## A Taxonomy of Parallel Strategies for Deduction (1999)

Venue: | Annals of Mathematics and Artificial Intelligence |

Citations: | 14 - 1 self |

### BibTeX

@ARTICLE{Bonacina99ataxonomy,

author = {Maria Paola Bonacina},

title = {A Taxonomy of Parallel Strategies for Deduction},

journal = {Annals of Mathematics and Artificial Intelligence},

year = {1999},

volume = {29},

pages = {223--257}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents a taxonomy of parallel theorem-proving methods based on the control of search (e.g., master-slaves versus peer processes), the granularity of parallelism (e.g., fine, medium and coarse grain) and the nature of the method (e.g., ordering-based versus subgoalreduction) . We analyze how the di#erent approaches to parallelization a#ect the control of search: while fine and medium-grain methods, as well as master-slaves methods, generally do not modify the sequential search plan, parallel-search methods may combine sequential search plans (multi-search) or extend the search plan with the capability of subdividing the search space (distributed search). Precisely because the search plan is modified, the latter methods may produce radically di#erent searches than their sequential base, as exemplified by the first distributed proof of the Robbins theorem generated by the Modified Clause-Di#usion prover Peers-mcd. An overview of the state of the field and directions...

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Citation Context ...mizing changes to the provers themselves, to the extent that provers communicate by writing and reading files. TECHS combines SPASS and DISCOUNT with the model-elimination tableau-based prover SETHEO =-=[76, 82]-=-. SETHEO and SPASS exchange subgoals (from SETHEO to SPASS) and lemmas (from SPASS to SETHEO), while SPASS and DISCOUNT exchange equations. Heuristics to select clauses to send include favoring unit l... |

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Citation Context ...ch modify the sequential notion of search plan, it proposes formal definitions of multi-search plan and distributed-search plan that match the surveyed strategies. Third, it covers many papers (e.g., =-=[1, 31, 30, 42, 40, 52, 55, 57, 65, 72, 74, 83, 91, 93, 95, 99, 102, 104]-=-) that have appeared since [21] was written in 1992, o#ering an up to date survey of the field. Last, the analysis in [21] emphasized the di#culty of parallelizing contraction-based strategies, and wa... |

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Citation Context ...dulation, AC-simplification, subsumption, and deletion by weight, with paramater max-weight equal to 30, 34, and 50, in the three proofs, respectively, for both provers. EQP used basic paramodulation =-=[7]-=-, for FWC # H and R # SWC, and plain paramodulation for SWC # FWC. This is the best choice for EQP. McCune also used basic paramodulation for FWC # H [80], plain paramodulation for SWC # FWC [80], and... |

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Citation Context ...so subgoal-reduction strategies. 2 Sequential theorem-proving strategies We begin by recalling basic concepts and terminology that will be referred to in the following. Surveys on the subject include =-=[12, 47, 53, 85, 11, 49, 84, 59, 24, 6, 34, 19]-=-, where the interested reader may find extensive bibliographies. 1 AND-parallelism may also be considered as parallelism at the term level, since the data accessed in parallel are the literals of a go... |

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Citation Context ...duction strategies [13, 8, 10]: the approach of [10], however, still needs to rely on the #-rule for universally quantified variables; the approach of [13, 8] resembles the instance-based approach of =-=[75]-=-: unification is used to generate instances, but subsumption cannot be applied to remove redundant instances. 7 Parallelism at the term, clause, and search level aim at representing fine-grain, medium... |

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Citation Context ...the whole problem, and the master process is responsible for merging their results. An example of this approach was the Team-Work method, originally conceived for ordering-based equational strategies =-=[39, 4, 5, 46, 44, 42]-=-, and later extended to a framework for applications in distributed artificial intelligence [45]. The master process, called supervisor, assigns to every slave the theoremproving problem, a time perio... |

31 | Hyper Tableaux | The Next Generation
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Citation Context .... Each process executes a strategy, develops a derivation and builds its own set of data. 6 In recent years, there has been some progress in the design of proof-confluent subgoal-reduction strategies =-=[13, 8, 10]-=-: the approach of [10], however, still needs to rely on the #-rule for universally quantified variables; the approach of [13, 8] resembles the instance-based approach of [75]: unification is used to g... |

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Citation Context ...theorem, as shown in the next section. 7 First distributed proof of the Robbins theorem The problem of proving that Robbins algebras are Boolean dates back to 1933, when E. V. Huntington demonstrated =-=[68, 69]-=- that the equation n(n(x) + y) + n(n(x) + n(y)) = x (H) is su#cient to present Boolean algebra, together with associativity and commutativity of +. Herbert Robbins conjectured that the equation n(n(x ... |

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Citation Context ...stributed proof reconstruction: the successful process can reconstruct the proof consulting only its data base. Modified Clause-Di#usion was implemented in Peers-mcd [17], with the Argonne prover EQP =-=[80]-=- as sequential base. Peers-mcd implements the ancestor-graph oriented (AGO) criteria to assign clauses to processes [14, 18]. An issue with the subdivision of the search space is to prevent the search... |

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Citation Context ...ch modify the sequential notion of search plan, it proposes formal definitions of multi-search plan and distributed-search plan that match the surveyed strategies. Third, it covers many papers (e.g., =-=[1, 31, 30, 42, 40, 52, 55, 57, 65, 72, 74, 83, 91, 93, 95, 99, 102, 104]-=-) that have appeared since [21] was written in 1992, o#ering an up to date survey of the field. Last, the analysis in [21] emphasized the di#culty of parallelizing contraction-based strategies, and wa... |

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Citation Context ...ause (another rule in Prolog terminology), this idea leads to trying clauses in parallel, that is OR-parallelism. The first prover based on this principle was PARTHENON [28], soon followed by PARTHEO =-=[88]-=-, in a tableau-based context, and METEOR [3], in the context of PTTP. In these systems, the shared structure is the stack of goal literals (e.g., in PARTHENON and the version of METEOR in shared memor... |

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Citation Context ...is idea leads to trying clauses in parallel, that is OR-parallelism. The first prover based on this principle was PARTHENON [28], soon followed by PARTHEO [88], in a tableau-based context, and METEOR =-=[3]-=-, in the context of PTTP. In these systems, the shared structure is the stack of goal literals (e.g., in PARTHENON and the version of METEOR in shared memory), each goal literal is a task, and the ess... |

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Citation Context ...ch modify the sequential notion of search plan, it proposes formal definitions of multi-search plan and distributed-search plan that match the surveyed strategies. Third, it covers many papers (e.g., =-=[1, 31, 30, 42, 40, 52, 55, 57, 65, 72, 74, 83, 91, 93, 95, 99, 102, 104]-=-) that have appeared since [21] was written in 1992, o#ering an up to date survey of the field. Last, the analysis in [21] emphasized the di#culty of parallelizing contraction-based strategies, and wa... |

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Citation Context ...theorem, as shown in the next section. 7 First distributed proof of the Robbins theorem The problem of proving that Robbins algebras are Boolean dates back to 1933, when E. V. Huntington demonstrated =-=[68, 69]-=- that the equation n(n(x) + y) + n(n(x) + n(y)) = x (H) is su#cient to present Boolean algebra, together with associativity and commutativity of +. Herbert Robbins conjectured that the equation n(n(x ... |

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17 |
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Citation Context ... it and conclude the search, but the strategy forbids such a move, because the generated refutation would not be linear. In analytic tableaux [89] (or equivalently the basic cut-free Gentzen system G =-=[89, 90]-=-), which may be considered as an ancestor of subgoal-reduction strategies, backtracking is not an issue, because after trying a proof attempt (e.g., a set of applications of the #-rule to instantiate ... |

16 | Planning for Distributed Theorem Proving: The Teamwork Approach
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Citation Context ...the whole problem, and the master process is responsible for merging their results. An example of this approach was the Team-Work method, originally conceived for ordering-based equational strategies =-=[39, 4, 5, 46, 44, 42]-=-, and later extended to a framework for applications in distributed artificial intelligence [45]. The master process, called supervisor, assigns to every slave the theoremproving problem, a time perio... |

16 |
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Citation Context ...ollowing two conditions #x#y x + y = x (FWC) #x#y n(x + y) = n(x) (SWC) termed First Winker Condition and Second Winker Condition, respectively, in [80], is su#cient to make a Robbins algebra Boolean =-=[97, 98]-=-, but such lemmas remained beyond the possibilities of automated theorem provers. In 1996 the automated prover EQP of William McCune proved that Robbins algebras are Boolean, as reported in [81]. The ... |

15 |
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Citation Context ...o signify that the clause belongs to a certain subproblem generated by splitting, and the Clause-Di#usion strategies of the next section. 6.2.4 Distributed-search strategies with peer processes DARES =-=[33]-=- can be considered an early approach with distributed-search and peer processes. The main idea was to subdivide the problem among the processes, and let each process ask the others for more clauses, i... |

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Citation Context ... SPTHEO [93], PSETHEO [99] and CPTHEO [57, 100], towards heterogeneous, multi-search systems. CPTHEO launches SETHEO and the resolution-based prover Delta, introduced as a pre-processor for SETHEO in =-=[87]-=-, in parallel. Since clauses generated by Delta are used by SETHEO as lemmas, CPTHEO is also a descendant of HPDS [92], with the master-slaves structure of HPDS replaced by peer processes in CPTHEO. A... |

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Citation Context .... It is the inverse of the problem we are studying in this survey: given a parallel plan #, find a sequential plan # # that simulates it. Previous surveys or collections on parallel deduction include =-=[54, 94, 21]-=-. The main classification criterion of [94] is whether the parallel components cooperate or compete to find a solution. This criterion appears to have been inspired by the classical distinction betwee... |

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Citation Context ...ollowing two conditions #x#y x + y = x (FWC) #x#y n(x + y) = n(x) (SWC) termed First Winker Condition and Second Winker Condition, respectively, in [80], is su#cient to make a Robbins algebra Boolean =-=[97, 98]-=-, but such lemmas remained beyond the possibilities of automated theorem provers. In 1996 the automated prover EQP of William McCune proved that Robbins algebras are Boolean, as reported in [81]. The ... |

13 |
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Citation Context .... Each process executes a strategy, develops a derivation and builds its own set of data. 6 In recent years, there has been some progress in the design of proof-confluent subgoal-reduction strategies =-=[13, 8, 10]-=-: the approach of [10], however, still needs to rely on the #-rule for universally quantified variables; the approach of [13, 8] resembles the instance-based approach of [75]: unification is used to g... |

13 |
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Citation Context ...gies they have been applied to. Reading the dotted lines in Figure 3 from top to bottom and from left to right, we see that parallelism at the term level was applied to parallel term rewriting (e.g., =-=[62, 48, 73, 71, 1, 2]-=-), and to contraction-based strategies (e.g., [29, 31, 72, 86]). AND-parallelism and ORparallelism were applied to subgoal-reduction strategies, in the context of PTTP (e.g., [28, 3] for OR-parallelis... |

12 |
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Citation Context ...tion, or how to keep a distributed data base inter-reduced. Several distributed global contraction schemes were proposed in [15, 22]. The scheme eventually implemented in both Aquarius [23] and Peers =-=[27]-=- was the simplest: every process retains the received inference messages so that its local data base mirrors the global one, and keeps its data base inter-reduced 13 . While the main idea was distribu... |

12 |
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Citation Context ...o, at least in principle, this might be done, because the signature is static, and all new terms are 7 Computing the completion of a set of ground equations consists in computing a congruence closure =-=[60]-=-. 11 formed with the given function symbols, but there are at least two di#culties. First, each equation needs its own variables, leading to the variable renaming problem mentioned above. Second, a ve... |

12 |
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Citation Context ...er parallel term rewriting as parallelism at the clause level; this may be the case for parallel rewriting machines (e.g., [62, 48, 73, 1, 2]) and parallel interpreters of functional languages (e.g., =-=[71]-=-). In the context of theorem proving, where the whole computation is better seen as a search, and each normalization, rather than each rewrite step, may constitute an inference, it is more natural to ... |

11 |
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Citation Context ...blem. This approach emerges from studying clause-level parallelizations of both ordering-based and subgoal-reduction strategies. For the ordering-based strategies, we consider the parallel prover ROO =-=[77, 78]-=- as a paradigmatic example. ROO was designed to parallelize the well-known Otter theorem prover [79]. The core of Otter's control is a main loop, which works with a list of clauses to be selected, cal... |

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Citation Context ...ly to other approaches based on parallelism at the clause level such as PARROT [70], which was a predecessor of ROO for expansion-oriented strategies, and the parallel implementation of completion in =-=[103]-=-, where each inference rule of completion was considered a type of task or transition. A more recent example in the instance-based family is the clause-level parallelization of hyper13 linking in [102... |

10 | On the modeling of search in theorem proving– towards a theory of strategy analysis
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Citation Context ...tood. In ordering-based theorem proving, the subdivision is much more di#cult, because the search space is generally an infinite, highly redundant and dynamic graph (because of pruning by contraction =-=[26]-=-). Slaves need to be aware of the contractions made by other slaves, which means the master may become a communication bottleneck. If the master alone is responsible for contraction, it may become a b... |

10 |
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Citation Context ...ther proof by trying another clause (another rule in Prolog terminology), this idea leads to trying clauses in parallel, that is OR-parallelism. The first prover based on this principle was PARTHENON =-=[28]-=-, soon followed by PARTHEO [88], in a tableau-based context, and METEOR [3], in the context of PTTP. In these systems, the shared structure is the stack of goal literals (e.g., in PARTHENON and the ve... |

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Citation Context ...the whole problem, and the master process is responsible for merging their results. An example of this approach was the Team-Work method, originally conceived for ordering-based equational strategies =-=[39, 4, 5, 46, 44, 42]-=-, and later extended to a framework for applications in distributed artificial intelligence [45]. The master process, called supervisor, assigns to every slave the theoremproving problem, a time perio... |

10 | P-SETHEO: Strategy parallelism in automated theorem proving
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