Math 2.0

Mathematical logic and foundations Problem

Post correspondence problem

Builds on StringDecidable language

Problem

Let Σ\Sigma be an alphabet with at least two symbols. An instance is a finite list of pairs (u1,v1),…,(un,vn)(u_1, v_1), \dots, (u_n, v_n) of strings over Σ\Sigma. A match is a finite nonempty sequence of indices i1,…,ik∈{1,…,n}i_1, \dots, i_k \in \set{1, \dots, n} such that ui1ui2⋯uik=vi1vi2⋯vik.u_{i_1} u_{i_2} \cdots u_{i_k} = v_{i_1} v_{i_2} \cdots v_{i_k}. Is there an algorithm that decides, for every instance, whether it has a match?

Remarks

Post proved in the paper that introduced the problem that no such algorithm exists: the set of instances with a match is not decidable. The set is recursively enumerable, since one can try all index sequences in order of length and stop when a match appears. The standard proof reduces the halting problem to it: the pairs are built so that a match spells out a halting computation of a given Turing machine. Example: the instance (a,ab),(bc,c)(a, ab), (bc, c) has the match 1,21, 2, since u1u2=abc=v1v2u_1 u_2 = abc = v_1 v_2. The instance (ab,a)(ab, a) alone has no match, since every concatenation of uu's is longer than the corresponding concatenation of vv's. Over a one-letter alphabet the problem is decidable, because only the lengths of the strings matter. The problem is also decidable for instances with n=2n = 2 pairs. Because the problem is purely about strings, it is a convenient source of undecidability for questions about grammars, for example whether two context-free grammars generate languages with a common string.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionFunction, concept, no evidence levelFunctionconceptMonoid, concept, no evidence levelMonoidconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionTuring machine, definition, no evidence levelTuring machinedefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionFree monoid, definition, no evidence levelFree monoiddefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionAlphabet, definition, no evidence levelAlphabetdefinitionString, definition, no evidence levelStringdefinitionFormal language, definition, no evidence levelFormal languagedefinitionDecidable language, definition, no evidence levelDecidable languagedefinitionPost correspondence problem, problem, no evidence levelPost correspondence p…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Function concept
  3. Monoid concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Turing machine definition
  8. Implication definition
  9. Predicate definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Codomain definition
  19. Injective function definition
  20. Free monoid definition
  21. Surjective function definition
  22. Bijective function definition
  23. Equinumerous sets definition
  24. Finite set definition
  25. Alphabet definition
  26. String definition
  27. Formal language definition
  28. Decidable language definition
  29. Post correspondence problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Post_correspondence_problem
About this page
Tag4ARVVS526T
URIm2://4ARVVS526T
Kindproblem
WritingW0 stub
MSC03D35, 68Q45, 68R15
StatusThis copy of the site has no build data. Run m2 status 4ARVVS526T.
AboutDecision problem
UsesString, Decidable language
Same as
Changed3 November 2026

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