Skip to main content
SpringerLink
Log in

More Undecidable Problems

  • Chapter
  • First Online:
Book cover Limits of Computation

Part of the book series: Undergraduate Topics in Computer Science ((UTICS))

  • 2726 Accesses

Abstract

In this chapter we will look at more undecidable problems. We show a famous result, Rices theorem, that any non-trivial purely semantical property of programs undecidable. The proof uses a reduction from the Halting problem. Such reductions and the reasoning principles they give rise to are investigated. The concept of a semi-decidable problem is introduced, and it is shown that the Halting problem is semi-decidable by means of the self-interpreter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Henry Gordon Rice (July 18, 1920–April 14, 2003) was an American logician and mathematician. He proved “his” theorem in his doctoral dissertation of 1951.

  2. 2.

    Program verification is actually a thriving discipline and quite considerable progress has been made in the last few decades [21] and has even reached giants like Facebook [4].

  3. 3.

    Wang (20 May 1921–13 May 1995) was born in China but later taught at Harvard where he was also the thesis advisor of Stephen Cook (of whom we will hear more later).

  4. 4.

    Problem reduction is a strategy that human beings or companies use all the time in an informal manner. For instance to “solve” the problem of getting from home to work, one could reduce it for instance to the problem of getting to the train station and on the right train, provided one already has a ticket.

  5. 5.

    That means that for different \(x\ne y\in A\) we can still have \(f(x)=f(y)\).

  6. 6.

    More precisely, “recursive” refers to the class of “partial recursive functions” on numbers (as introduced by Kleene in the late 1930s) which one can show are also a reasonable notion of computation equivalent to all the one we have presented.

  7. 7.

    Again, note that this does not mean we cannot write a program that might produce the answer for a specific input value. If this input is finite then a brute force search will always work. For instance, we can clearly write a program that decides whether a floor of a given fixed size (say \(5\times 5\)) can be tiled with given, say k, tile types since there are \(k^{25}\) possible ways to tile such a floor. Note that there is a combinatorial explosion of the number of possible tilings but this does not concern us in computability. It will concern us in the part about complexity.

  8. 8.

    Haskell and ML are such programming languages but they exclude the type of all types.

  9. 9.

    It may also return false positives, i.e. accept programs that are not type sound, this happens e.g. in the case of division-by-zero.

  10. 10.

    For a good textbook on types see [16].

  11. 11.

    The characteristic function \(\chi \) of a set A of elements of type T is of type \(T\rightarrow \mathbb {B}\) and defined by \(\chi (d) = d\in A\).

  12. 12.

    Tibor Radó (June 2, 1895–December 29, 1965) was a Hungarian mathematician who moved to the USA and worked in Computer Science later in his life. The publication on the Busy Beaver function is maybe his most famous one [19].

  13. 13.

    This function is named after Wilhelm Friedrich Ackermann (29 March 1896–24 December 1962), a German mathematician, and Rózsa Péter, born Politzer, (17 February 1905–16 February 1977), a Hungarian mathematician.

References

  1. Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Z. Phonetik, Sprachwiss. Kommunikationsforsch. 14, 143–192 (1961)

    Google Scholar 

  2. Berger, R.: Undecidability of the domino problem. Mem. AMS 66, 1–72 (1966)

    MathSciNet  MATH  Google Scholar 

  3. Brady, A.H.: The determination of the value of Rado’s noncomputable function \(\Sigma (k)\) for four-state turing machines, Mathematics of Computation, 40(162), 647–665 (1983)

    Google Scholar 

  4. Calcagno, C., Distefano, D., Dubreil, J., Gabi, D., Hooimeijer, P., Luca, M., O’Hearn, P., Papakonstantinou, I., Purbrick, J., Rodriguez, D.: Moving fast with software verification. In: Havelund, K., Holzmann, G., Joshi, R. (eds.) NASA Formal Methods. Lecture Notes in Computer Science, pp. 3–11. Springer International Publishing (2015)

    Google Scholar 

  5. Cantor, D.C.: On the ambiguity problem in Backus systems. J. ACM 9(4), 477–479 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chomsky, N., Schützenberger, M.P.: The algebraic theory of context-free languages. In: Braffort, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, pp. 118–161. North Holland, Amestradam (1963)

    Chapter  Google Scholar 

  7. Culik II, K.: An aperiodic set of 13 Wang tiles. Discrete Math. 160(1–3), 245–251 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Floyd, R.W.: On ambiguity of phrase structure languages. Commun. ACM 5(10), 526–532 (1962)

    Article  MATH  Google Scholar 

  9. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  10. Lin, S., Radó, T.: Computer studies of Turing machine problems. J. ACM 12(2), 196–212 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  11. Markoff, A.: On the impossibility of certain algorithms in the theory of associative system. Academ. Sci. URSS. 55, 83–586 (1947)

    MathSciNet  MATH  Google Scholar 

  12. Matiyasevich, Y.V.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)

    MATH  Google Scholar 

  13. Marxen, H., Bundtrock, J.: Attacking the Busy Beaver 5. Bull. EATCS 40, 247–251 (1990)

    MATH  Google Scholar 

  14. Michel, P.: The Busy Beaver Competition: a historical survey. Available via DIALOG. arXiv:0906.3749 (2012). Accessed 5 June 2015

  15. Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Proc. Steklov Inst. Math. (in Russian). 44, 1–143 (1955)

    Google Scholar 

  16. Pierce, B.: Types and Programming Languages. MIT Press, Cambridge (2002)

    MATH  Google Scholar 

  17. Post, E.: Recursive unsolvability of a problem of thue. J. Symb. Logic 12(1), 1–11 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  18. Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt.(English translation: Stansifer, R.: Presburger’s Article on Integer Arithmetic: Remarks and Translation (Technical Report). TR84-639. Ithaca/NY. Dept. of Computer Science, Cornell University) Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, 1929, pp. 92–101 (1930)

    Google Scholar 

  19. Radó, T.: On non-computable functions. Bell Syst. Tech. J. 41(3), 877–884 (1962)

    Article  MathSciNet  Google Scholar 

  20. Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Trans. Amer. Math. Soc. 74, 358–366 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  21. Silva, V.D., Kroening, D., Weissenbacher, G.: A survey of automated techniques for formal software verification. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 27(7), 1165–1178 (2008)

    Article  Google Scholar 

  22. Soare, R.I.: The history and concept of computability. In: Griffor, E.R. (ed.) Handbook of Computability Theory, pp. 3–36. North-Holland, Amestradam (1999)

    Chapter  Google Scholar 

  23. Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. Sec. Ser. 42, 230–265 (1936)

    MathSciNet  MATH  Google Scholar 

  24. Wells, J.: Typability and type-checking in system F are equivalent and undecidable. Ann. Pure Appl. Logic 98(1–3), 111–156 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, School of Engineering and Informatics, University of Sussex, Brighton, UK

    Bernhard Reus

Authors
  1. Bernhard Reus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernhard Reus .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Reus, B. (2016). More Undecidable Problems. In: Limits of Computation. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-27889-6_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-27889-6_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-27887-2

  • Online ISBN: 978-3-319-27889-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation