Post correspondence problem
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| Post correspondence problem | |
|---|---|
| Name | Post correspondence problem |
| Inventor | Emil Post |
| Year | 1946 |
| Field | Computability theory |
| Related | Turing machine, Hilbert's tenth problem, Rice's theorem |
Post correspondence problem is a decision problem in Computability theory introduced by Emil Post in 1946 concerning matching sequences of finite strings. It is a classical example used alongside Turing machine, Church–Turing thesis, Alan Turing-era results to illustrate undecidability, and it connects to examples across Mathematical logic, Automata theory, Formal language theory, and Recursive function theory.
Definition
The Post correspondence problem is stated for two finite lists of nonempty finite words over a common alphabet: given lists A = (a1, a2, ..., an) and B = (b1, b2, ..., bn), determine whether there exists a nonempty sequence of indices i1, i2, ..., ik such that the concatenation ai1 ai2 ... aik equals bi1 bi2 ... bik. Emil Post formulated this construction in correspondence with problems explored in Decision problem discussions contemporary to Hilbert's problems and motivations related to Kurt Gödel-style incompleteness results. The formulation is often presented when teaching reductions from Turing machine halting questions, or when demonstrating hardness results comparable to those in Hilbert's tenth problem and contributions by Alonzo Church.
Examples
Simple examples illustrate solvable and unsolvable instances: for small alphabets and short lists one can hand-construct matching sequences, a technique taught alongside exercises referencing Noam Chomsky-inspired grammars, John von Neumann-style automata constructions, and standard textbook instances used by authors linked to Michael Sipser and Hopcroft and Ullman. Classic pedagogical instances relate to encodings of computations of a given Turing machine or simulation of rewriting systems studied by A. A. Markov (mathematician) and Andrey Kolmogorov. Counterexamples and crafted solvable cases are sometimes compared in lectures referencing David Hilbert and Emil Post historical notes.
Decidability and Undecidability
Post proved that the general Post correspondence problem is undecidable by a reduction from the halting problem for Turing machines, aligning the result with undecidability results from Alonzo Church and Alan Turing. The undecidability is a central result in Computability theory curricula and is frequently presented alongside reductions to or from Rice's theorem, Word problem (group theory), and insolvability results connected to Hilbert's tenth problem. The undecidable nature implies there is no algorithm, uniform across all instances, that always decides existence of a matching sequence, a consequence taught in courses influenced by Kurt Gödel and later expositions by Martin Davis and Hilary Putnam.
Reductions and Related Problems
PCP serves as a source problem for reductions proving undecidability of numerous problems: reductions target the emptiness problem for intersection of Context-free grammar languages, reachability problems in Petri nets, tiling problems as in Wang tile-based proofs, conjugacy problems in Semigroup and Monoid theory contexts, and decision problems in Combinatorial group theory such as the Word problem (group theory). The problem is related to the Mortality problem for matrix semigroups, constrained tiling formulations used by Hao Wang, and to questions in Symbolic dynamics and Cellular automaton reachability studied by researchers connected to John Conway and Stephen Wolfram.
Variants and Extensions
Many variants alter alphabet size, allow bounded or synchronizing constraints, or demand circular matching; these extensions have been analyzed in literature connected to Context-free grammar restrictions, Regular language intersections, and bounded PCP instances appearing in works tied to Eugene W. Myers and contributors influenced by Michael O. Rabin. Notable variants include the Post correspondence problem with a designated starting tile, the marked PCP variant used in reductions to problems about Matrix mortality and Trace monoid equations, and two-dimensional variants that relate to Wang tile tiling and decidability questions studied in the tradition of Hao Wang and Berger.
Complexity and Restricted Cases
While general PCP is undecidable, a variety of restricted instances are decidable and have complexity classifications connected to problems in Automata theory and Formal language theory. For example, PCP over a unary alphabet reduces to periodicity questions treated in results associated with Fine and Wilf-style theorems and can be decided via algorithms related to finite automata constructions found in texts by Hopcroft and Ullman and Michael Sipser. Bounded PCP (limiting the number of tiles) is decidable and its complexity ties to NP and PSPACE results discussed in complexity theory literature involving Stephen Cook and Richard Karp. Other restrictions yielding decidability include bounding the number of tiles, restricting tile morphisms to length-preserving forms, or placing constraints that align instances with solvable decision problems in Combinatorics on words and results attributed to Jean Berstel and Maxime Crochemore.