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International Conference on Implementation and Application of Automata

CIAA 2021: Implementation and Application of Automata pp 14–24Cite as

Say No to Case Analysis: Automating the Drudgery of Case-Based Proofs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12803))

Abstract

I present an argument that long, tedious proofs requiring a human to check many cases should be replaced by an algorithm, so a computer can do the work instead.

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Notes

  1. 1.

    An example is “This statement has no proof in Peano arithmetic with less than \(10^{100}\) symbols.”.

  2. 2.

    The Fibonacci representation of a natural number \(n\) is a finite binary string \(a_1 a_2 \cdots a_t\) such that \(n = \sum _{1\le i \le t} a_i F_{t+2-i}{} \), and \(a_i a_{i+1} = 0{ for}1 \le i <t\).

  3. 3.

    For example, “Beware of bugs in the above code; I have only proved it correct, not tried it.”– Donald Knuth.

References

  1. Aigner, M., Ziegler, G.M.: Proofs from THE BOOK, 5th edn. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44205-0

    Book  MATH  Google Scholar 

  2. Appel, K., Haken, W.: Every Planar Map is Four-Colorable. Contemporary Mathematics, vol. 98. American Mathematical Society (1989)

    Google Scholar 

  3. Bruyère, V., Hansel, G., Michaux, C., Villemaire, R.: Logic and \(p\)-recognizable sets of integers. Bull. Belg. Math. Soc. 1,191–238 (1994), corrigendum, Bull. Belg. Math. Soc. 1 (1994), 577

    Google Scholar 

  4. Charlier, E., Domaratzki, M., Harju, T., Shallit, J.: Composition and orbits of language operations: finiteness and upper bounds. Int. J. Comput. Math. 90, 1171–1196 (2013)

    Article  MathSciNet  Google Scholar 

  5. Cilleruelo, J., Luca, F.: Every positive integer is a sum of three palindromes (2016). preprint available at https://arxiv.org/abs/1602.06208v1

  6. Cilleruelo, J., Luca, F., Baxter, L.: Every positive integer is a sum of three palindromes. Math. Comp. 87, 3023–3055 (2018)

    Article  MathSciNet  Google Scholar 

  7. Currie, J.D., Saari, K.: Least periods of factors of infinite words. RAIRO Info. Theor. Appl. 43, 165–178 (2009)

    Article  MathSciNet  Google Scholar 

  8. Dean, R.A.: A sequence without repeats on \(x, x^{-1}, y, y^{-1}\). Amer. Math. Monthly 72, 383–385 (1965)

    Article  MathSciNet  Google Scholar 

  9. Dyson, F.: What do you believe is true even though you cannot prove it? (2005). https://www.edge.org/response-detail/11675

  10. Entringer, R.C., Jackson, D.E., Schatz, J.A.: On nonrepetitive sequences. J. Combin. Theory Ser. A 16, 159–164 (1974)

    Article  MathSciNet  Google Scholar 

  11. Fici, G., Zamboni, L.Q.: On the least number of palindromes contained in an infinite word. Theoret. Comput. Sci. 481, 1–8 (2013)

    Article  MathSciNet  Google Scholar 

  12. Fleischer, L., Shallit, J.: Automata, palindromes, and reversed subwords (2020). Manuscript under submission

    Google Scholar 

  13. Gosper Jr., R.W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 75, 40–42 (1978)

    Article  MathSciNet  Google Scholar 

  14. Heule, M.: Schur number five. In: Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), pp. 6598–6606. AAAI Press (2018)

    Google Scholar 

  15. Heule, M.J.H., Kullmann, O., Marek, V.W.: Solving and verifying the boolean Pythagorean triples problem via cube-and-conquer. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 228–245. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_15

    Chapter  MATH  Google Scholar 

  16. Kawsumarng, S., Khemaratchatakumthorn, T., Noppakaew, P., Pongsriiam, P.: Sumsets associated with Wythoff sequences and Fibonacci numbers. Period. Math. Hung. 82(1), 98–113 (2020). https://doi.org/10.1007/s10998-020-00343-0

    Article  MathSciNet  MATH  Google Scholar 

  17. Madhusudan, P., Nowotka, D., Rajasekaran, A., Shallit, J.: Lagrange’s theorem for binary squares. In: Potapov, I., Spirakis, P., Worrell, J. (eds.) 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), pp. 18:1–18:14. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2018)

    Google Scholar 

  18. Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Symposium on Switching and Automata Theory (SWAT), pp. 125–129. IEEE Computer Society (1972)

    Google Scholar 

  19. Mousavi, H.: Automatic theorem proving in Walnut (2016). arxiv preprint. http://arxiv.org/abs/1603.06017

  20. Petkovšek, M., Wilf, H., Zeilberger, D.: \(A=B\). A. K. Peters (1996)

    Google Scholar 

  21. Rajasekaran, A., Shallit, J., Smith, T.: Additive number theory via automata theory. Theor. Comput. Syst. 64, 542–567 (2020)

    Article  MathSciNet  Google Scholar 

  22. Rajasekaran, A., Smith, T., Shallit, J.: Sums of palindromes: an approach via automata. In: Niedermeier, R., Vallée, B. (eds.) 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), pp. 54:1–54:12. Leibniz International Proceedings in Informatics, Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2018)

    Google Scholar 

  23. Rampersad, N., Shallit, J.: Words avoiding reversed subwords. J. Combin. Math. Combin. Comput. 54, 157–164 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Shallit, J.: Sumsets of Wythoff sequences, Fibonacci representation, andbeyond (2021). Period. Math. Hung., to appear

    Google Scholar 

  25. Tefera, A.: What is a Wilf-Zeilberger pair? Not. Am. Math. Soc. 57, 508–509 (2010)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jeffrey Shallit

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  1. Jeffrey Shallit
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Correspondence to Jeffrey Shallit .

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Editors and Affiliations

  1. University of Bremen, Bremen, Germany

    Sebastian Maneth

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Shallit, J. (2021). Say No to Case Analysis: Automating the Drudgery of Case-Based Proofs. In: Maneth, S. (eds) Implementation and Application of Automata. CIAA 2021. Lecture Notes in Computer Science(), vol 12803. Springer, Cham. https://doi.org/10.1007/978-3-030-79121-6_2

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  • DOI: https://doi.org/10.1007/978-3-030-79121-6_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-79120-9

  • Online ISBN: 978-3-030-79121-6

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