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### Termination for direct sums of left-linear complete term rewriting systems

Y. Toyama, J. W. Klop, H. P. Barendregt
1995 Journal of the ACM
A term rewriting system is called complete if it is confluent and terminating.  ...  We prove that completeness of TRSs is a "modular" property (meaning that it stays preserved under direct sums), provided the constituent TRSs are left-linear.  ...  Termination for the Direct Sum In this section, we will prove the main result, that is, the termination property for the direct sum R b © P w of left-linear and complete R b, R w.  ...

### Termination for the direct sum of left-linear term rewriting systems [chapter]

Yoshihito Toyama, Jan Willem Klop, Hendrik Pieter Barendregt
1989 Lecture Notes in Computer Science
It is shown that two term rewriting systems both are left-linear and complete if and only if the direct sum of these systems is so.  ...  A b s t r a c t The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols.  ...  Two term rewriting systems R0 and R1 are left-linear and complete iff the direct sum R0 (~ R1 is so.  ...

### Page 5759 of Mathematical Reviews Vol. , Issue 97I [page]

1997 Mathematical Reviews
(NL-NIJM; Nijmegen) Termination for direct sums of left-linear complete term rewriting systems. (English summary) J. Assoc. Comput. Mach. 42 (1995), no. 6, 1275-1304.  ...  So completeness of TRSs is a modular property that is, it is preserved under direct sums of TRSs—provided the underlying TRSs are left-linear. Peter R. J.  ...

### Modular termination of r-consistent and left-linear term rewriting systems

Manfred Schmidt-Schauß, Massimo Marchiori, Sven Eric Panitz
1995 Theoretical Computer Science
A modular property of term rewriting systems is one that holds;for the direct sum of two disjoint term rewriting systems iff it holds for every involved term rewriting system.  ...  Since confluence implies r-consistency, this constitutes a generalisation of the theorem of Toyama, Klop, and Barendregt on the modularity of termination for confluent and left-linear term rewriting systems  ...  Acknowledgements We thank Bernhard Gramlich, Jan Willem Klop and Enno Ohlebusch for valuable comments on a draft of this paper.  ...

### Algebraic semantics and complexity of term rewriting systems [chapter]

Tohru Naoi, Yasuyoshi Inagaki
1989 Lecture Notes in Computer Science
Termination for direct sums of left-linear complete term rewrite systems. On the Church-Rosser property for the direct sum of term rewriting systems. J. ACM, 34(1):128-143, 1987. V. van Oostrom.  ...  Decidability of termination for semi-constructor TRSs, left-linear shallow TRSs and related systems.  ...

### Page 2565 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews
In this paper, the author shows that UN~ is modular for left- linear term rewriting systems.  ...  The author shows that (i) decidability of termination is not a modular property for finite left-linear monadic TRSs, (ii) decid- ability of confluence and decidability of completeness of (left- or right  ...

### Modular term rewriting systems and the termination

Masahito Kurihara, Ikuo Kaji
1990 Information Processing Letters
Toyama for discussing the definitions  ...  The direct sum of two term rewriting systems (TRSs) [2, 8] R, and R, is confluent if R, and R, are confluent [lo] , but the direct sum is not necessarily terminating even if R, and R, are terminating  ...  [ll] ; that is, termination is not a "modular" [6] property for general term rewriting systems.  ...

### On modularity in infinitary term rewriting

Jakob Grue Simonsen
2006 Information and Computation
modularity of top-termination and normalization for left-linear systems.  ...  We study modular properties in strongly convergent infinitary term rewriting.  ...  Acknowledgments The author gratefully acknowledges the insightful comments of the referees and their contribution to the presentation of the paper.  ...

### Computing with rewrite systems

Nachum Dershowitz
1985 Information and Control
Rewrite systems may be used for straightforward computation by simplifying terms.  ...  Term-rewriting systems, that is, sets of directed equations, provide a paradigm of computation with particularly simple syntax and semantics.  ...  discussions, and the referees for their valuable suggestions.  ...

### Modular proofs for completeness of hierarchical term rewriting systems

M.R.K.Krishna Rao
1995 Theoretical Computer Science
A property P of term rewriting systems is modular if the following holds: two rewriting systems ~o and ~?  ...  Therefore, techniques employed in proving the modularity results for direct sums and constructor sharing systems are not applicable for hierarchical combinations.  ...  The author is grateful to the referees for many suggestions in improving the presentation very much.  ...

### Primitive Rewriting [chapter]

Nachum Dershowitz
2005 Lecture Notes in Computer Science
It comes as no surprise that rewrite systems have the same computational power as the other basic models. 2 Moreover, rewrite systems may be restricted in various ways, including left-linearity, orthogonality  ...  Another natural candidate for proofs regarding term rewriting is Recursion Theory, a direction we promote in this contribution.  ...  Acknowledgement I thank the referees for their critical reading and constructive suggestions.  ...

### Unique Normal Forms and Confluence of Rewrite Systems: Persistence

Rakesh M. Verma
1995 International Joint Conference on Artificial Intelligence
We also prove the confluence of the union (function symbols can be shared) of a nonlin ear system with a left-linear system under fairly general conditions.  ...  In this paper, we study two funda mental concepts, uniqueness of normal forms and confluence, for nonlinear systems in the ab sence of termination.  ...  Lemma 20 If the Ir-disjoint union R of a left-linear system R 1 and any system R 2 is nonoverlapping and semi-terminating, then R is persistent.  ...

### Page 3206 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews
Standard or extended (i.e. modulo a theory) completion of term rewriting systems may abort or succeed for a given input.  ...  Basically, termination of a rewrite system R is proved by means of termination of a simplified rewrite system S and an auxiliary rewrite system U connecting R and S.  ...

### Modularity of simple termination of term rewriting systems with shared constructors

Masahito Kurihara, Azuma Ohuchi
1992 Theoretical Computer Science
Ohuchi, Modularity of simple termination of term rewriting systems with shared constructors, Theoretical Computer Science 103 (1992) 273-282.  ...  A term rewriting system is simply terminating if there exists a simplification ordering showing its termination.  ...  Toyama and anonymous referees for their helpful comments. This work is partially supported by the Grants from Ministry of Education, Science and Culture of Japan, #02249104 and #01460253.  ...

### Page 5013 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews
of r-consistent and left-linear term rewriting systems.  ...  Summary: “A modular property of term rewriting systems is one that holds for the direct sum of two disjoint term rewriting systems iff it holds for every involved term rewriting system.  ...
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