Abstract
This essay narrates some of the influences that Alasdair Urquhart has had on computer science at the intersection of automated theorem proving, temporal logic and lattice theory—topics that have no obvious relationship to one another. I illustrate this by showing how Allen’s temporal relations are represented in a system for constraint logic programming over intervals and how the combination of a linear-resolution theorem prover and an interval constraint satisfaction system are connected via a lattice-theoretical semantic model. I also speculate on whether some of Thich Nhat Hanh’s meditation teachings might have been one of the sources of Alasdair’s creative inspirations and preternatural capacity for lateral thinking.
Second Reader I. Düntsch Fujian Normal University & Brock University
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References
Allen, J. F. (1981). An interval-based representation of temporal knowledge. IJCAI, 81, 221–226.
Cleary, J. G. (1987). Logical arithmetic. Future Computing Systems, 2(2), 125–149.
Cook, S. A. (1971). The complexity of theorem-proving procedures. Proceedings of the third annual ACM symposium on Theory of computing (pp. 151–158).
Covington, M. A., Nute, D., & Vellino, A. (1996). Prolog Programming in Depth. Hoboken: Prentice-Hall, Inc.
Dowling, W. F., & Gallier, J. H. (1984). Linear-time algorithms for testing the satisfiability of propositional horn formulae. The Journal of Logic Programming, 1(3), 267–284.
Düntsch, I. (2005). Relation algebras and their application in temporal and spatial reasoning. Artificial Intelligence Review, 23(4), 315–357.
Grätzer, G. (1978). General Lattice Theory. Basel: Birkhäuser.
Haken, A. (1985). The intractability of resolution. Theoretical Computer Science, 39, 297–308.
Hanh, T. N. (2010). The Sun My Heart: Reflections on Mindfulness, Concentration, and Insight (2nd ed.). Berkeley: Parallax Press.
Hill, P., Lloyd, J., & Lloyd, J. W. (1994). The Gödel Programming Language. Cambridge: MIT press.
Older, W. J., & Vellino, A. (1993). Constraint arithmetic on real intervals. In Benhamou, F. & Colmerauer, A. (Eds.), Constraint Logic Programming: Selected Research (pp. 175–195). Cambridge: MIT Press.
Reckhow, R. A. (1975). On the lengths of proofs in the propositional calculus. Ph.D. thesis, University of Toronto.
Rescher, N., & Urquhart, A. (1971). Temporal Logic (Vol. 3). Berlin: Springer Science & Business Media.
Siekmann, J., & Wrightson, G. (1983). Automation of Reasoning Vol. 2: Classical Papers on Computational Logic 1967–1970. Berlin, Heidelberg: Springer.
Tseitin, G. S. (1970). On the complexity of derivation in propositional calculus. In Siekmann, J., Wrightson, G. (Eds.), Automation of Reasoning Vol. 2: Classical Papers on Computational Logic 1967–1970 (pp. 466–483). Berlin: Springer.
Urquhart, A. (1983). Relevant implication and projective geometry. Logique et Analyse, 26(103/104), 345–357.
Urquhart, A. (1984). The undecidability of entailment and relevant implication. Journal of Symbolic Logic, 49(4), 1059–1073.
Urquhart, A. (1987). Hard examples for resolution. Journal of the ACM (JACM), 34(1), 209–219.
Urquhart, A. (1995). The complexity of propositional proofs. Bulletin of Symbolic Logic, 1(4), 425–467.
van Emden, M. H. (2010). Integrating interval constraints into logic programming. Preprint, https://arxiv.org/abs/1002.1422.
van Emden, M. H., & Kowalski, R. A. (1976). The semantics of predicate logic as a programming language. Journal of the ACM (JACM), 23(4), 733–742.
Vellino, A. (1993). The relative complexity of analytic tableaux and sl-resolution. Studia Logica, 52(2), 323–337.
Acknowledgements
I would like to acknowledge the long-time influence of my former colleague William Older at Bell-Northern Research, who, next to Alasdair, was a prominent mentor and showed me by example that one can attend to the abstract mathematics of a problem and without losing track of the practical details of the assembler code that implements its solution. I am very grateful to Ivo Düntsch and Alasdair Urquhart for their detailed comments, suggestions and corrections to drafts of this paper as well as to the editors for their Herculean efforts in putting this whole volume together—in the midst of a global pandemic, no less.
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Appendix
Appendix
Alasdair wrote the letter shown in Figs. 19.4, 19.5, 19.6, 19.7 in the summer of 1984 as I was nearing a first draft of my thesis. He had been studying contradictions generated by Tseitin graphs (discussed in Sect. 19.3) and was looking for a simple way of counting the resolution steps needed to eliminate a variable (edge in a Tseitin graph) to show that resolution proofs for them must grow exponentially. He eventually proved his conjecture (stated on p. 3 of the letter) using his insight that the critical truth- value assignments are an exponential function of the cyclomatic number of Tseitin graphs.
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Vellino, A. (2022). Satisfiability, Lattices, Temporal Logic and Constraint Logic Programming on Intervals. In: Düntsch, I., Mares, E. (eds) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Outstanding Contributions to Logic, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-71430-7_19
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