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Revisiting MU-Puzzle. A Case Study in Finite Countermodels Verification

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Reachability Problems (RP 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11123))

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Abstract

In this paper we consider well-known MU puzzle from Goedel, Escher, Bach: An Eternal Golden Braid book by D. Hofstadter, as an infinite state safety verification problem for string rewriting systems. We demonstrate fully automated solution using finite countermodels method (FCM). We highlight advantages of FCM method and compare it with alternatives methods using regular invariants.

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Notes

  1. 1.

    They also notice that Hofstadter was aware about the decision procedure, but never formally wrote a proof.

  2. 2.

    We used Prover9 and Mace4 version 0.5 (December 2007) [9] running on AMD A6-3410MX APU 1.60Ghz, RAM 4 GB, Windows 7 Enterprise.

  3. 3.

    In a reasonably defined partial order. Instead of a partial order \(\le \) motivated by the iterative finite model building procedure, one may consider a partial order defined by inclusion of corresponding languages.

References

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Alexei Lisitsa

Authors
  1. Alexei Lisitsa
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Correspondence to Alexei Lisitsa .

Editor information

Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Igor Potapov

  2. Aix-Marseille University, Marseille, France

    Pierre-Alain Reynier

Appendix

Appendix

To the Proof of Proposition 4. We present here all minimal countermodels found by Mace4 for all formulae \(T_{MIU} \wedge C_{IV}\rightarrow \phi '_{d}\) with \(\phi '_{d}\) being two disjunct subformulae of \(\varphi _{d}^{C_{IV}} \equiv \exists x \exists y T (M * x * M *y) \vee \exists x T(I*x) \vee \exists x T(U * x) \vee \exists x (T(M*x) \wedge C(x))\). Notice that all of them are less wrt \(\le \) than a minimal countermodel for \(T_{MIU} \wedge C_{IV}\rightarrow \varphi _{d}^{C_{IV}}\) whose domain size is 8.

(1) \(\phi '_{d} \equiv \exists x \exists y T (M * x * M *y) \vee \exists x T(I*x)\)

figure h

(2) \(\phi '_{d} \equiv \exists x \exists y T (M * x * M *y) \vee \exists x T(U*x)\)

figure i

(3) \(\phi '_{d} \equiv \exists x \exists y T (M * x * M *y) \vee \exists x (T(M*x) \wedge C(x))\)

figure j

(4) \(\phi '_{d} \equiv \exists x T(I*x) \vee \exists x T(U*x)\)

figure k

(5) \(\phi '_{d} \equiv \exists x T(I*x) \vee \exists x (T(M*x) \wedge C(x))\)

figure l

(6) \(\phi '_{d} \equiv \exists x T(U*x) \vee \exists x (T(M*x) \wedge C(x))\)

figure m

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Lisitsa, A. (2018). Revisiting MU-Puzzle. A Case Study in Finite Countermodels Verification. In: Potapov, I., Reynier, PA. (eds) Reachability Problems. RP 2018. Lecture Notes in Computer Science(), vol 11123. Springer, Cham. https://doi.org/10.1007/978-3-030-00250-3_6

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  • DOI: https://doi.org/10.1007/978-3-030-00250-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00249-7

  • Online ISBN: 978-3-030-00250-3

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