## A Uniform Procedure for Converting Matrix Proofs into Sequent-Style Systems (2000)

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Venue: | Journal of Information and Computation |

Citations: | 10 - 7 self |

### BibTeX

@ARTICLE{Kreitz00auniform,

author = {Christoph Kreitz and Stephan Schmitt},

title = {A Uniform Procedure for Converting Matrix Proofs into Sequent-Style Systems},

journal = {Journal of Information and Computation},

year = {2000},

volume = {162},

pages = {2000}

}

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### Abstract

We present a uniform algorithm for transforming machine-found matrix proofs in classical, constructive, and modal logics into sequent proofs. It is based on unified representations of matrix characterizations, of sequent calculi, and of prefixed sequent systems for various logics. The peculiarities of an individual logic are described by certain parameters of these representations, which are summarized in tables to be consulted by the conversion algorithm.

### Citations

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(Show Context)
Citation Context ... especially in scientific applications. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic, theorem provers based on resolution =-=[23, 31]-=- and the connection method [5, 6, 14, 4] have demonstrated that logical deduction can be simulated sufficiently well on a computer. Recently the characterizations of logical validity, which underly th... |

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182 |
Isabelle: The next 700 theorem provers
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(Show Context)
Citation Context ...easoning, as illustrated in Figure 1. A proof task to be solved by the ATP system will be extracted by the user or a strategy of the application system (e.g. Mathematica [30], NuPRL [7], and Isabelle =-=[20]-=-). After selecting the particular logic (classical, intuitionistic, or modal) a uniform proof procedure will create a machine-proof (e.g. a matrix proof in non-normal form). This proof will be convert... |

148 |
Automated Theorem Proving
- Bibel
- 1987
(Show Context)
Citation Context ...ons. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic, theorem provers based on resolution [23, 31] and the connection method =-=[5, 6, 14, 4]-=- have demonstrated that logical deduction can be simulated sufficiently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intui... |

128 |
SETHEO: A High-Performance Theorem Prover
- Letz, Schumann, et al.
- 1992
(Show Context)
Citation Context ...ons. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic, theorem provers based on resolution [23, 31] and the connection method =-=[5, 6, 14, 4]-=- have demonstrated that logical deduction can be simulated sufficiently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intui... |

94 |
On matrices with connections
- Bibel
- 1981
(Show Context)
Citation Context ...ons. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic, theorem provers based on resolution [23, 31] and the connection method =-=[5, 6, 14, 4]-=- have demonstrated that logical deduction can be simulated sufficiently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intui... |

87 |
Automated Deduction in Nonclassical Logics
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- 1990
(Show Context)
Citation Context ...iently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intuitionistic logic and the modal logics K, K4, D, D4, T, S4, and S5 =-=[17, 28, 29, 27]-=-. On this basis the existing proof methods have been extended accordingly [18, 19] in order to develop a coherent theorem prover that can deal with a variety of logics and many applications requiring ... |

85 | A resolution calculus for modal logics
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- 1988
(Show Context)
Citation Context ...iently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intuitionistic logic and the modal logics K, K4, D, D4, T, S4, and S5 =-=[17, 28, 29, 27]-=-. On this basis the existing proof methods have been extended accordingly [18, 19] in order to develop a coherent theorem prover that can deal with a variety of logics and many applications requiring ... |

84 |
Intuitionistic Logic Model Theory and Forcing
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- 1969
(Show Context)
Citation Context ...classical logic (C), intuitionistic logic (J), and the modal logics K, K4, D, D4, T, S4, S5 with their cumulative, varying and constant domain variants concerning the Kripke-semantics of these logics =-=[9, 10]-=-. In our representations we shall separate the aspects common to all these logics from those that depend on a particular logic. This separation allows a compact presentation of each calculus and is cr... |

63 |
The Foundations of Mathematics
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(Show Context)
Citation Context ...ified presentation we shall provide schematic rules containing parameters whose values describe the individual logics. To achieve a compact representation we adopt a notation based on tableau systems =-=[3, 10] and encode a sequent Γ ⊢ -=-∆ by a set S of signed formulae, which we call the associated set. Formally, the associated set of Γ ⊢ ∆ is defined as S=SΓ ∪S∆ where SΓ ={〈F, 1〉 | F ∈ Γ } and S∆={〈F, 0〉 | F... |

43 |
Transforming Matings into Natural Deduction Proofs
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- 1980
(Show Context)
Citation Context ...erstood by experts of the envisaged application they must be transformed into a more readable form. Techniques for a humanly comprehensible presentation of machine-found proofs have been described in =-=[1, 32, 2, 21, 15, 16]-=-. They transform classical resolution- or matrix proofs into natural calculi such as LK or N K [11]. Other approaches try to present them even in natural language [8]. In [24] we have developed an alg... |

42 | A resolution theorem prover for intuitionistic logic - Tammet - 1996 |

29 | A connection based proof method for intuitionistic logic
- Otten, Kreitz
- 1995
(Show Context)
Citation Context ...ly these systems, have been extended to intuitionistic logic and the modal logics K, K4, D, D4, T, S4, and S5 [17, 28, 29, 27]. On this basis the existing proof methods have been extended accordingly =-=[18, 19]-=- in order to develop a coherent theorem prover that can deal with a variety of logics and many applications requiring mathematical reasoning. The use of automated theorem proving (ATP) in practical ap... |

26 | On transforming intuitionistic matrix proofs into standard-sequent proofs
- Schmitt, Kreitz
(Show Context)
Citation Context ...bed in [1, 32, 2, 21, 15, 16]. They transform classical resolution- or matrix proofs into natural calculi such as LK or N K [11]. Other approaches try to present them even in natural language [8]. In =-=[24]-=- we have developed an algorithm for converting intuitionistic matrix proofs into the sequent calculus LJ to support program development in interactive proof assistants [13]. In this paper we present a... |

24 |
Matrix proof methods for modal logics
- Wallen
- 1987
(Show Context)
Citation Context ...iently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intuitionistic logic and the modal logics K, K4, D, D4, T, S4, and S5 =-=[17, 28, 29, 27]-=-. On this basis the existing proof methods have been extended accordingly [18, 19] in order to develop a coherent theorem prover that can deal with a variety of logics and many applications requiring ... |

22 |
Proof Methods for Modal and Intuitionistic
- Fitting
- 1983
(Show Context)
Citation Context ...not yet be reduced. These wait-labels again depend on the underlying logic. Methodically we proceed in two steps. Because of a similarity between matrix calculi and Fitting’s prefixed tableau system=-=s [10]-=- we first show how to convert a matrix proof into a prefixed sequent proof. This algorithm, which we present in Section 3, only requires unified representations of the calculi. Afterwards we extend ou... |

22 |
Structuring Computer Generated Proofs
- Lingenfelder
- 1990
(Show Context)
Citation Context ...erstood by experts of the envisaged application they must be transformed into a more readable form. Techniques for a humanly comprehensible presentation of machine-found proofs have been described in =-=[1, 32, 2, 21, 15, 16]-=-. They transform classical resolution- or matrix proofs into natural calculi such as LK or N K [11]. Other approaches try to present them even in natural language [8]. In [24] we have developed an alg... |

21 | Proof Transformations in Higher-Order Logic
- Pfenning
- 1987
(Show Context)
Citation Context ...erstood by experts of the envisaged application they must be transformed into a more readable form. Techniques for a humanly comprehensible presentation of machine-found proofs have been described in =-=[1, 32, 2, 21, 15, 16]-=-. They transform classical resolution- or matrix proofs into natural calculi such as LK or N K [11]. Other approaches try to present them even in natural language [8]. In [24] we have developed an alg... |

18 | Guiding Program Development Systems by a Connection Based Proof Strategy
- Kreitz, Otten, et al.
- 1996
(Show Context)
Citation Context ...in natural language [8]. In [24] we have developed an algorithm for converting intuitionistic matrix proofs into the sequent calculus LJ to support program development in interactive proof assistants =-=[13]-=-. In this paper we present a transformation procedure that allows a uniform treatment of classical, constructive, and modal logics. It will support the integration of automated theorem proving into a ... |

14 |
Implementing Mathematics with the NuPRL proof development system
- al
- 1986
(Show Context)
Citation Context ...areas of logical reasoning, as illustrated in Figure 1. A proof task to be solved by the ATP system will be extracted by the user or a strategy of the application system (e.g. Mathematica [30], NuPRL =-=[7]-=-, and Isabelle [20]). After selecting the particular logic (classical, intuitionistic, or modal) a uniform proof procedure will create a machine-proof (e.g. a matrix proof in non-normal form). This pr... |

9 | Instantiation of existentially quantified variables in inductive specification proofs - Pientka, Kreitz - 1998 |

7 |
Automated reasoning contributes to mathematics and logic
- al
- 1990
(Show Context)
Citation Context ... especially in scientific applications. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic, theorem provers based on resolution =-=[23, 31]-=- and the connection method [5, 6, 14, 4] have demonstrated that logical deduction can be simulated sufficiently well on a computer. Recently the characterizations of logical validity, which underly th... |

4 |
More on the problem of finding a mapping between clause representation and natural-deduction representation
- Andrews
- 1991
(Show Context)
Citation Context |

4 |
The problem of finding a mapping between clause representation and natural-deduction representation
- Wos
- 1990
(Show Context)
Citation Context |

2 |
A Uniform Proof Procedure for Classical and
- Otten, Kreitz
- 1996
(Show Context)
Citation Context ...ly these systems, have been extended to intuitionistic logic and the modal logics K, K4, D, D4, T, S4, and S5 [17, 28, 29, 27]. On this basis the existing proof methods have been extended accordingly =-=[18, 19]-=- in order to develop a coherent theorem prover that can deal with a variety of logics and many applications requiring mathematical reasoning. The use of automated theorem proving (ATP) in practical ap... |

2 | Deleting redundancy in proof reconstruction
- Schmitt, Kreitz
- 1998
(Show Context)
Citation Context ... However, avoiding the decomposition problem and hence, search behavior during the transformation demands the integration of additional knowledge from the proof search into the reconstruction process =-=[25, 26]-=-. Future work will involve combining our transformation algorithm with a proof procedure for various non-classical logics in order to guide the derivation of proofs in one of the existing generic tool... |

1 |
Uniform algorithms for proof reconstruction in classical and non-classical logics
- Schmitt
- 1998
(Show Context)
Citation Context ... (p, q), if q=pp ′ for some |p ′ | ≥ 1, and q �� r for any prefix r in S. The complete treatise of this problem as well as correctness proofs for the modified prefixed sequent calculi can be=-= found in [26]. 3.-=-2 The Algorithm TOTAL The uniform transformation algorithm TOTAL is presented in Figure 2. It takes as input a logic L and a partial reduction ordering ∝ ⋆ that represents a matrix proof in L. The... |

1 | A Uniform Proof Procedure for Classical and Non-Classical Logics KI-96, LNAI 1137 - Springer - 1995 |