## Map calculus: Initial application scenarios and experiments based on Otter (1998)

Citations: | 6 - 6 self |

### BibTeX

@MISC{Aureli98mapcalculus:,

author = {Fabiola Aureli and Eugenio Omodeo and Marco Temperini},

title = {Map calculus: Initial application scenarios and experiments based on Otter},

year = {1998}

}

### OpenURL

### Abstract

Properties of a few familiar structures (natural numbers, nested lists, lattices) are formally specified in Tarski-Givant's map calculus, with the aim of bringing to light new translation techniques that may bridge the gap between first-order predicate calculus and the map calculus. It is also highlighted to what extent a state-of-the-art theorem-prover for first-order logic, namely Otter, can be exploited not only to emulate, but also to reason about, map calculus.

### Citations

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Citation Context ...itnesses the existence of corresponding lines of reasoning in L \Theta . One can moreover load into Otter, along withs\Theta , a set E of map equalities, e.g. those in the upper part of Figure 1 (cf. =-=[9]-=-, p.184 fol.), and derive theorems of \Theta \Theta (E), such as those in the lower part of the same figure. The very shape of the equalities ins\Theta is the result of us having carried out a number ... |

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Citation Context ...tituting predicates for variables) are sentences logically provable in L \Theta . This indicates that we can use an automated deduction tool conceived for first-order logic, Otter to be specific (cf. =-=[14]-=-), to experiment with L \Theta . Although Otter cannot directly produce derivations of L \Theta , once the equalities that forms\Theta are loaded into Otter, whatever chain of inference steps can be d... |

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Citation Context ...pecified (see Sec.5 and following). In particular, the formalization of a classical example (a line-editor, cf. Sec.6) enlightens the connection between map calculus and algebraic specifications (cf. =-=[26, 8]-=-). Exercise of this nature, based on paper and pencil for the time being, is aimed at bringing to light translation techniques that may effectively bridge the gap between firstorder predicate calculus... |

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Citation Context ...ll it--- that Peirce and Schroder had contributed to create and which short after the appearance of the Principia fell into general oblivion, was already signaled as an anomaly by Tarski in 1941 (cf. =-=[23]-=-). Even more anomalous it should be considered today, since the subsequent work of Tarski and others (cf. [24]) made it clear that map calculus had no inner weaknesses preventing it from becoming the ... |

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Citation Context ...pecified (see Sec.5 and following). In particular, the formalization of a classical example (a line-editor, cf. Sec.6) enlightens the connection between map calculus and algebraic specifications (cf. =-=[26, 8]-=-). Exercise of this nature, based on paper and pencil for the time being, is aimed at bringing to light translation techniques that may effectively bridge the gap between firstorder predicate calculus... |

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Citation Context ...of contemporary logic (cf. [3, 12, 25, 4]). Only a few people are aware that very important prePrincipia milestones were laid down by C.S. Peirce and E. Schroder and culminated in the monumental work =-=[21, 22] on the Al-=-gebra der Logik . The "rather capricious line of historical development" of the algebraic form of logic ---the map calculus, as we will call it--- that Peirce and Schroder had contributed to... |

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Citation Context ...lae occur. These schematic variables can be regarded as universally quantified second-order variables, and as such can be treated by higher-order proof assistants, Isabelle to mention one (cf., e.g., =-=[15, 17, 18]-=-). A similar advantage can be gained by using first-order variables to represent predicate expressions of L \Theta : virtually any theorem prover or proof assistant can then be used to deal with theor... |

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Citation Context ... compare the results of our experiments with the work of others. Otter is attractive in this respect, because it has been the system underlying experiments of the kind we have in mind, as reported in =-=[19, 20]. Mo-=-reover, the fact that Otter encompasses full first-order logic paves the way to combined reasoning tactics that, e.g., perform resolution of 1l ffi P ffi 1l=1l against P=��. a. P ffi '=P right uni... |

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Citation Context ...pia fell into general oblivion, was already signaled as an anomaly by Tarski in 1941 (cf. [23]). Even more anomalous it should be considered today, since the subsequent work of Tarski and others (cf. =-=[24]-=-) made it clear that map calculus had no inner weaknesses preventing it from becoming the frame for an omnicomprehensive deductive system such as set theory. The rehabilitation of map calculus from it... |

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Citation Context ...lae occur. These schematic variables can be regarded as universally quantified second-order variables, and as such can be treated by higher-order proof assistants, Isabelle to mention one (cf., e.g., =-=[15, 17, 18]-=-). A similar advantage can be gained by using first-order variables to represent predicate expressions of L \Theta : virtually any theorem prover or proof assistant can then be used to deal with theor... |

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Citation Context ... compare the results of our experiments with the work of others. Otter is attractive in this respect, because it has been the system underlying experiments of the kind we have in mind, as reported in =-=[19, 20]. Mo-=-reover, the fact that Otter encompasses full first-order logic paves the way to combined reasoning tactics that, e.g., perform resolution of 1l ffi P ffi 1l=1l against P=��. a. P ffi '=P right uni... |

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Citation Context ... a singleton, and hence makes ; and U 2 the only possible values for each map expression P . Figures 1 and 3, to be commented later on, show 1 This is typical of algebraic specifications---see, e.g., =-=[10]-=-. more sophisticated examples. We postpone to Sec.4 the definition of a deductive machinery for L \Theta . L + is a variant version of a first-order dyadic predicate language: an atomic formula of L +... |

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Citation Context ...Boole's Laws of thought (1854), Frege's Begriffsschrifts(1879), and the Whitehead-Russell's Principia Mathematica (1910) have been three major milestones in the development of contemporary logic (cf. =-=[3, 12, 25, 4]). On-=-ly a few people are aware that very important prePrincipia milestones were laid down by C.S. Peirce and E. Schroder and culminated in the monumental work [21, 22] on the Algebra der Logik . The "... |

24 |
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Citation Context ... Neumann, Godel, and Bernays to discover a finite axiomatization for set theory (which, by the way, is sometimes preferred to Zermelo-Fraenkel as a basis for experimentation with theorem provers, cf. =-=[5, 27, 19]-=-). In their formulation of set theory, variables range over the subclasses of the universe of sets, some of which are sets whereas others are not. We will exploit these ideas in the next section, to p... |

24 |
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Citation Context ... a general technique that enables one to reduce to three the number of variables in any sentence ff of the language of a strong theory, by suitably exploiting the map expressions that describe conju5 =-=[11]-=- makes the following example: resorting to four variables may seem essential to express the existence of four distinct entities in the domain U of discourse, but in the theory of strict total ordering... |

21 |
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Citation Context ...Boole's Laws of thought (1854), Frege's Begriffsschrifts(1879), and the Whitehead-Russell's Principia Mathematica (1910) have been three major milestones in the development of contemporary logic (cf. =-=[3, 12, 25, 4]). On-=-ly a few people are aware that very important prePrincipia milestones were laid down by C.S. Peirce and E. Schroder and culminated in the monumental work [21, 22] on the Algebra der Logik . The "... |

15 |
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Citation Context ...lae occur. These schematic variables can be regarded as universally quantified second-order variables, and as such can be treated by higher-order proof assistants, Isabelle to mention one (cf., e.g., =-=[15, 17, 18]-=-). A similar advantage can be gained by using first-order variables to represent predicate expressions of L \Theta : virtually any theorem prover or proof assistant can then be used to deal with theor... |

13 |
Nineteenth Century Roots of Algebraic Logic and Universal Algebra”, in Hajnal Andréka
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Citation Context ... it from becoming the frame for an omnicomprehensive deductive system such as set theory. The rehabilitation of map calculus from its disrepute (whose historical causes are skillfully investigated in =-=[1]-=-) has reopened the opportunity, dismissed for decades, to put together complementary virtues of the map calculus and of first-order predicate logic. This paper gives a contribution in this direction. ... |

8 | On a question of A. Salomaa: The equational theory of regular expressions over a singleton alphabet is not finitely based, Theoret - Aceto, Fokkink, et al. |

5 |
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(Show Context)
Citation Context ..., defined as follows: Definition. A sentence ff of L + is said to be in k variables ( k a natural number) if no subformula ' of ff involves more than k distinct free variables. As for expressibility, =-=[13]-=- and [24] classify many sentences as shown in Figure 5. Of these, the ones in the first group are provable in the most varied theories of sets; those in the second group can be interpreted in lattice ... |

4 |
Machine support of relational computations: The Kiel RELVIEW system
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- 1997
(Show Context)
Citation Context ...ough we are eagerly following this approach in order to play, and experiment with, specifications written in the map language, we have in mind to invert the approach in the long run. Like others (cf. =-=[2, 7]-=-), we believe that the map calculus deserves an autonomous and effective instrumentation to be put to the service of first-order reasoning, and of automated reasoning in general. Acknowledgements We a... |

4 | On existentially quantified conjunctions of atomic formulae of L
- Cantone, Cavarra, et al.
- 1997
(Show Context)
Citation Context ...e seems to be no way out of this lack of semantic completeness of the derivability notion for L \Theta . 2 In spite of this shortcoming, L \Theta proves adequate as a support for number theories (cf. =-=[6]-=-) as well as for full-blown theories of sets; in these contexts, it can fairly compete with predicate calculus. In short (as we will see through an example in Sec.5 and will discuss again at the begin... |

3 |
Specifiche formali di proprietà di relazioni: esempi
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- 1997
(Show Context)
Citation Context ...er language employing at most three distinct individual variables. As mentioned, techniques for translating formulae of such a language to the map calculus, and vice-versa, have been studied in depth =-=[16]-=-. Starting with this section and through the sequel of this paper, we expound an interesting case-study: the encoding of the lattice theory in the map calculus. The main aim of this study is to show h... |

2 |
LIBRA: A Lazy Interpreter of Binary Relational Algebra
- Dwyer
- 1995
(Show Context)
Citation Context ...ough we are eagerly following this approach in order to play, and experiment with, specifications written in the map language, we have in mind to invert the approach in the long run. Like others (cf. =-=[2, 7]-=-), we believe that the map calculus deserves an autonomous and effective instrumentation to be put to the service of first-order reasoning, and of automated reasoning in general. Acknowledgements We a... |