84] can be reduced to (strong) normalisation of the ....

[Article contains additional citation context not shown here]

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In [NM94, pp. 379--392]. Full version available as Technical Report, CS-R9501, CWI, January 1995.

....above, this is a little surprising since the class of OHRSs properly contains the class of ISs. The main difference between our approach and the one in [AL94] is that we essentially put to use the decomposition of HRSs into a rule component and a substitution (simply typed calculus) component ([Oos94, OR]) In a HRS rewrite step, the rule component takes care of the destruction and creation of function symbols and the substitution component takes care of the erasure and duplication of parts of the term. Example 1. Consider the term rewriting rule f(X; Y ) g(Y; Y ) The step f(h; i) g(i; i) ....

Vincent van Oostrom and Femke van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In [NM94, pp. 379--392]. Full version available as Technical Report, CS-R9501, CWI, January 1995.

....binding) there exist several conceptually similar, but notationaly often quite different proposals. Long ago, the first general higher order format was introduced by Klop [Klo80] under the name of Combinatory Reduction Systems. Since then, several other interesting formalisms have been introduced [Kha92,Nip93,Wol93,OR94,Tak93]. This paper is based on the notion of Expression Reduction System introduced by the first author [Kha92] but our results also apply to the other higher order formats. Often it is of interest to have the possibility to put restrictions on the generation of either the terms or the rewrite relation ....

....FD proof is based on Nederpelt Klop s method [Ned73,Klo80] for reducing strong normalization to weak normalization. It is similar in structure to, but simpler than Klop s original confluence proof for orthogonal CRSs [Klo80] and we think not more difficult than other existing confluence proofs [vR93,Nip93,OR94,Mel93]. The idea of orthogonality is that contraction of a redex does not destroy others (in whatever way) but rather leaves a number of their residuals. A prerequisite for the definition of residual is the notion of descendant allowing to trace subterms during a reduction. Whereas this is simple in ....

Van Oostrom V., van Raamsdonk F. Weak orthogonality implies confluence: the higher-order case. In: Proc. of the 3 rd International Symposium on Logical Foundations of Computer Science, LFCS'94, LNCS, vol. 813, Nerode A., Matiyasevich Yu. V., eds. St. Petersburg, 1994, p. 379-392.

No context found.

V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In A. Nerode, editor, Logical Foundations of Computer Science, pages 379--392. Springer LNCS 813, 1994.

No context found.

V. v. Oostrom and F. v. Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In [78, pp. 379--392], 1994.

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