## Consolution as a Framework for Comparing Calculi (1994)

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Venue: | JOURNAL OF SYMBOLIC COMPUTATION |

Citations: | 14 - 10 self |

### BibTeX

@ARTICLE{Baumgartner94consolutionas,

author = {Peter Baumgartner and Ulrich Furbach},

title = {Consolution as a Framework for Comparing Calculi},

journal = {JOURNAL OF SYMBOLIC COMPUTATION},

year = {1994},

volume = {16}

}

### Years of Citing Articles

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### Abstract

In this paper, stepwise and nearly stepwise simulation results for a number of first-order proof calculi are presented and an overview is given that illustrates the relations between these calculi. For this purpose, we modify the consolution calculus in such a way that it can be instantiated to resolution, tableaux model elimination, a connection method and Loveland's model elimination.

### Citations

276 |
Symbolic Logic and Mechanical Theorem Proving
- Chang
- 1973
(Show Context)
Citation Context ...mple. y Here we are not compatible with (Eder, 1991). What he calls "derivation" is called "refutation" by us; this was done to be compatible with subsequent calculi and to standar=-=d text books (e.g. (Chang and Lee, 1973-=-)). 6 P. Baumgartner and U. Furbach P Q ? ? ? ? ? An extension step can be performed by selecting one branch of the tree, say the one leading to P , and an input clause that contains a literal which c... |

224 |
Automated Theorem Proving: A Logical Basis
- Loveland
- 1978
(Show Context)
Citation Context ... refined consolution method, which can then be shown to simulate stepwise the model elimination calculus. Model elimination was originally introduced by Loveland (cf. (Loveland, 1968; Loveland, 1969; =-=Loveland, 1978-=-)). It can be seen as a restricted form of linear resolution (see (Loveland, 1978) and section 7). However, model elimination and its linear proof format was invented before linear resolution, which c... |

149 |
Automated Theorem Proving
- Bibel
- 1987
(Show Context)
Citation Context ...eaux model elimination, a connection method and Loveland's model elimination. 1. Introduction In (Eder, 1991) the consolution method is introduced as a bridge between the idea of matrix methods (see (=-=Bibel, 1987-=-)) and resolution. Eder defines a consolution calculus together with a construction which yields a consolution refutation for every resolution refutation. This gives both a demonstration that resoluti... |

100 | A Prolog technology theorem prover: Implementation by an extended Prolog compiler
- Stickel
- 1988
(Show Context)
Citation Context ...ertain model elimination calculus (model elimination as defined by Loveland), a connection method calculus and a tableau model elimination calculus on which proof procedures like PTTP (Stickel, 1989; =-=Stickel, 1988-=-) and SETHEO (Letz et al., 1992) are based. The result for resolution and consolution was shown in (Eder, 1991). Altogether we will prove the diagram from figure 1, where the A ! B means that calculus... |

88 |
First Order Logic
- Smullyan
- 1971
(Show Context)
Citation Context ...lausal tableau calculus" is chosen as a starting point for a comparison of various calculi and proof procedures. It corresponds exactly to Beth's or Smullyan's analytic tableaux calculus (Beth, 1=-=959; Smullyan, 1968-=-) when restricted to propositional formulas which are given in clause form. This calculus can be further restricted, such that tableau expansions are allowed only when the clause used for the extensio... |

69 |
First Order Logic and Automated Theorem Proving
- Fitting
- 1996
(Show Context)
Citation Context ...ollow the lines from (Letz et al., 1992) and define the inference rules as tree-transforming operators. Then the calculus is much in the spirit of semantic tableaux with unification for clauses (see (=-=Fitting, 1990)), but with an-=- important restriction. This restriction will be explained below and justifies using the new name --- "tableau model elimination" --- instead of qualifying it as "analytical tableaux fo... |

64 |
The Foundations of Mathematics
- Beth
- 1959
(Show Context)
Citation Context ...1993) the "clausal tableau calculus" is chosen as a starting point for a comparison of various calculi and proof procedures. It corresponds exactly to Beth's or Smullyan's analytic tableaux =-=calculus (Beth, 1959-=-; Smullyan, 1968) when restricted to propositional formulas which are given in clause form. This calculus can be further restricted, such that tableau expansions are allowed only when the clause used ... |

57 | A Prolog technology theorem prover: A new exposition and implementation
- Stickel
- 1992
(Show Context)
Citation Context ...tep by step a certain model elimination calculus (model elimination as defined by Loveland), a connection method calculus and a tableau model elimination calculus on which proof procedures like PTTP (=-=Stickel, 1989-=-; Stickel, 1988) and SETHEO (Letz et al., 1992) are based. The result for resolution and consolution was shown in (Eder, 1991). Altogether we will prove the diagram from figure 1, where the ¢ £ ¤ mean... |

54 |
Mechanical Theorem Proving by Model Elimination
- Loveland
- 1968
(Show Context)
Citation Context ...ilding on this, we define a more refined consolution method, which can then be shown to simulate stepwise the model elimination calculus. Model elimination was originally introduced by Loveland (cf. (=-=Loveland, 1968-=-; Loveland, 1969; Loveland, 1978)). It can be seen as a restricted form of linear resolution (see (Loveland, 1978) and section 7). However, model elimination and its linear proof format was invented b... |

17 | A model elimination calculus with built-in theories
- Baumgartner
- 1992
(Show Context)
Citation Context ...ll become clear in the next section. The next theorem shows that sequence consolution is a framework to express tableau model elimination. Together with the completeness of tableau model elimination (=-=Baumgartner, 1992-=-) the theorem also yields completeness of sequence consolution. THEOREM 3.1. (Sequence Consolution Simulates Tableau Model Elimination) Let M be a matrix and (P 1 ; : : : ; P n ) a tableau model elimi... |

17 |
A unifying view of some linear Herbrand procedures
- Loveland
- 1972
(Show Context)
Citation Context ...imination is not possible. We are confident that our approach, using sequence consolution as a framework calculus, helps to understand the common ground and the differences between those calculi. In (=-=Loveland, 1972-=-), the relation between ME and several other calculi is investigated. As a main result, which is also contained and proved in a similar form in (Loveland, 1978), it demonstrated that ME and a certain ... |

12 |
Consolution and its relation with resolution
- Eder
- 1991
(Show Context)
Citation Context ...pose, we modify the consolution calculus in such a way that it can be instantiated to resolution, tableaux model elimination, a connection method and Loveland's model elimination. 1. Introduction In (=-=Eder, 1991-=-) the consolution method is introduced as a bridge between the idea of matrix methods (see (Bibel, 1987)) and resolution. Eder defines a consolution calculus together with a construction which yields ... |

12 |
A sequent-style model elimination strategy and a positive refinement
- Plaisted
- 1990
(Show Context)
Citation Context ...eduction procedure (Prawitz, 1970) are discussed, and it is demonstrated at the ground level that under certain circumstances there is a 1-to-1 relationship between matrix reduction and ME. Plaisted (=-=Plaisted, 1990-=-) has obtained a mapping (on the ground level) from model elimination to a sequent calculus (see e.g. (Gallier, 1987)). Using this setting he shows that it suffices to apply reduction steps to leaves ... |

10 |
A proof procedure with matrix reduction
- Prawitz
- 1970
(Show Context)
Citation Context ...ans of an example, and this example can also be used to show the same negative result for tableau model elimination. Also in that paper the linked conjunct procedure and a matrix reduction procedure (=-=Prawitz, 1970-=-) are discussed, and it is demonstrated at the ground level that under certain circumstances there is a 1-to-1 relationship between matrix reduction and ME. Plaisted (Plaisted, 1990) has obtained a ma... |

5 |
A Unified Approach to Theory Reasoning. Forschungsbericht 15/92
- Baumgartner, Furbach, et al.
- 1992
(Show Context)
Citation Context ...n figure 2): M(()) = () M(a i K) = (a ) \Delta M(K) M(b i K) = ((b 1 i ); : : : ; (b k i )) ffl M(K); where b i = (b 1 i ; : : : ; b k i ) 2 Such degenerated, linear shaped trees are called ferns in (=-=Baumgartner et al., 1992-=-). More formally, ferns are trees, where every inner node lies on one single path from the root to a leaf. The following lemma can be shown after rigorous formalization by induction on the length of c... |

4 |
A Comparative Study of Several Proof Procedures
- Bibel
- 1982
(Show Context)
Citation Context ...on proof is a linear input refutation and to us it is not clear if and how this approach works for non-linear resolution proofs as well. Bibel gave a comparative study of several proof procedures in (=-=Bibel, 1982). There, -=-a number of proof procedures are compared with three versions of the connection method (which were called "matrix methods" at that time). Bibel gave a theorem which establishes a relation be... |

4 |
Relative Complexities of First Order Languages
- Eder
- 1992
(Show Context)
Citation Context ...ered as a connection method. Clearly, for our purpose we have to commit ourselves to a single such "connection calculus". Here we will define a connection calculus that is very similar to th=-=e one in (Eder, 1992-=-). 18 P. Baumgartner and U. Furbach THE CONNECTION METHOD AND TABLEAU MODEL ELIMINATION The example in section 3 can serve to illustrate the connection method in relation to tableau model elimination.... |

4 |
SETHEO: A High-Performace Theorem Prover
- Letz, Schumann, et al.
- 1992
(Show Context)
Citation Context ...lculus (model elimination as defined by Loveland), a connection method calculus and a tableau model elimination calculus on which proof procedures like PTTP (Stickel, 1989; Stickel, 1988) and SETHEO (=-=Letz et al., 1992-=-) are based. The result for resolution and consolution was shown in (Eder, 1991). Altogether we will prove the diagram from figure 1, where the A ! B means that calculus A can be simulated stepwise by... |

3 |
A Simplified Version for the Model Elimination Theorem Proving Procedure
- Loveland
- 1969
(Show Context)
Citation Context ...we define a more refined consolution method, which can then be shown to simulate stepwise the model elimination calculus. Model elimination was originally introduced by Loveland (cf. (Loveland, 1968; =-=Loveland, 1969-=-; Loveland, 1978)). It can be seen as a restricted form of linear resolution (see (Loveland, 1978) and section 7). However, model elimination and its linear proof format was invented before linear res... |

2 |
First-Order Proof Calculi and Proof Procedures for Automated Deduction
- Letz
- 1993
(Show Context)
Citation Context ...uence consolution for these investigations, we obtained several new results and formal proofs of folklore-like theorems. Consolution as a Framework for Comparing Calculi 29 RELATION TO OTHER WORK In (=-=Letz, 1993) the &quo-=-t;clausal tableau calculus" is chosen as a starting point for a comparison of various calculi and proof procedures. It corresponds exactly to Beth's or Smullyan's analytic tableaux calculus (Beth... |

1 |
Consolution as a Framework for Comparing Calculi 31
- Gallier
- 1987
(Show Context)
Citation Context ...and thus can be removed. Second, disjuncts may be shortened; for example, (Qs:P) may be replaced by Q . This corresponds in some sense to the "weakening" rule in Gentzen's sequent calculus (=-=see e.g. (Gallier, 1987) for the -=-sequent calculus). The shortening step is sound since it preserves non-tautologyhood. However, it may cause incompleteness by throwing away the "wrong" literal, i.e. the literal that contrib... |

1 |
Vergleichende qualitative and quantitative Untersuchungen von Beweisverfahren
- Schreiber
- 1974
(Show Context)
Citation Context ... the same argumentation we can conclude that the tableau model elimination calculus exponentially simulates resolution. Negative results with respect to stepwise simulatability are also contained in (=-=Schreiber, 1974-=-). He relates an early presentation of the connection method according to Bibel and two variants of the resolution calculus (binary resolution with factoring and linear resolution). By means of exampl... |

1 |
A Prolog Technology Theorem Prover: ANew Exposition and Implementation
- Stickel
- 1989
(Show Context)
Citation Context ...tep by step a certain model elimination calculus (model elimination as defined by Loveland), a connection method calculus and a tableau model elimination calculus on which proof procedures like PTTP (=-=Stickel, 1989-=-; Stickel, 1988) and SETHEO (Letz et al., 1992) are based. The result for resolution and consolution was shown in (Eder, 1991). Altogether we will prove the diagram from figure 1, where the A ! B mean... |