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The inevitability of inconsistent abstract spaces

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Abstract

Abstraction has been widely used in automated deduction; a major problem with its use is that the abstract space can be inconsistent even though the ground space is consistent. We show that, under certain very weak conditions true of practically all the abstractions used in the past (but true also of a much wider class of abstractions), this problem cannot be avoided.

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References

  1. Bledsoe, W. W. and Tyson, M., ‘The UT interactive prover’. Technical report, Mathematics Department, University of Texas (1975) ATP-17.

  2. Doyle, R. J., ‘Constructing and refining causal explanations from an inconsistent domain theory’, inProc. Fifth National Conference on Artificial Intelligence, pp. 538–544, Philadelphia (1986) AAAI.

  3. Giunchiglia, F. and Giunchiglia, E., ‘Building complex derived inference rules: A decider for the class of prenex universal-existential formulas’, inProc. 7th European Conference on Artificial Intelligence (1988) pp. 607–609. Extended version available as DAI Research Paper 359, Dept. of Artificial Intelligence, Edinburgh.

  4. Green, C., ‘Application of theorem proving to problem solving’, inProc. 1st IJCAI Conference, p. 219–239, International Joint Conference on Artificial Intelligence (1969).

  5. Giunchiglia, F. and Walsh, T., ‘Abstract theorem proving’, inProc. IJCAI 89, (1989) pp. 372–377. IRST Technical Report 8902-03. Also available as DAI Research Paper No. 430, University of Edinburgh.

  6. Giunchiglia, F. and Walsh, T., ‘Theorem proving with definitions’, inProc. of the 7th Conference of the Society for the Study of Artificial Intelligence and Simulation of Behaviour (1989) pp. 175–183. Also available as IRST Technical Report 8901-03 and DAI Research Paper No. 429, Dept. of Artificial Intelligence, Edinburgh.

  7. Giunchiglia, F. and Walsh, T., ‘A theory of abstraction’,Artificial Intelligence Journal 56(2–3), 323–390 (1992). Also IRST-Technical Report 9001-14, IRST, Trento, Italy.

    Google Scholar 

  8. Hobbs, J. R., ‘Granularity’, inProc. 9th IJCAI Conference, pp. 432–435. International Joint Conference on Artificial Intelligence (1985).

  9. Mendelson, E.,Introduction to Mathematical Logic, Van Nostrand Reinhold (1964).

  10. McCarthy, J. and Hayes, P., ‘Some philosophical problems from the standpoint of artificial intelligence’, in B. Meltzer and D. Michie (Eds.),Machine Intelligence, pp. 463–502. Edinburgh University Press (1969).

  11. Nilsson, N. J.,Principles of Artificial Intelligence, Tioga Publishing Co. (1980).

  12. Newell, A. and Simon, H. A.,Human Problem Solving, Prentice-Hall (1972).

  13. Newell, A., Shaw, J. C. and Simon, H. A., ‘Empirical explorations of the logic theory machine’, in H. Fiegenbaum and K. Feldman (Eds.),Computers & Thought, pp. 134–152, McGraw-Hill (1963).

  14. Plaisted, D. A., ‘Abstraction mappings in mechanical theorem proving’, inProc. 5th Conference on Automated Deduction, pp. 264–280 (1980).

  15. Plaisted, D. A., ‘Theorem proving with abstraction’,Artificial Intelligence,16, 47–108 (1981).

    Google Scholar 

  16. Plaisted, D. A., ‘Abstraction using generalization functions’, inProc. 8th Conference on Automated deduction, pp. 365–376 (1986).

  17. Plummer, D., ‘Grazing: Controlling the use of rewrite rules’, Ph.D. thesis, Dept. of Artificial Intelligence, University of Edinburgh (1987).

  18. Prawitz, D.,Natural Deduction-A Proof Theoretical Study. Almquist and Wiksell, Stockholm (1965).

    Google Scholar 

  19. Prawitz, D., ‘Ideas and results in proof theory’, in J. E. Fenstad (Ed.),Proc. 2nd Scandinavian Logic Symposium. North Holland (1971).

  20. Rich, E.,Artificial Intelligence. McGraw-Hill, New York (1983).

    Google Scholar 

  21. Sacerdoti, E. D., ‘Planning in a hierarchy of abstraction spaces’,Artificial Intelligence 5, 115–135 (1974).

    Google Scholar 

  22. Tenenberg, J. D., ‘Preserving consistency across abstraction mappings’, inProc. 10th IJCAI Conference, pp. 1011–1014 (1987).

  23. Tenenberg, J. D., ‘Abstraction in planning’, Ph.D. Thesis, Computer Science Department, University of Rochester (1988). Also TR 250.

  24. Unruh, A. and Rosenbloom, P., ‘Abstraction in problem solving and learning’, inProc. 11th IJCAI Conference (1989) pp. 681–687.

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Authors and Affiliations

  1. Mechanized Reasoning Group, IRST, Povo, 38100, Trento, Italy

    Fausto Giunchiglia

  2. DIST, University of Genoa, Genoa, Italy

    Fausto Giunchiglia

  3. Department of Artificial Intelligence, University of Edinburgh, 80 South Bridge, EH1 1HN, Edinburgh, Scotland

    Toby Walsh

Authors
  1. Fausto Giunchiglia
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  2. Toby Walsh
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Giunchiglia, F., Walsh, T. The inevitability of inconsistent abstract spaces. J Autom Reasoning 11, 23–41 (1993). https://doi.org/10.1007/BF00881899

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  • DOI: https://doi.org/10.1007/BF00881899

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