### Citations

1102 |
A machine-oriented logic based on the resolution principle,"
- Robinson
- 1965
(Show Context)
Citation Context ...pment in this area. Analytic cut trees provide a further development. It has been recently suggested by Oppacher and Suen [13] and Fitting [7] that tableaux provide a viable alternative to Robinson’s =-=[14]-=- resolution as a basis for automated deduction. From the viewpoint of efficiency such a suggestion has some limitations. For, by a result of [4] tableaux cannot p-simulate resolution and, as shown in ... |

483 |
First-order logic and automated theorem proving.
- Fitting
- 1996
(Show Context)
Citation Context ...i’s [6] analytic elimination trees seem to be the only development in this area. Analytic cut trees provide a further development. It has been recently suggested by Oppacher and Suen [13] and Fitting =-=[7]-=- that tableaux provide a viable alternative to Robinson’s [14] resolution as a basis for automated deduction. From the viewpoint of efficiency such a suggestion has some limitations. For, by a result ... |

361 | The relative efficiency of propositional proof systems.
- Cook, Reckhow
- 1979
(Show Context)
Citation Context ...losure conditions. The main advantage of analytic cut trees over tableaux consists in their efficiency. Relative efficiency of proof systems can be studied in terms of the notion of psimulation ([4], =-=[5]-=-): a proof system S is said to p-simulate a proof system S ′ if there exists a polynomial p(n) such that, for any sentence ϕ, ifϕhas an S-proof of complexity less than or equal to n, then has an S ′-p... |

353 |
First-Order Logic.
- Smullyan
- 1968
(Show Context)
Citation Context ...t formula is a subformula of at least one formula in the conclusion. Although cuts cannot be eliminated from the resulting system, the subformula property still holds. As it is well known, Smullyan’s =-=[16]-=- tableaux are a streamlined version of Kleene’s [11] cut-free sequent calculus G4 (see [2] for details). Is there a similarly streamlined counterpart of Smullyan’s analytic cut system? In this paper w... |

303 |
Investigations into logical deduction
- Gentzen
- 1969
(Show Context)
Citation Context ...d analytic cut trees are intrinsically more efficient than any cut-free system. Keywords: Analytic cut; subformula property; resolution 1 Introduction In introducing his calculus of sequents Gentzen (=-=[9]-=- pp. 85-6) suggested that, if one were not interested in cut elimination, nine out of his fourteen operational inference rules could be replaced by axioms. Apparently he saw no way of replacing the re... |

298 |
Symbolic Logic and Mechanical Theorem Proving,
- Chang, Lee
- 1973
(Show Context)
Citation Context ...rule is not valid. 8 Relationship of analytic cut trees to resolution In this section we show that sequent analytic cut trees can p-simulate resolution. For background on resolution see Chang and Lee =-=[3]-=- or Gallier [8]. Definition 8.1 1. We expand formulas by adding formulas of the form ϕ1 ∨ ... ∨ ϕn, for any finite (possibly empty) list of formulas ϕ1, ..., ϕn. Wecallϕ1 ∨ ... ∨ ϕn the generalized di... |

189 |
Logic for Computer Science { Foundations of Automatic Theorem Proving
- Gallier
- 1986
(Show Context)
Citation Context ...id. 8 Relationship of analytic cut trees to resolution In this section we show that sequent analytic cut trees can p-simulate resolution. For background on resolution see Chang and Lee [3] or Gallier =-=[8]-=-. Definition 8.1 1. We expand formulas by adding formulas of the form ϕ1 ∨ ... ∨ ϕn, for any finite (possibly empty) list of formulas ϕ1, ..., ϕn. Wecallϕ1 ∨ ... ∨ ϕn the generalized disjunction of ϕ1... |

183 |
The province of logic
- Kneale
- 1956
(Show Context)
Citation Context ...al inference rules, but such a restriction was removed by Smullyan through his analytic cut system [15]. Inference rules exactly corresponding to Smullyan’s [15] axioms were also considered by Kneale =-=[12]-=-. Gentzen’s ([9] p. 69) basic aim was to show that every purely logical proof can be reduced to a definite, though not unique, normal form, hence he considered cut elimination as primary and the subfo... |

84 |
Mathematical Logic,
- Kleene
- 1967
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Citation Context ... the conclusion. Although cuts cannot be eliminated from the resulting system, the subformula property still holds. As it is well known, Smullyan’s [16] tableaux are a streamlined version of Kleene’s =-=[11]-=- cut-free sequent calculus G4 (see [2] for details). Is there a similarly streamlined counterpart of Smullyan’s analytic cut system? In this paper we introduce such a counterpart in the form of analyt... |

63 | The taming of the cut. Classical refutations with analytical cut.
- D’Agostino, Mondadori
- 1994
(Show Context)
Citation Context ...of systems from which cuts cannot be eliminated but satisfying the subformula property is still a relatively unexplored one. Apart from Smullyan’s [15] analytic cut system, D’Agostino and Mondadori’s =-=[6]-=- analytic elimination trees seem to be the only development in this area. Analytic cut trees provide a further development. It has been recently suggested by Oppacher and Suen [13] and Fitting [7] tha... |

53 |
Proof theory and logical complexity
- Girard
- 1987
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Citation Context ...lassical meaning of logical operations, it can be argued that they do not capture the essentials of classical semantics. Indeed, there is a 3-valued semantics, specifically the one defined in Girard (=-=[10]-=- pp. 168-169), in which all tableau rules are valid but the cut rule is not valid. 8 Relationship of analytic cut trees to resolution In this section we show that sequent analytic cut trees can p-simu... |

45 |
On the Lengths of Proofs in the Propositional Calculus
- Reckhow
- 1976
(Show Context)
Citation Context ...ral closure conditions. The main advantage of analytic cut trees over tableaux consists in their efficiency. Relative efficiency of proof systems can be studied in terms of the notion of psimulation (=-=[4]-=-, [5]): a proof system S is said to p-simulate a proof system S ′ if there exists a polynomial p(n) such that, for any sentence ϕ, ifϕhas an S-proof of complexity less than or equal to n, then has an ... |

41 |
HARP: A tableau-based theorem prover
- Oppacher, Suen
- 1988
(Show Context)
Citation Context ...tino and Mondadori’s [6] analytic elimination trees seem to be the only development in this area. Analytic cut trees provide a further development. It has been recently suggested by Oppacher and Suen =-=[13]-=- and Fitting [7] that tableaux provide a viable alternative to Robinson’s [14] resolution as a basis for automated deduction. From the viewpoint of efficiency such a suggestion has some limitations. F... |

19 |
Analytic cut
- Smullyan
- 1968
(Show Context)
Citation Context ...rules could be replaced by axioms. Apparently he saw no way of replacing the remaining five operational inference rules, but such a restriction was removed by Smullyan through his analytic cut system =-=[15]-=-. Inference rules exactly corresponding to Smullyan’s [15] axioms were also considered by Kneale [12]. Gentzen’s ([9] p. 69) basic aim was to show that every purely logical proof can be reduced to a d... |

2 |
Semantic entailment and formal derivability. Medelingen von de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde
- Beth
- 1955
(Show Context)
Citation Context ...bleaux, but are also uniformly more efficient than the latter.7. COMPUTATIONAL INEFFICIENCY OF TABLEAUX 745 Remark 7.3 The fact that tableaux cannot p-simulate analytic cut trees contradicts Beth’s (=-=[1]-=- p. 23]) original motivation in introducing tableaux, that they would allow to construct proofs which were remarkably concise and could even be proved to be the shortest possible ones. Example 7.4 The... |

2 | Using full first order logic as a programming language. In Rendiconti del Seminario Matematico Universit`a e Politecnico di Torino. Fascicolo Speciale
- Cellucci
- 1987
(Show Context)
Citation Context ...e eliminated from the resulting system, the subformula property still holds. As it is well known, Smullyan’s [16] tableaux are a streamlined version of Kleene’s [11] cut-free sequent calculus G4 (see =-=[2]-=- for details). Is there a similarly streamlined counterpart of Smullyan’s analytic cut system? In this paper we introduce such a counterpart in the form of analytic cut trees, i.e. trees of formulas g... |

1 |
Tableau systems for first order number theory and certain higher order theories
- Toledo
- 1975
(Show Context)
Citation Context ...lausible if by a roundabout proof we mean one which contains no redundancies. That the inefficiency of tableaux stems from an unnecessary duplication of subtrees in tableaux is supported by Toledo’s (=-=[17]-=- Appendix 1) result that any closed tableau T for Γ containing cuts can be converted into a closed cut-free tableau T ′ for Γ such that, if in calculating the complexity of tableaux we ignore duplicat... |