## A Parallel Completion Procedure for Term Rewriting Systems (1992)

Venue: | In Conference on Automated Deduction |

Citations: | 12 - 7 self |

### BibTeX

@INPROCEEDINGS{Yelick92aparallel,

author = {Katherine A. Yelick and Stephen J. Garland},

title = {A Parallel Completion Procedure for Term Rewriting Systems},

booktitle = {In Conference on Automated Deduction},

year = {1992},

pages = {109--123},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

We present a parallel completion procedure for term rewriting systems. Despite an extensive literature concerning the well-known sequential Knuth-Bendix completion procedure, little attention has been devoted to designing parallel completion procedures. Because naive parallelizations of sequential procedures lead to over-synchronization and poor performance, we employ a transition-based approach that enables more effective parallelizations. The approach begins with a formulation of the completion procedure as a set of transitions (in the style of Bachmair, Dershowitz, and Hsiang) and proceeds to a highly tuned parallel implementation that runs on a shared memory multiprocessor. The implementation performs well on a number of standard examples.

### Citations

509 |
word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ...ell on a number of standard examples. 1 Introduction We describe a parallel completion procedure for term rewriting systems. A sequential completion procedure was formulated first by Knuth and Bendix =-=[13]-=-. Extensions, modifications, and applications to algebra, theorem proving, and data type induction are described by Buchberger [3] and Dershowitz [5]. Performance is an important factor that limits th... |

455 | The existence of refinement mappings
- Abadi, Lamport
- 1991
(Show Context)
Citation Context ...Note that there is no fairness requirement between any of the other actions. 2 The axioms in Figures 3 and 4 can be viewed as an implementation of Figure 1, and standard refinement mapping techniques =-=[1]-=- applied to prove correctness. In these methods, the set of possible executions of the implementation are shown to be a subset of the executions of the specification, with a refinement mapping used to... |

268 | Orderings for term rewriting systems
- Dershowitz
- 1982
(Show Context)
Citation Context ...undecidable problem. The most common approach to proving that a system is noetherian is to use a reduction ordering on terms, i.e., a monotonic well-founded ordering that is stable under substitution =-=[4]-=-. If ? is a reduction ordering and l ? r for every rule l ! r in R, then R is noetherian. Completion procedures generally employ a fixed reduction ordering and halt with failure if this ordering is no... |

137 |
Complete sets of reductions for some equational theories
- Peterson, Stickel
- 1981
(Show Context)
Citation Context ...st. Some generalizations allow function symbols in R that are not in E, requiring only that $ R be a conservative extension of $ E . Others include completion modulo equations (such as commutativity) =-=[17]-=-. Although these generalizations fall in the class addressed by our approach, we do not consider them in this paper. Solving the completion problem involves generating additional rules, if necessary, ... |

132 | A guide to LP, the Larch prover
- Garland, Guttag
- 1991
(Show Context)
Citation Context ...sizes larger grained parallelism; it allows parallel rewriting during internormalization, but does not intentionally schedule multiple rewrites of the same term. Some sequential completion procedures =-=[11, 14, 15]-=- also divide the state into components, either to achieve better performance or to facilitate reasoning about correctness. The data structures used for our state components differ from these, however,... |

118 |
Orderings for equational proofs
- Bachmair, Dershowitz, et al.
- 1986
(Show Context)
Citation Context ...ion 3 describes the issues that arise in finding good parallel solutions. Section 4 presents transition axioms for a completion procedure using the inference rules of Bachmair, Dershowitz, and Hsiang =-=[2]-=-. Section 5 transforms these axioms into ones suitable for parallel implementation. Section 6 describes the implementation. Section 7 describes its performance. Sections 8 and 9 describe related work ... |

62 | On the sequential nature of unification - Dwork, Kanellakis, et al. - 1984 |

29 | Debugging Larch Shared Language Specifications
- Garland, Guttag, et al.
- 1990
(Show Context)
Citation Context ...rocessors were about 65% of those in Figure 5. Our parallel completion procedure is consistently faster, even when running on one processor, than the completion procedure provided by the Larch Prover =-=[12]-=-. While this comparison is somewhat unfair to the Larch Prover, which has more functionality, the comparison does show that our parallel implementation is fast enough to be of practical use. These per... |

23 | Completion and its applications
- Dershowitz
- 1989
(Show Context)
Citation Context ...rocedure was formulated first by Knuth and Bendix [13]. Extensions, modifications, and applications to algebra, theorem proving, and data type induction are described by Buchberger [3] and Dershowitz =-=[5]-=-. Performance is an important factor that limits the applicability of completion procedures, and of term rewriting systems in general. We show how parallelism can lead to significantly better performa... |

18 | The computation of GrÃ¶bner bases on a shared memory multiprocessor
- Vidal
- 1990
(Show Context)
Citation Context ...uccessfully is Buchberger's algorithm for computing Grobner Bases [3]. Like us, Ponder observed performance problems when using a straightforward parallelization of Buchberger's algorithm [18]. Vidal =-=[22]-=- redesigned the data structures and rearranged the top-level procedure to get superior speed-ups. His procedure does internormalization in a separate sequential phase. It is therefore simpler than our... |

13 |
1985], A procedure for automatically proving the termination of a set of rewrite rules
- Detlefs, Forgaard
(Show Context)
Citation Context ...ir ability to resist failure is one of the qualities by which they are judged. Completion procedures can be made failure resistant by allowing the reduction ordering to be enlarged in restricted ways =-=[8]-=-. Completion procedures can be made unfailing by leaving some equations unordered and restricting the application of rewrite rules [2]. Because these generalizations do not raise interesting new quest... |

12 | Parallel algorithms for term matching - Dwork, Kanellakis - 1988 |

8 |
Evaluation of "Performance Enhancements" in Algebraic Manipulation Systems
- Ponder
- 1988
(Show Context)
Citation Context ...rallelized successfully is Buchberger's algorithm for computing Grobner Bases [3]. Like us, Ponder observed performance problems when using a straightforward parallelization of Buchberger's algorithm =-=[18]-=-. Vidal [22] redesigned the data structures and rearranged the top-level procedure to get superior speed-ups. His procedure does internormalization in a separate sequential phase. It is therefore simp... |

8 | Using abstraction in explicitly parallel programs
- Yelick
- 1991
(Show Context)
Citation Context ...rnormalization and one after critical pairing. Unnecessary synchronization points are artifacts of basing parallel programs on sequential programs. Hence we use a different, transition-based approach =-=[23]-=- that allows parallelism both between and within critical pairing and internormalization. 3.3 Correctness Achieving performance while maintaining correctness requires care in choosing algorithms and d... |

7 | Parallelizing the closure computation in automated deduction
- Slaney, Lusk
- 1990
(Show Context)
Citation Context ...d for our state components differ from these, however, because they are concurrent and have more control information to allow for the additional executions that come from parallelism. Slaney and Lusk =-=[20]-=- describe a parallel closure computation, which divides the problem of completing R = fr 1 ; : : : ; r ng into tasks that compute critical pairs betweensr i and fr 1 ; : : : ; r i g, adding any result... |

6 | History and basic features of the critical pair / completion procedure - Buchberger - 1987 |

6 |
Completion procedures as transition rules + control
- Lescanne
- 1989
(Show Context)
Citation Context ...sizes larger grained parallelism; it allows parallel rewriting during internormalization, but does not intentionally schedule multiple rewrites of the same term. Some sequential completion procedures =-=[11, 14, 15]-=- also divide the state into components, either to achieve better performance or to facilitate reasoning about correctness. The data structures used for our state components differ from these, however,... |

5 | Tight complexity bounds for term matching problems - Verma, Ramakrishnan - 1992 |

4 | Optimal speedups for parallel pattern matching in trees - Ramesh, Ramakrishnan - 1987 |

3 |
an implementation of completion procedures as sets of transitions rules
- Orme
- 1990
(Show Context)
Citation Context ...sizes larger grained parallelism; it allows parallel rewriting during internormalization, but does not intentionally schedule multiple rewrites of the same term. Some sequential completion procedures =-=[11, 14, 15]-=- also divide the state into components, either to achieve better performance or to facilitate reasoning about correctness. The data structures used for our state components differ from these, however,... |

2 |
Private communication
- Dershowitz
- 1990
(Show Context)
Citation Context ... true if r 1 is older than r 2 and the left sides of r 1 and r 2 are renamings of each other. Using the age of rules is mentioned in [2], and the validity of this solution was confirmed by Dershowitz =-=[6]-=-. Any procedure that performs a fair interleaving of these actions solves the completion problem. CP fairness ensures both that every equation appearing in E is State Components E : set[equation] + or... |

2 |
An abstract machine for concurrent term rewriting
- Dershowitz, Lindenstrauss
- 1990
(Show Context)
Citation Context ...as processors are added. This does not guarantee scalability beyond a small number of processors, but neither does it provide evidence against scalability. 8 Related Work Dershowitz and Lindenstrauss =-=[7]-=- describe an abstract concurrent machine for rewriting a term by a set a rules, and they use it to compare various rewriting strategies. The parallelism in their model is very fine grained. They assoc... |

2 |
Doing algebra with REVE
- Martin
- 1986
(Show Context)
Citation Context ...to Ursula Martin. The time required to complete group56 depends dramatically on the order in which critical pairs are computed, because one particular critical pair eliminates most of the other rules =-=[16]-=-. A round-robin scheduler, which executes each transition procedure once before going on to the next, seems to exhibit super-linear speed-ups for this example, gaining a factor of eight with two proce... |