## Automated theorem proving: mapping logic into AI (1986)

Venue: | Proceedings of the International Symposium on Methodologies for Intelligent Systems |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Loveland86automatedtheorem,

author = {D. W. Loveland},

title = {Automated theorem proving: mapping logic into AI},

booktitle = {Proceedings of the International Symposium on Methodologies for Intelligent Systems},

year = {1986},

pages = {214--229},

publisher = {Press}

}

### OpenURL

### Abstract

ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace "for-mal " by "mechanical " we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not always the methodologies, ATP has been considered to be within the domain of AI. We explore the methodologies of ATP, including the logics that underlie the theorem provers, and discuss some of the mechanisms that utilize these logics. These include term rewriting systems, mathematical induction, inductionless induction and even mixed integer programming. The ATP field, via resolution, has even provided the foundation for an exciting AI and database programming language, PROLOG. We conclude with a new method for extending the PROLOG system to work with non-Horn clause sets within a positive logic format, particularly simple for "slightly non-Horn " clause sets.

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Citation Context ...of new clauses, often with a very positive effect. The work with resolution emphasizes the importance of interaction between the automated provers and the user. Resolution is a machine-oriented logic =-=[21]-=- yet the significant proofs involve, at the stage, definite participation by humans. This is true of almost all provers we discuss. For example, in AURA it includes inference rule selection (globally)... |

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Citation Context ...ller have shown that every typed k-calculus formula is provable in the theory if and only if it has a "tautologous development", a generalized form of Herbrand's theorem upon which resolution depends =-=[35]-=-. However, the unification (due to Huet) used in the procedure arising from this theorem is much more complex than for first-order logic; the algorithm may not terminate. Nevertheless, a concise proof... |

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Citation Context ... (See [5] for one version of natural deduction, somewhat distinct from Gentzen's original formulation.) Resolution, a refutation logic covering the same ground, is also used of course. (See [6], [7], =-=[8]-=- or [9] for extensive coverage here). Although many AIers are familiar with the several styles of logics for the predicate calculus, and one example does little to enlighten the others, we can display... |

56 |
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Citation Context ...rse, the purpose of LT was to simulate production of P.M.'s proofs (rather, the proof style), and thus study the mathematician's reasoning forms. Recent work in the propositional calculus by Jeroslow =-=[12]-=- has dealt more carefully with determining satisfiability (computationally equivalent to validity) using known mixed integer programming techniques. He shows that a method that includes branch-and-bou... |

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Citation Context ...se systems, the underlying logic for PROLOG. (See [39].) There is much activity in the area of logic programming, and extending PROLOG in particular. Two important extensions are into parallel PROLOG =-=[40]-=- and introducing higher-order logic constructs, i.e. )~-conversion [41]. Reintroducing the occurs-check in unification so that PROLOG is sound again but still fast is receiving attention [42], [43]. A... |

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Citation Context ...man-oriented theorem provers, such as those of Bledsoe (see [23]). Suppes has developed interactive theorem provers for teaching logic and set theory courses that use his version of natural deduction =-=[24a]-=-, [24b]. The idea of Suppes' systems is to have the students find the major points of the proof and let the computer automatically fill in the details. Besides first-order logic rules of inferences, s... |

39 |
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(Show Context)
Citation Context ...OG [40] and introducing higher-order logic constructs, i.e. )~-conversion [41]. Reintroducing the occurs-check in unification so that PROLOG is sound again but still fast is receiving attention [42], =-=[43]-=-. A primary reason why PROLOG has been so successful is that so much of what one wishes to do can be expressed in Horn classes. However, some problems of interest demand a larger setting, where severa... |

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Citation Context ... validity) using known mixed integer programming techniques. He shows that a method that includes branch-and-bound techniques relates closely to the theorem-proving method DP of Davis-Putnam found in =-=[18]-=- which is computationally more efficient than the original version of DP that uses a resolution-type rule. An interesting feature of the mathematical programming technique studied by Jeroslow is the u... |

29 |
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(Show Context)
Citation Context ... there has been u movement towards using a graph form of resolution where literals (occasionally, clauses) are graph nodes and arcs connect potentially complementary literals. (See, e.g., [17], [18], =-=[19]-=-, [20].) Besides givingquick access to the complementary literals the organization induced a tighter control on the introduction of new clauses, often with a very positive effect. The work with resolu... |

20 |
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(Show Context)
Citation Context ...applications. A very intriguing discovery regarding equational theories with induction ("inductive domains") was made by David Musser about 1979 [32] and further investigated by others such as Goguen =-=[33]-=-. It can be called "inductionless induction" and, very roughly, allows an induction proof to be replaced by a consistency proof. Applicable to "fully specified" theories, where variable-free terms are... |

17 |
The occur-check problem in prolog
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Citation Context ...l PROLOG [40] and introducing higher-order logic constructs, i.e. )~-conversion [41]. Reintroducing the occurs-check in unification so that PROLOG is sound again but still fast is receiving attention =-=[42]-=-, [43]. A primary reason why PROLOG has been so successful is that so much of what one wishes to do can be expressed in Horn classes. However, some problems of interest demand a larger setting, where ... |

16 |
Empirical explorations of the Logic Theory Machine. A case study in heuristic
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Citation Context ...ssible in substance within the finitely axiomatized first order set theory. ATP Systems. The earliest theorem proving system to attract attention was the Logic Theorist (LT) of Newell, Shaw and Simon =-=[11]-=-, about 1957. LT used the formal system of Principia Mathematics (P.M.), a Hilbert-type system where modus ponens (MP) is a primary inference rule. This style system is now rarely used because MP can ... |

16 |
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(Show Context)
Citation Context ...Andrews TPS for higher order logic, which in internal mode has a resolution connection graph flavor, but where conversion to a natural deduction format is performed for presentation to the user. (See =-=[22]-=-.) It is also involved in the design of human-oriented theorem provers, such as those of Bledsoe (see [23]). Suppes has developed interactive theorem provers for teaching logic and set theory courses ... |

16 |
A simplified problem reduction format
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(Show Context)
Citation Context ...uehner [44], itself rooted in the model elimination procedure. Another interesting extension of the problem reduction method that processes Horn clause logic is a natural deduction system of Plaisted =-=[45]-=-. While these systems do handle any problem having a first-order formulation, there is a price to pay in speed and complexity of machinery. For example, for the PTTP it is necessary to present (or hav... |

15 |
A case study of theorem proving by the Knuth-Bendix method discovering that x 3 =x implies ring commutativity
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(Show Context)
Citation Context ... in ATP. For example, most of the theorem provers that reside within program verification systems use term rewriting systems. The power of the Knuth-Bendix approach was demonstrated when Mark Stickel =-=[26]-=- proved that x a = x implies ring commutativity, a challenge problem given by Bledsoe in 1977 [27]. (This challenge theorem was also proved by Veroff using the AURA system at Argonne.) In Stickel's ap... |

10 |
Logic machine architecture: kernel functions
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Citation Context ...ng logic circuits. See [9]. An interesting recent paper [15] presents one view of ATP and lists some open research problems. Also from the same group is the LMA system (for Logic Machine Architecture =-=[16]-=- which provides tools so others can build their own theorem prover and includes a ready-made prover with the full capability of AURA. Within the last decade or so there has been u movement towards usi... |

10 |
Some Automatic Proofs in Analysis
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(Show Context)
Citation Context ...here conversion to a natural deduction format is performed for presentation to the user. (See [22].) It is also involved in the design of human-oriented theorem provers, such as those of Bledsoe (see =-=[23]-=-). Suppes has developed interactive theorem provers for teaching logic and set theory courses that use his version of natural deduction [24a], [24b]. The idea of Suppes' systems is to have the student... |

6 |
editors) . Automated Theorem Proving: After 25 Years
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(Show Context)
Citation Context ...a strong formal basis, see [1]. Some important systems omitted here are discussed there. Someone wishing to learn more about the ATP field may well want to start by surveying the papers in the volume =-=[2]-=- in which the above paper appears. The Underlying Logics. Since various logics are used in ATP systems, it is important to understand the relationship of these logics. Figure 1 attempts to portray the... |

5 |
Student use of an interactive theorem prover
- Suppes
- 1984
(Show Context)
Citation Context ...ented theorem provers, such as those of Bledsoe (see [23]). Suppes has developed interactive theorem provers for teaching logic and set theory courses that use his version of natural deduction [24a], =-=[24b]-=-. The idea of Suppes' systems is to have the students find the major points of the proof and let the computer automatically fill in the details. Besides first-order logic rules of inferences, signific... |

4 |
Automated theorem proving. Vieweg Verlag
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(Show Context)
Citation Context ...5] for one version of natural deduction, somewhat distinct from Gentzen's original formulation.) Resolution, a refutation logic covering the same ground, is also used of course. (See [6], [7], [8] or =-=[9]-=- for extensive coverage here). Although many AIers are familiar with the several styles of logics for the predicate calculus, and one example does little to enlighten the others, we can display some f... |

2 |
Proof checking, theorem proving and program verification
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- 1983
(Show Context)
Citation Context ...orem and similar classical theorems, the prover has been used to show the Turing completeness of the PURE LISP programming language and the correctness of the RSA public key encryption algorithm. See =-=[31]-=- for a short overview of this important project and for references regarding these and other applications. A very intriguing discovery regarding equational theories with induction ("inductive domains"... |

1 |
Automated theorem proving: a quarter century review
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(Show Context)
Citation Context ...upport from Army Research Office Grant No. DAAG29-84K-0073. 214sFor another survey of ATP, focusing on ATP systems emphasizing human simulation as well as systems stressing a strong formal basis, see =-=[1]-=-. Some important systems omitted here are discussed there. Someone wishing to learn more about the ATP field may well want to start by surveying the papers in the volume [2] in which the above paper a... |

1 |
Computation-oriented reduction of predicate calculus to propositional logic
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(Show Context)
Citation Context ...g technique studied by Jeroslow is the use of computationally fast linear programming to catch many inconsistent formulas. Jeroslow is attempting to extend these techniques to first-order logic. (See =-=[13]-=-.) Most mechanized theorem provers deal with the predicate calculus. About fifteen years ago one could safely say that most predicate calculus theorem provers were resolution based, but it may not be ... |

1 |
Automated reasoning: basic research problems. Report ANL/MCSTM-67
- Wos
- 1986
(Show Context)
Citation Context ...ng with equality, has assisted these investigators in discovering previously open mathematical problems, plus practical problems such as designing logic circuits. See [9]. An interesting recent paper =-=[15]-=- presents one view of ATP and lists some open research problems. Also from the same group is the LMA system (for Logic Machine Architecture [16] which provides tools so others can build their own theo... |

1 |
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- 1975
(Show Context)
Citation Context ...decade or so there has been u movement towards using a graph form of resolution where literals (occasionally, clauses) are graph nodes and arcs connect potentially complementary literals. (See, e.g., =-=[17]-=-, [18], [19], [20].) Besides givingquick access to the complementary literals the organization induced a tighter control on the introduction of new clauses, often with a very positive effect. The work... |

1 |
The Markgraf Karl refutation procedure (Full
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- 1981
(Show Context)
Citation Context ... has been u movement towards using a graph form of resolution where literals (occasionally, clauses) are graph nodes and arcs connect potentially complementary literals. (See, e.g., [17], [18], [19], =-=[20]-=-.) Besides givingquick access to the complementary literals the organization induced a tighter control on the introduction of new clauses, often with a very positive effect. The work with resolution e... |

1 |
Resolution takes more than polynomial time
- Haken
- 1983
(Show Context)
Citation Context ...ems very natural it turned out to be very difficult to show that if a broader class of proofs were permitted, corresponding to standard resolution, that one still had problems with huge proofs. Haken =-=[21a]-=- has now shown exponential sized proofs exist for standard resolution. It is still open whether introducing new variables, in the flavor of our discussion of controlling input size, can limit proof si... |

1 |
Non-resolution theorem proving. Artif. Intell
- Bledsoe
- 1977
(Show Context)
Citation Context ...use term rewriting systems. The power of the Knuth-Bendix approach was demonstrated when Mark Stickel [26] proved that x a = x implies ring commutativity, a challenge problem given by Bledsoe in 1977 =-=[27]-=-. (This challenge theorem was also proved by Veroff using the AURA system at Argonne.) In Stickel's approach associative/commutative unification and other augmentation were used to speed the convergen... |

1 |
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(Show Context)
Citation Context ...been used in applications from intelligent databases to 223sexpert systems. It is available for microcomputers as well as large mainframes, and is at the core of the Fifth Generation Computer Project =-=[38]-=-. As previously mentioned, the technology of PROLOG is that of a linear input resolution theorem prover. However, its application as a general context-free language parser, as a generalized database i... |

1 |
Linear resolution with collection function
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(Show Context)
Citation Context ... only because it compares itself to modern implementations of PROLOG but because the first implementation of a PROLOG prototype by Colmerauer et al. was based on SL-resolution of Kowalski and Kuehner =-=[44]-=-, itself rooted in the model elimination procedure. Another interesting extension of the problem reduction method that processes Horn clause logic is a natural deduction system of Plaisted [45]. While... |

1 | A Prolog for non-Horn programs. Forthcoming report - Loveland |