D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993. Frederic Blanqui, Claude Kirchner, Colin Riba A Proofs of Section 3.2 We begin by proving the well-foundedness of #.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we present primary origins for the higher order case in Section 3, and extensions to secondary origins in Section 4. In Sections 5 and 6 we mention related work and draw some conclusions. 2. Higher Order Term Rewriting For the definition of Higher Order Term Rewriting Systems (HRSs) we follow [26, 22, 24]. The main difference from the first order case is that terms in HRSs are constructed according to the simply typed calculus [7] 2.1 The Simply Typed Calculus The set of type symbols T consists of elementary type symbols from T 0 and of functional type symbols (ff fi) where ff; fi 2 T . We ....

....representations for origins containing more structure than the (simple) sets of paths could be fruitful. Nipkow s definition of higher order TRSs requires the rewrite rules to satisfy several syntactic constraints [22] We have discussed origins using the more liberal setting of Wolfram [26]. Obviously, the same origins can be established for Nipkow s HRSs. The nicer matching behavior of Nipkow s HRSs will probably have a favorable effect on the origins. The mapping 19 between Nipkow s HRSs and Klop s combinatory reduction systems (CRSs) 18] as described in [24] can be the basis for ....

D.A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....that the unification of a linear pattern with an arbitrary second order term that shares no variables with it is decidable. He also proves the decidability of a few extensions of this problem. There are still skeletons that do not satisfy the definition of a relaxed higherorder pattern. Wolfram [13] presents a terminating procedure for higher order matching. Whether or not this procedure is complete is not yet known. If it is complete then general higher order matching is decidable. The results in this paper say little, if anything, about the decidability of higher order matching when the ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993. This article was processed using the L A T E X macro package with LLNCS style

....either by enriching rst order rewriting with higher order capabilities or by adding to calculus algebraic features allowing one, in particular, to deal with equality in an ecient way. In the rst case, we nd the works on CRS [KvOvR93] XRS [Pag98] and other higher order rewriting systems [Wol93, NP98] in the second case the works on combination of calculus with term rewriting [Oka89, BT88, GBT89, JO97] to mention only a few. Our previous works on the control of term rewriting [KKV95, Vit94, BKKR01] led us to introduce the calculus. Indeed we realized that the tool that is needed ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....KORSO [10] make use of it, while Ergo [15] uses higher order uni cation. However, there do also exist many successful transformation systems that avoid it completely, for example Kids [16] The two standard objections to its use relate to eciency (even second order matching is known to be NP hard [17]) and the need to impose a speci c typing discipline. Our algorithm operates on untyped terms, which eliminates the second issue; the rst is signi cant but thus far we have been able to obtain adequate performance for our needs with an implementation that still has plenty of room for ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....we have not seen such a uniform presentation before. Moreover we do think the resulting presentation is elegant and the analysis required and performed is useful. It would be an interesting exercise to give a similar analysis of higher order equational logics [1] comprising higher order rewriting [8] and their termination models [7] Another direction would be to generalize the other well known result of [3] the characterisation of equational varieties as varieties which are closed under building subalgebras, homomorphic images, and direct products. Acknowledgments Thanks are due to ETL, ....

D.A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. CUP, 1993.

....we have not seen such a uniform presentation before. Moreover we do think the resulting presentation is elegant and the analysis required and performed is useful. It would be an interesting exercise to give a similar analysis of higher order equational logics [1] comprising higher order rewriting [8] and their termination models [7] Another direction would be to generalize the other well known result of [3] the characterisation of equational varieties as varieties which are closed under building subalgebras, homomorphic images, and direct products. Acknowledgments Thanks are due to ETL, ....

D.A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. CUP, 1993.

....area is currently very active. Klop introduced in [15] the Combinatory Reduction Systems (CRS) an early reference for Khasidashvili s Expression Reduction Systems (ERS ) is [14] Nipkow introduces Higher Order Rewrite Systems (HRS) in [16] Wolfram defines Higher Order Term Rewrite Systems (see [23]) van Oostrom and van Raamsdonk introduce Higher Order term Rewriting Systems in [21] We choose in this work to use ERS because their syntax and semantics are simple and natural (they allow for example to write fi reduction in calculus as usual while CRS do not) and the correspondence between ....

David Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....de Bruijn metaterms, which are the syntactical objects used to express any general higher order rewrite system formulated in a de Bruijn context. Many higher order rewrite systems (HORS) exist and work in the area is currently very active: CRS [14] ERS [12] CERS [13] HRS [15] the systems in [22] and [20] We choose in this work to use ERS because their syntax and semantics are simple and natural (they allow for example to write fi reduction in calculus as usual while CRS do not) and the correspondence between ERS and HRS has already been established [21] We shall begin with (a ....

D. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....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, ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993. 28

....[4, 7] is false. The conjecture was that, in a model of the simply typed calculus with only finitely many elements at each type, definability (by a closed term of the calculus) is decidable. This conjecture had been shown to imply many things, for example, Statman [7] see also Wolfram s book [8]) has shown it implies the decidability of pure higher order pattern matching (a problem that remains open at the time of writing) and is equivalent to higher order pattern matching with ffi functions. The proof of undecidability given here uses encodings of semi Thue systems as definability ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993. 13

....offered by higher order matching, there also exist very successful transformation systems that do not incorporate its use, for example Kids [23] and APTS [19] There are two significant objections to the use of higher order matching. First, even second order matching is known to be NP hard [7, 25], so a truly efficient implementation is out of the question. Second, higher order matching algorithms are restrictive, in particular in the typing discipline that they require. In this paper, we have demonstrated how that second objection can be eliminated, by giving an algorithm that operates on ....

....with the MAG system [10] there seem to be a handful of techniques to deal with failed matches (for instance raising a rule by introducing explicit abstractions) so we feel that the disadvantage is not too serious. There is a wealth of related work on higher order matching and unification [5, 7, 11, 12, 13, 16, 18, 24, 25], to name just a few. One important concept identified in some of these works (in particular [16, 18] is that of a restricted notion of higher order pattern. To wit, a restricted pattern is a normal term where every occurrence of a free function variable is applied to a list of distinct local ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....(Section 1.2) which do have variable types. All systems in Barendregt s cube have variable types. 2 Statman s definability conjecture would have implied the decidability of the higher order matching, but R. Loader has proved [Loa93] the undecidability of the lambda definability. D. Wolfram [Wol93] has suggested a higher order matching algorithm and conjectured its finite termination. The lower complexity bound for the higher order matching problem in is the tower 2 2 Delta Delta Delta 2 of height c Delta n= log(n) where n is the length of equation and c 0 [Vor97] Note that the ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science Series. Cambridge

....offered by higher order matching, there also exist very successful transformation systems that do not incorporate its use, for example Kids [23] and APTS [19] There are two significant objections to the use of higher order matching. First, even second order matching is known to be NP hard [7, 25], so a truly efficient implementation is out of the question. Second, higher order matching algorithms are restrictive, in particular in the typing discipline that they require. In this paper, we have demonstrated how that second objection can be eliminated, by giving an algorithm that operates on ....

....with the MAG system [9] there seem to be a handful of techniques to deal with failed matches (for instance raising a rule by introducing explicit abstractions) so we feel that the disadvantage is not too serious. There is a wealth of related work on higher order matching and unification [5, 7, 11 13, 16, 18, 24, 25], to name just a few. One important concept identified in some of these works (in particular [16, 18] is that of a restricted notion of higher order pattern. To wit, a restricted pattern is a normal term where every occurrence of a free function variable is applied to a list of distinct local ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....we observe that the matching power (or complexity of matching, depending on one s point of view) of HRSs is already present in CRSs, making a natural encoding of the former into the latter possible. This is due to the pattern condition of HRSs. For HRSs not satisfying the pattern condition (cf. [Wol93]) this is no longer the case and an encoding seems to be not straightforward anymore (even if we would lift some of the restrictions on left hand sides of CRS rules) The next proposition shows that although CRSs and HRSs have the same matching power, HRSs have more rewrite power , i.e. they can ....

....also Rosen [Ros73] ffl CS = Contraction Scheme. Introduced by Aczel [Acz78] ffl (a) reductions were introduced by Hindley [Hin78] ffl CRS = Combinatory Reduction System. Introduced by Klop [Klo80] ffl HOTRS = Higher Order Term Rewriting System. Introduced by Wolfram in his PhD thesis, see [Wol93]. ffl ERS = Expression Reduction System. Introduced by Khasidashvili [Kha90] ffl (I)IN = Intuitionistic) Interaction Net. Introduced by Lafont [Laf90] ffl HRS = Higher order Rewrite System. They were introduced by Nipkow [Nipa] 5. Discussion 22 ffl (D)IS = Discrete) Interaction System. ....

D.A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....These systems essentially are TRSs with bound variables. In [12, 13] the formalism of Higher order Rewrite Systems (HRS) is described, which is very similar to CRSs in essence, but rather different in presentation. A precise comparison is given in [17] A more general setting is given in [18]. Quite other approaches can be found in [4, 9] Two important issues concerning rewrite systems are termination and confluence. For results about local confluence of HRSs and confluence of orthogonal HRSs the reader is referred to [12] and [13] respectively. In [11, ch. II.3] the confluence of ....

....possible rewrite steps or all possible redexes. This semantical approach is more convenient than a syntactical technique. The aim of this paper is to generalise this semantical characterisation of termination for TRSs to one for HRSs. The definition of Higher order Rewriting we use, is close to [18], so it is an extension of [12, 13] The main result is that such a generalisation is possible. The interpretation of terms can be extended to the interpretation of higher order terms. The orderings and the notion of strictness can also be generalised. The techniques to achieve this are similar to ....

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D.A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge tracts in theoretical computer science. Cambridge University Press, Cambridge, 1993.

.... comprises all of: first order term rewriting systems (TRSs [DJ, Klo] and lambda calculi [Bar84, Bar] as well as Combinatory Reduction Systems (CRSs [Klo80, KOR93] Interaction Systems (ISs [AL94, AL96] Higher order Rewriting Systems (HRSs [Nipa, MN94] and Higher Order Term Rewriting Systems ([Wol93]) This was shown in [ORb] which contains an extensive overview of HORSs. For a comparison of higher order formats, see [ORa] and for extensions to conditional orthogonal higher order term rewriting, see [Tak, KO95] Net and Graph Rewriting Similarly to the previous paragraph, we sketch here by ....

D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....present primary origins for the higher order case in Section 3, and extensions to secondary origins in Section 4. In Sections 5 and 6 we mention some related work and draw some conclusions. 2 Higher Order Term Rewriting For the definition of Higher Order Term Rewriting Systems (HRSs) we follow [Wol93, Nip91, OR93] The main difference with the first order case that terms in HRSs are constructed according to the simply typed calculus [Chu40] 2.1 The Simply Typed Calculus The set of type symbols T consists of elementary type symbols from T 0 , and, if ff; fi 2 T , of functional type ....

....representations for origins containing more structure than the (simple) sets of paths could be fruitful. Nipkow s definition of higher order TRSs requires the rewrite rules to satisfy several syntactic constraints [Nip91] We have discussed origins using the more liberal setting of Wolfram [Wol93] Obviously, the same origins can be established for Nipkow s HRSs. The nicer matching behavior of Nipkow s HRSs will probably have a favorable effect on the origins. The mapping between Nipkow s HRSs and Klop s combinatory reduction systems (CRSs) Klo80] as described in [OR93] can be the basis ....

D.A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993.

....j where the x are numerical variables, and the C j are numerical constants. Terms are constructed in the proof to represent numerical constants, and disagreement pairs are constructed to represent the equations of any instance of Hilbert s Tenth Problem. The encoding used below is a reformulation [24] in simply typed calculus of Goldfarb s [10] If C denotes a natural number c 0, it is represented by the term cA, where c is the curried typed Church numeral (ty: t( Delta Delta Delta t z c (y) Delta Delta Delta)x:G(A; x) and (A) x) y) G) and ....

D.A. Wolfram. The Clausal Theory of Types. Volume 21 of Cambridge Tracts in Theoretical Computer Science, (Cambridge, 1993).

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D. A. Wolfram. The Clausal Theory of Types, volume 21 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1993. Frederic Blanqui, Claude Kirchner, Colin Riba A Proofs of Section 3.2 We begin by proving the well-foundedness of #.

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