## Automatic Proofs and Counterexamples for Some Ortholattice Identities (1998)

### Cached

### Download Links

- [info.mcs.anl.gov]
- [www.cs.unm.edu]
- [ftp.ese-metz.fr]
- DBLP

### Other Repositories/Bibliography

Venue: | Information Processing Letters |

Citations: | 23 - 2 self |

### BibTeX

@ARTICLE{Mccune98automaticproofs,

author = {William Mccune},

title = {Automatic Proofs and Counterexamples for Some Ortholattice Identities},

journal = {Information Processing Letters},

year = {1998},

volume = {65},

pages = {285--291}

}

### OpenURL

### Abstract

This note answers questions on whether three identities known to hold for orthomodular lattices are true also for ortholattices. One identity is shown to fail by MACE, a program that searches for counterexamples, an the other two are proved to hold by EQP, an equational theorem prover. The problems, from work in quantum logic, were given to us by Norman Megill. Keywords: Automatic theorem proving, ortholattice, quantum logic, theory of computation. 1 Introduction An ortholattice is an algebra with a binary operation (join) and a unary operation 0 (complement) satisfying the following (independent) set of identities. x y = (x 0 y 0 ) 0 (definition of meet) x y = y x (x y) z = x (y z) x (x y) = x x 00 = x x (y y 0 ) = y y 0 Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. From these identities one can...

### Citations

308 | Otter 3.0 reference manual and guide
- McCune
- 1994
(Show Context)
Citation Context ... statements without many variables, and it usually cannot find large models. Even with these limitations, it is a valuable complement to our theorem provers. (Our more well-known theorem prover Otter =-=[5, 8]-=- does not seem as effective as EQP for lattice-like problems because Otter lacks associative-commutative unification and several valuable paramodulation strategies. See [7] for several examples of usi... |

97 | A Davis-Putnam program and its application to finite first-order model search: Quasigroup existence problems
- McCune
- 1994
(Show Context)
Citation Context ...strengths is that associativity and commutativity of binary operations are built into the inference rules. This feature makes EQP perform well on many problems involving lattice-like structures. MACE =-=[4]-=- is a program that searches for finite models of first-order statements. In practice, it is limited to fairly simple statements without many variables, and it usually cannot find large models. Even wi... |

64 |
Basic paramodulation and superposition
- Bachmair, Ganzinger, et al.
- 1992
(Show Context)
Citation Context ...ir of equations was selected, then the oldest pair of equations was selected, and so on. In other words, the strategy was four parts shortest-first to one part breadth-first. ffl Basic paramodulation =-=[1]-=-. This restriction on application of the inference rule prevented substitution of terms that arise from instantiation alone. Such inferences are redundant. ffl Prime paramodulation. This restriction p... |

50 | Otter - the CADE-13 competition incarnations
- McCune, Wos
- 1997
(Show Context)
Citation Context ... statements without many variables, and it usually cannot find large models. Even with these limitations, it is a valuable complement to our theorem provers. (Our more well-known theorem prover Otter =-=[5, 8]-=- does not seem as effective as EQP for lattice-like problems because Otter lacks associative-commutative unification and several valuable paramodulation strategies. See [7] for several examples of usi... |

27 | Automated Deduction in Equational Logic and Cubic Curves
- McCune, Padmanabhan
- 1996
(Show Context)
Citation Context ...own theorem prover Otter [5, 8] does not seem as effective as EQP for lattice-like problems because Otter lacks associative-commutative unification and several valuable paramodulation strategies. See =-=[7]-=- for several examples of using Otter for this type of problem.) 2 Equation E1 I had no intuition about whether E1 could be proved for ortholattices. Hence, I put the programs to work in parallel, with... |

24 | 33 basic test problems: A practical evaluation of some paramodulation strategies
- McCune
- 1997
(Show Context)
Citation Context ...is on using various versions of the modus ponens rule with quantum logic. Such inference rules may affect computational efficiency of systems based on quantum logic. 1.1 The Programs EQP and MACE EQP =-=[6]-=- is an automated theorem-proving program for statements in first-order equational logic. It has several strategies for applying equational reasoning and searching for proofs. One of its strengths is t... |

15 |
Binary orthologic with modus ponens is either orthomodular or distributive
- Pavicic, Megill
- 1998
(Show Context)
Citation Context ...((asb 0 )s(asb)))s(a 0s((a 0sb)s(a 0sb 0 ))))) = 1 (E2) (((a 0sb)s(a 0sb 0 ))s(as(a 0sb))) 0s(a 0sb) = 1 (E3) These three equations arose in work on quantum logic by Mladen Pavici'c and Norman Megill =-=[9, 10]-=-. Each equation was known to hold for orthomodular lattices, but it was unknown whether any of them holds for ortholattices. Megill asked whether any of Argonne's automated deduction programs could be... |

13 |
Orthomodularity is not Elementaryā€¯, The
- Goldblatt
- 1984
(Show Context)
Citation Context ...space (non-abelian, non-distributive). 2 However, working with quantum logic has been difficult; for example, decidability of the word problem for orthomodular lattices is unknown. R. Goldblatt states=-=[2]-=-: ... further evidence of the intractability of quantum logic. It is perhaps the first example of a natural and significant logic that leaves the usual methods defeated. The work of Pavici'c and Megil... |

3 |
Orthomodular structures and physical theory
- Marlow
- 1978
(Show Context)
Citation Context ...ked if our programs could prove it. Quantum logic is meant to describe probability of events in quantum dynamics (cf. Boolean algebra describes probability in classical dynamics). A. R. Marlow states =-=[3]-=-: Quantum theory is simply the replacement in standard probability theory of event-as-subset-of-a-set (abelian, distributive) by event-assubspace -of-a-Hilbert-space (non-abelian, non-distributive). 2... |

1 |
Correspondence by electronic mail
- Megill, Sept
- 1997
(Show Context)
Citation Context ...((asb 0 )s(asb)))s(a 0s((a 0sb)s(a 0sb 0 ))))) = 1 (E2) (((a 0sb)s(a 0sb 0 ))s(as(a 0sb))) 0s(a 0sb) = 1 (E3) These three equations arose in work on quantum logic by Mladen Pavici'c and Norman Megill =-=[9, 10]-=-. Each equation was known to hold for orthomodular lattices, but it was unknown whether any of them holds for ortholattices. Megill asked whether any of Argonne's automated deduction programs could be... |

1 |
Correspondence by electronic mail. Fortunately, two of the larger equations (associativity of meet and x (y y ) = y y ) could be omitted, because they depend on the other axioms and lemmas; this allowed the search to run in less than 70 megabytes
- Megill, Sept
- 1997
(Show Context)
Citation Context ...s((asb 0 )s(asb)))s(a 0s((a 0sb)s(a 0sb 0 ))))) = 1 (E2) (((a 0sb)s(a 0sb 0 ))s(as(a 0sb))) 0s(a 0sb) = 1 (E3) These three equations arose in work on quantum logic by Norman Megill and Mladen Pavacic =-=[6]-=-. Each equation was known to hold for orthomodular lattices, but it was unknown whether any of them holds for ortholattices. Megill asked whether any of Argonne's automated deduction programs could be... |