Invariants, Patterns and Weights for Ordering Terms

We prove that any simplification order over arbitrary terms is an extension of an order by weight, by considering a related monadic term algebra called the spine. We show that any total ground-stable simplification order on the spine lifts to an order on the full term algebra. Conversely, under certain restrictions, a simplification ordering on the term algebra defines a weight function on the spine, which in turn can be lifted to a weight order on the original ground terms which contains the original order. We investigate the Knuth?Bendix and polynomial orders in this light. We provide a general framework for ordering terms by counting embedded patterns, which gives rise to many new orderings. We examine the recursive path order in this context.

[1]  Ursula Martin,et al.  The Classification of Polynomial Orderings on Monadic Terms , 2001, Applicable Algebra in Engineering, Communication and Computing.

[2]  H. Brown,et al.  Computational Problems in Abstract Algebra , 1971 .

[3]  Dieter Hofbauer.  Termination Proofs by Multiset Path Orderings Imply Primitive Recursive Derivation Lengths , 1992, Theor. Comput. Sci..

[4]  Ursula Martin,et al.  The order types of termination orderings on monadic terms, strings and multisets , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[5]  Pierre Lescanne,et al.  Termination of Rewriting Systems by Polynomial Interpretations and Its Implementation , 1987, Sci. Comput. Program..

[6]  Joachim Steinbach,et al.  Simplification Orderings: Histrory of Results , 1995, Fundam. Informaticae.

[7]  Ursula Martin,et al.  A Note on Division Orderings on Strings , 1990, Inf. Process. Lett..

[8]  Teo Mora,et al.  The Gröbner Fan of an Ideal , 1988, J. Symb. Comput..

[9]  Ursula Martin,et al.  A Geometrical Approach to Multiset Orderings , 1989, Theor. Comput. Sci..

[10]  Joachim Steinbach.  Generating Polynomial Orderings , 1994, Inf. Process. Lett..

[11]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[12]  Ursula Martin,et al.  The order types of termination orderings on monadic terms, strings and monadic terms, strings and multisets , 1993 .

[13]  Ursula Martin,et al.  On the Diversity of Orderings on Strings , 1995, Fundam. Informaticae.

[14]  Nachum Dershowitz.  Orderings for Term-Rewriting Systems , 1979, FOCS.

[15]  Hans Zantema,et al.  Termination of Term Rewriting: Interpretation and Type Elimination , 1994, J. Symb. Comput..

[16]  Péter Pröhle,et al.  Scott's Conjecture is True, Position Sensitive Weights , 1997, RTA.

[17]  Andreas Weiermann,et al.  Term Rewriting Theory for the Primitive Recursive Functions , 1997, Ann. Pure Appl. Log..

[18]  Jean H. Gallier,et al.  What's So Special About Kruskal's Theorem and the Ordinal Gamma0? A Survey of Some Results in Proof Theory , 1991, Ann. Pure Appl. Log..

[19]  Nachum Dershowitz,et al.  Orderings for term-rewriting systems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[20]  Ursula Martin,et al.  Linear Interpretations by Counting Patterns , 1993, RTA.

[21]  Pierre Lescanne,et al.  Decomposition Ordering as a Tool to Prove the Termination of Rewriting Systems , 1981, IJCAI.

[22]  Dieter Hofbauer.  Termination Proofs by Multiset Path Orderings Imply Primitive Recursive Derivation Lengths , 1990, ALP.

[23]  Jean Gallier,et al.  Ann. Pure Appl. Logic , 1997 .

[24]  B. Gramlich.  Simple Termination Is Diicult , 1993 .

[25]  Elizabeth A. Scott,et al.  Weights for Total Division Orderings on Strings , 1994, Theor. Comput. Sci..

[26]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[27]  Ursula Martin.  Theorem Proving with Group Presentations: Examples and Questions , 1996, CADE.