## Shortest axiomatizations of implicational S4 and S5

Venue: | the Notre Dame Journal of Formal Logic (NDJFL |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Ernst_shortestaxiomatizations,

author = {Zachary Ernst and Branden Fitelson and Kenneth Harris and Larry Wos},

title = {Shortest axiomatizations of implicational S4 and S5},

booktitle = {the Notre Dame Journal of Formal Logic (NDJFL},

year = {}

}

### OpenURL

### Abstract

Abstract. Shortest possible axiomatizations for the implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.

### Citations

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Citation Context ...or the minor premise α. All proofs reported here were discovered with the assistance of the automated reasoning program c45_revised.tex; 10/10/2001; 9:50; p.1s2 Ernst, Fitelson, Harris, and Wos Otter =-=[9]-=-. The extensive role of automated reasoning in the present research is discussed in Section 4. 2. Axiomatic C4 We begin with appropriate background to place the question we answer in perspective. 2.1.... |

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Citation Context ...c C4 and C5, automated reasoning strategies played a crucial role. In particular, we relied heavily on William McCune’s automated reasoning program Otter [9], H. Zhang and J. Zhang’s model finder SEM =-=[24]-=-, and John Slaney’s c45_revised.tex; 10/10/2001; 9:50; p.9s10 Ernst, Fitelson, Harris, and Wos model finder MaGIC [15]. Here, we outline the approach used to obtain these results and briefly discuss s... |

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Citation Context ...e same purpose, we used Veroff’s hints strategy in which the researcher provides attractive patterns, patterns that are keyed upon themselves and also on patterns that subsume or are subsumed by them =-=[17]-=-. We restricted the program’s reasoning by placing bounds on the complexity of retained conclusions and on the number of distinct letters occurring in such. Purity (of the circles of proofs) was achie... |

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Citation Context ...Cune’s program to discover extremely complex and difficult proofs. For example, Otter has yielded proofs consisting of 200 applications of condensed detachment for theorems of significant depth. (See =-=[23]-=- and [22] for information on the solution of challenge problems using Otter, as well as for open problems.) We believe that further results regarding axiomatizations for more complex logics await futu... |

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Citation Context ...ogram to discover extremely complex and difficult proofs. For example, Otter has yielded proofs consisting of 200 applications of condensed detachment for theorems of significant depth. (See [23] and =-=[22]-=- for information on the solution of challenge problems using Otter, as well as for open problems.) We believe that further results regarding axiomatizations for more complex logics await future advanc... |

14 | Searching for Circles of Pure Proofs
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Citation Context ...7.7 8. CpCCqCqrCqr D.2.8 9. CCpCqCqrCsCpCqr D.9.2 10. CpCCqCrsCCqrCqs D.10.10 11. CCpCqrCCpqCpr ∗ This circle of proofs (1) ⇒ (3) ⇒ (2) ⇒ (1) has the additional property of being pure—in the sense of =-=[19]-=- and [20]. That is, (i) the proof of (1) ⇒ (3) does not make use of (2), (ii) the proof of (3) ⇒ (2) does not make use of (1), and (iii) the proof of (2) ⇒ (1) does not make use of (3). We think this ... |

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Citation Context ...implication, single axiom, automated reasoning 1. Background and Conventions The implicational fragments of the modal logics S4 and S5 have been studied extensively over the years (see, for instance, =-=[8]-=-, [7], [1], [5], and [11]). Following tradition, we use the labels “C4” and “C5” to denote the strict implicational fragments of S4 and S5, respectively. Prior [14, Appendix I] reports a variety of Hi... |

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Citation Context ...g, we used the resonance strategy, which enables the researcher to provide patterns (formulas or equations) that are treated as attractive because of their functional shape (ignoring their variables) =-=[18]-=-. For the same purpose, we used Veroff’s hints strategy in which the researcher provides attractive patterns, patterns that are keyed upon themselves and also on patterns that subsume or are subsumed ... |

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Citation Context ...ngle axiom, automated reasoning 1. Background and Conventions The implicational fragments of the modal logics S4 and S5 have been studied extensively over the years (see, for instance, [8], [7], [1], =-=[5]-=-, and [11]). Following tradition, we use the labels “C4” and “C5” to denote the strict implicational fragments of S4 and S5, respectively. Prior [14, Appendix I] reports a variety of Hilbert-style axi... |

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Citation Context ...riety of logics, with equal success. For instance, we have used this approach to discover the shortest known basis for the implicational fragment of the logic RM (first axiomatized by Meyer and Parks =-=[12]-=-, [13]) [4], and McCune and Veroff have successfully used a similar approach to find short single axioms in lattice theory [10]. Currently, however, an exhaustive search such as the one used in the pr... |

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Citation Context ...n, single axiom, automated reasoning 1. Background and Conventions The implicational fragments of the modal logics S4 and S5 have been studied extensively over the years (see, for instance, [8], [7], =-=[1]-=-, [5], and [11]). Following tradition, we use the labels “C4” and “C5” to denote the strict implicational fragments of S4 and S5, respectively. Prior [14, Appendix I] reports a variety of Hilbert-styl... |

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Citation Context ...cation, single axiom, automated reasoning 1. Background and Conventions The implicational fragments of the modal logics S4 and S5 have been studied extensively over the years (see, for instance, [8], =-=[7]-=-, [1], [5], and [11]). Following tradition, we use the labels “C4” and “C5” to denote the strict implicational fragments of S4 and S5, respectively. Prior [14, Appendix I] reports a variety of Hilbert... |

5 |
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Citation Context ...of logics, with equal success. For instance, we have used this approach to discover the shortest known basis for the implicational fragment of the logic RM (first axiomatized by Meyer and Parks [12], =-=[13]-=-) [4], and McCune and Veroff have successfully used a similar approach to find short single axioms in lattice theory [10]. Currently, however, an exhaustive search such as the one used in the present ... |

4 |
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Citation Context ...ed.tex; 10/10/2001; 9:50; p.2sShortest Axiomatizations of Implicational S4 and S5 3 for C4. The shortest of these include the following 2-basis, which contains only 20 symbols and 9 occurrences of C. =-=(2)-=- CpCqq CCpCqrCCpqCsCpr So far, we have found six such 2-bases, and we know that there are at most eight. We suspect that there exist exactly six. We have eliminated all other 2-bases of this complexit... |

4 |
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Citation Context ...s [14, Appendix II]). That is, the notation “D.a.b” (appearing to the left of each line in our deductions) is used to denote the most general possible result of detachment (i.e., condensed detachment =-=[6]-=-) with a, or some substitution in a, for the major premise Cαβ, and with b, or some substitution in b, for the minor premise α. All proofs reported here were discovered with the assistance of the auto... |

2 |
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(Show Context)
Citation Context ...m, automated reasoning 1. Background and Conventions The implicational fragments of the modal logics S4 and S5 have been studied extensively over the years (see, for instance, [8], [7], [1], [5], and =-=[11]-=-). Following tradition, we use the labels “C4” and “C5” to denote the strict implicational fragments of S4 and S5, respectively. Prior [14, Appendix I] reports a variety of Hilbert-style axiomatizatio... |

2 |
and the Moufang identity problem
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Citation Context ...CCqCqrCqr D.2.8 9. CCpCqCqrCsCpCqr D.9.2 10. CpCCqCrsCCqrCqs D.10.10 11. CCpCqrCCpqCpr ∗ This circle of proofs (1) ⇒ (3) ⇒ (2) ⇒ (1) has the additional property of being pure—in the sense of [19] and =-=[20]-=-. That is, (i) the proof of (1) ⇒ (3) does not make use of (2), (ii) the proof of (3) ⇒ (2) does not make use of (1), and (iii) the proof of (2) ⇒ (1) does not make use of (3). We think this circle of... |

1 |
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Citation Context ... about how this exhaustive search was conducted and how the matrices and bases were discovered. Our automated reasoning strategies also yielded the following new 21-symbol (10-C) single axiom for C4: =-=(3)-=- CCpCCqCrrCpsCCstCuCpt As noted earlier, the question of the existence of a single axiom for C4 has been a long-standing open problem in the axiomatics of modal logic [2, page 83]. We have ruled out a... |

1 | A concise axiomatization of RM
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(Show Context)
Citation Context ...ics, with equal success. For instance, we have used this approach to discover the shortest known basis for the implicational fragment of the logic RM (first axiomatized by Meyer and Parks [12], [13]) =-=[4]-=-, and McCune and Veroff have successfully used a similar approach to find short single axioms in lattice theory [10]. Currently, however, an exhaustive search such as the one used in the present study... |

1 |
Yet another single law for lattices. Forthcoming in Algebra Universalis
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(Show Context)
Citation Context ...icational fragment of the logic RM (first axiomatized by Meyer and Parks [12], [13]) [4], and McCune and Veroff have successfully used a similar approach to find short single axioms in lattice theory =-=[10]-=-. Currently, however, an exhaustive search such as the one used in the present study is prohibitively time consuming when applied to c45_revised.tex; 10/10/2001; 9:50; p.11s12 Ernst, Fitelson, Harris,... |

1 |
Matrix Generator for Implication Connectives
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(Show Context)
Citation Context ...s automated reasoning program Otter [9], H. Zhang and J. Zhang’s model finder SEM [24], and John Slaney’s c45_revised.tex; 10/10/2001; 9:50; p.9s10 Ernst, Fitelson, Harris, and Wos model finder MaGIC =-=[15]-=-. Here, we outline the approach used to obtain these results and briefly discuss some of the automated reasoning strategies. 1. First, we wrote computer programs to generate a large list of candidate ... |

1 |
Strict implication in a sequence of extensions of S4
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- 1981
(Show Context)
Citation Context ...s far as we know, no 2-axiom bases for C4 were ever reported in the literature. Moreover, no single axiom for C4 has been discovered; indeed, this is stated as an open problem in [2, page 83]. Ulrich =-=[16]-=- has shown that C4 is also the strict implicational fragment of each modal logic between S4 and S4.3; hence, our bases are new and shortest bases for the strict implicational fragments of these extens... |

1 |
The strategy of cramming. preprint ANL/MCS-P898-0801, Argonne National Laboratory
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- 2001
(Show Context)
Citation Context ... sought proofs of minimal length. With cramming, one instructs the program to rely heavily on chosen steps of a proof with the objective of cramming most or all of them into another proof of interest =-=[21]-=-. Obvious changes were made when we searched for axiom bases with more than a single formula. For example, in contrast to the study of a possible single axiom where its denial was placed in what is ca... |