## A Prolog Technology Theorem Prover: Implementation by an Extended Prolog Compiler (1987)

Venue: | Journal of Automated Reasoning |

Citations: | 100 - 2 self |

### BibTeX

@ARTICLE{Stickel87aprolog,

author = {Mark E. Stickel},

title = {A Prolog Technology Theorem Prover: Implementation by an Extended Prolog Compiler},

journal = {Journal of Automated Reasoning},

year = {1987},

volume = {4},

pages = {353--380}

}

### Years of Citing Articles

### OpenURL

### Abstract

A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full first-order predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, the model-elimination reduction rule that is added to Prolog inferences to make the inference system complete, and depth-first iterative-deepening search instead of unbounded depth-first search to make the search strategy complete. A Prolog technology theorem prover has been implemented by an extended Prolog-to-LISP compiler that supports these additional features. It is capable of proving theorems in the full first-order predicate calculus at a rate of thousands of inferences per second. 1 This is a revised and expanded version of a paper presented at the 8th International Conference on Automated Deduction, Oxford, England, July 1986, and is to appear in Journal of Automated Reasoning. This research was supported by the Defense Advanced Research Projects Agency under Co...

### Citations

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Citation Context ...ed depth-first search strategy. For theorem proving, Prolog's unbounded depth-first search strategy must be replaced by some complete search strategy, such as breadth-first search or the A* algorithm =-=[26]-=-. However, the arbitrary choice of a complete search strategy may result in losing much of the efficiency of Prolog implementations. In particular, adopting breadth-first search or the A* algorithm wo... |

488 |
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Citation Context ...ing Prolog technology in theorem proving is being developed at Argonne National Laboratory [5]. The Argonne researchers plan to use Prolog technology (an implementation of the Warren abstract machine =-=[37]-=-) to reimplement their powerful interactive theorem-proving system LMA+ITP [21, 22, 23]. The work on PTTP has tried to extend Prolog to general-purpose theorem proving 29 in such a way as to get the h... |

412 | Depth-first iterative-deepening: An optimal admissible tree search
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Citation Context ... obvious solution is to run with increasing depth bounds; first one tries to find a proof with depth 1, then depth 2, and so on, until a proof is found. This is called depth-first iterative deepening =-=[13]-=- (we also called it staged or consecutively bounded depth11 first search [34, 35]). The effect is similar to breadth-first search except that results from earlier levels are recomputed rather than sto... |

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Citation Context ...er of inferences is guaranteed to decrease (in any search space in which the combined clauses are used). At the end of the table, we also provide results for the problems that appear in Chang and Lee =-=[8]-=-, pp. 298--305. All nine problems were solved in a total of a little more than one second. 10 Related Work There are several other efforts to design and construct Prolog-like systems that perform soun... |

224 |
Automated Theorem Proving: A Logical Basis
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(Show Context)
Citation Context ...at a good approach to building a PTTP is to employ a complete inference system that is also an input procedure. Probably the simplest is [an affirmative form of ] the model elimination (ME) procedure =-=[16, 17, 20]-=-. 3 The ME procedure requires only the addition of the following inference operation to Prolog to constitute a complete inference system for the first-order predicate calculus: If the current goal mat... |

111 |
Theorem proving via general matings
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Citation Context ... caching provides makes his system competitive with PTTP in total solution time for some problems. TUM's PROTHEO (Prolog-like theorem prover) is a Prolog-like extension based on the connection method =-=[4, 1]-=-. It uses depth-bounded search, lemmas to eliminate redundant computations, and has some bottom-up as well as top-down reasoning capabilities. PTTP is also somewhat related to other efforts (too numer... |

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Citation Context ...rminating unification. If the values of X and Y in the first example are unified later, the unification would not terminate unless a unification algorithm capable of handling infinite terms were used =-=[10]-=-. Although applying the occurs check in logic programming can be quite costly, it is less likely to be too expensive in theorem proving, since the huge terms sometimes generated in logic programming a... |

39 |
A Prolog technology theorem prover
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(Show Context)
Citation Context ...find a proof with depth 1, then depth 2, and so on, until a proof is found. This is called depth-first iterative deepening [13] (we also called it staged or consecutively bounded depth11 first search =-=[34, 35]-=-). The effect is similar to breadth-first search except that results from earlier levels are recomputed rather than stored. Thus, when searching is done to depth n, level n \Gamma 1 results are being ... |

32 |
Refutation Graphs
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31 |
The sharing of structure in theorem-proving programs
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Citation Context ...by applying the unifying substitution to the parent clauses. This is far more expensive in both time and space than Prolog inference. The second less frequently used method involves structure sharing =-=[6]-=-, in which a resolvent is represented by the parents plus the unifying substitution. Whenever the resolvent must be examined (e.g., for printing or resolution with another clause), it is traversed, wi... |

31 |
Non-Horn clause logic programming without contrapositives
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Citation Context ...eated in one of the terms, i.e., when the term is linear, no occurs check will be necessary for the entire unification operation. There are alternative methods to assuring sound unification. Plaisted =-=[28]-=- has suggested an elegant method, which we currently use in another version of PTTP, of transforming clauses to isolate parts that may require unification with the occurs check. Repeated occurrences o... |

27 |
A Parallel Implementation of Iterative-Deepening A
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Citation Context ...20 or below, or when search of a single level simply takes too long) then methods that partition the search space for a single level must be employed to take advantage of all the available processors =-=[29]-=-. This is akin to the OR-parallel execution of standard Prolog programs. 6 Existing Refinements The changes made in unification, the inference system, and the search strategy are all sufficient to cre... |

26 |
A simplified format for the model elimination procedure
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(Show Context)
Citation Context ...at a good approach to building a PTTP is to employ a complete inference system that is also an input procedure. Probably the simplest is [an affirmative form of ] the model elimination (ME) procedure =-=[16, 17, 20]-=-. 3 The ME procedure requires only the addition of the following inference operation to Prolog to constitute a complete inference system for the first-order predicate calculus: If the current goal mat... |

19 |
An implementation of the model elimination proof procedure
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Citation Context ...culty. We used the Wilson and Minker study [39] as a source of problems. 7 They took Problems 1--9 from Reboh et al. [31], problems 10--19 from Michie et al. [25], problems 20--25 from Fleisig et al. =-=[12]-=-, problems 26--58 from Wos [41], and problems 59--86 from Lawrence and Starkey [15]. This problem set has also been used to test the Markgraf Karl Refutation Procedure connection-graph resolution theo... |

16 |
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Citation Context ...to deferred heads. When no goals or deferred heads remain, the query has been solved. Plaisted has implemented a new theorem prover using a new method based on his simplified problem reduction format =-=[27]-=-. His system also does not require contrapositives. Plaisted emphasizes the unnatural backward-chaining searches that can result from the use of contrapositives. For example, solving and(a,b) by and(X... |

15 |
An analysis of consecutively bounded depth-first search with applications in automated deduction
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Citation Context ...find a proof with depth 1, then depth 2, and so on, until a proof is found. This is called depth-first iterative deepening [13] (we also called it staged or consecutively bounded depth11 first search =-=[34, 35]-=-). The effect is similar to breadth-first search except that results from earlier levels are recomputed rather than stored. Thus, when searching is done to depth n, level n \Gamma 1 results are being ... |

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Citation Context ...(); undo-unify; if unify(args,args2) then q(. . . ,cont); undo-unify; if unify(args,args3) then r(. . . ,s(. . . ,cont)); undo-unify end This approach to Prolog compilation is also described by Cohen =-=[9]-=-. We discuss below Prolog's deficiencies for general-purpose theorem proving and examine the manner in which they are dealt with by the current PTTP implementation. 3 Sound Unification The first obsta... |

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Resolution, Refinements, and Search Strategies: A Comparative Study
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Citation Context ... operation, and indefinite answers. 9 Performance It is never an easy task to find a large number of problems with suitable accessibility, variety, and difficulty. We used the Wilson and Minker study =-=[39]-=- as a source of problems. 7 They took Problems 1--9 from Reboh et al. [31], problems 10--19 from Michie et al. [25], problems 20--25 from Fleisig et al. [12], problems 26--58 from Wos [41], and proble... |

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Citation Context .... For example, a Prolog technology theorem prover could play a role like that of the Terminator [2] in the Markgraf Karl Refutation Procedure or of an implementation of the linked inference principle =-=[40]-=- in the Argonne theorem provers. It is our hope that, with PTTP, we have helped establish a new standard of performance for theorem-proving programs working on problems comparable in difficulty to our... |

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6 |
The hole in goal trees: some guidance from resolution theory. Advance P_R_E_ Q_ SS!!!SeS
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Citation Context ...at a good approach to building a PTTP is to employ a complete inference system that is also an input procedure. Probably the simplest is [an affirmative form of ] the model elimination (ME) procedure =-=[16, 17, 20]-=-. 3 The ME procedure requires only the addition of the following inference operation to Prolog to constitute a complete inference system for the first-order predicate calculus: If the current goal mat... |

5 |
Automated Theorem Proving. Friedr. Vieweg
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Citation Context ... caching provides makes his system competitive with PTTP in total solution time for some problems. TUM's PROTHEO (Prolog-like theorem prover) is a Prolog-like extension based on the connection method =-=[4, 1]-=-. It uses depth-bounded search, lemmas to eliminate redundant computations, and has some bottom-up as well as top-down reasoning capabilities. PTTP is also somewhat related to other efforts (too numer... |

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Paths to high-performance automated theorem proving
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Citation Context ...lete deduction, as well as in deemphasis of its use as a programming language. Another valuable method for using Prolog technology in theorem proving is being developed at Argonne National Laboratory =-=[5]-=-. The Argonne researchers plan to use Prolog technology (an implementation of the Warren abstract machine [37]) to reimplement their powerful interactive theorem-proving system LMA+ITP [21, 22, 23]. T... |

5 |
The Markgraf Karl Refutation Procedure
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Citation Context ...om Wos [41], and problems 59--86 from Lawrence and Starkey [15]. This problem set has also been used to test the Markgraf Karl Refutation Procedure connection-graph resolution theorem-proving program =-=[7, 30]-=-. 7 The technical-report version of their article includes a listing of the problems. Their tabulated results show problems named EX5-T1, EX5-T2, LS76, and LS86, but the listing contains only a single... |

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Citation Context ...plementation in Prolog rather than as a Prolog extension makes it much slower than PTTP. Other inference systems that are complete extensions of Prolog for non-Horn clauses have been proposed by Eder =-=[11]-=-, Loveland [18, 19], Plaisted [28] and a group at Technische Universitat Munchen (TUM) [3]. In Eder's system, the non-Horn clause PQR S is represented by the clause r; s !- p, q, and only the literals... |

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Experimental tests of resolution based theorem-proving strategies
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Citation Context ...k Problems 1--9 from Reboh et al. [31], problems 10--19 from Michie et al. [25], problems 20--25 from Fleisig et al. [12], problems 26--58 from Wos [41], and problems 59--86 from Lawrence and Starkey =-=[15]-=-. This problem set has also been used to test the Markgraf Karl Refutation Procedure connection-graph resolution theorem-proving program [7, 30]. 7 The technical-report version of their article includ... |

2 |
Study of automatic theorem-proving programs
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Citation Context ... to find a large number of problems with suitable accessibility, variety, and difficulty. We used the Wilson and Minker study [39] as a source of problems. 7 They took Problems 1--9 from Reboh et al. =-=[31]-=-, problems 10--19 from Michie et al. [25], problems 20--25 from Fleisig et al. [12], problems 26--58 from Wos [41], and problems 59--86 from Lawrence and Starkey [15]. This problem set has also been u... |

2 |
An experiment in programming with full first-order logic
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Citation Context ...one second. 10 Related Work There are several other efforts to design and construct Prolog-like systems that perform sound and complete deduction for the full-first order predicate calculus. F-Prolog =-=[36]-=- is very similar in its interfaces and basic algorithms to PTTP. It includes unification with the occurs check, the reduction operation (solving a goal in the context of a set of assumptions 27 that i... |

1 |
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Citation Context ...her inference systems that are complete extensions of Prolog for non-Horn clauses have been proposed by Eder [11], Loveland [18, 19], Plaisted [28] and a group at Technische Universitat Munchen (TUM) =-=[3]-=-. In Eder's system, the non-Horn clause PQR S is represented by the clause r; s !- p, q, and only the literals r and s are matched in resolution operations. Non-Horn proofs involving case analysis are... |

1 |
MKRP: a performance test by working mathematicians
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Citation Context ...om Wos [41], and problems 59--86 from Lawrence and Starkey [15]. This problem set has also been used to test the Markgraf Karl Refutation Procedure connection-graph resolution theorem-proving program =-=[7, 30]-=-. 7 The technical-report version of their article includes a listing of the problems. Their tabulated results show problems named EX5-T1, EX5-T2, LS76, and LS86, but the listing contains only a single... |

1 | Automated theorem proving: Mapping logic into AI
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- 1986
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Citation Context ... Prolog rather than as a Prolog extension makes it much slower than PTTP. Other inference systems that are complete extensions of Prolog for non-Horn clauses have been proposed by Eder [11], Loveland =-=[18, 19]-=-, Plaisted [28] and a group at Technische Universitat Munchen (TUM) [3]. In Eder's system, the non-Horn clause PQR S is represented by the clause r; s !- p, q, and only the literals r and s are matche... |

1 |
Nonclausal Logic Programming
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Citation Context ...t of the !- connective. 5 Wilkins [38] developed the first nonclausal version of the model elimination procedure. Nonclausal formulas are also a vital feature of the TABLOG logic programming language =-=[24]-=-. If the query is nonclausal, its negation must be included among the assertions even when only definite answers are sought. For example, the proof of P (a)sP (b) from P (a)sP (b) requires the clauses... |

1 |
Chess 4.5---the Northwestern University University chess program
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Citation Context ... second time, level n \Gamma 2 results for the third time, and results at level 1 for the nth time. Despite a long history of depth-first iterative-deepening search, notably in chess-playing programs =-=[33]-=-, the strategy remained unanalyzed and unadvocated in theorem-proving and problem-solving applications until quite recently. Because of the exponential growth in the size of the search space as the de... |

1 |
QUEST: a non-clausal theorem proving system
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- 1973
(Show Context)
Citation Context ...e provided nonclausal assertions that are transformed automatically into proper inputs for PTTP, i.e., into contrapositives with a single literal as the first argument of the !- connective. 5 Wilkins =-=[38]-=- developed the first nonclausal version of the model elimination procedure. Nonclausal formulas are also a vital feature of the TABLOG logic programming language [24]. If the query is nonclausal, its ... |