Sudan University of Science and Technology

By: Haitham Abdel Moniem

Undecidable Problems

Supervisor: Dr. Mohamed El-Hafiz

Recursive and Recursively Enumerable Languages

• The language accepted by Turing machines are called recursively enumerable (RE), and the subset of RE languages that are accepted by a TM that always halts are called recursive.

Complements of Recursive and RE Languages

• The recursive languages are closed under complementation, and if a language and its complement are both RE, then both languages are actually recursive. Thus, the complement of an RE-but-not-recursive language can never be RE.

Decidability and Undecidability

• “Decidable” is synonym for “recursive” although we tend to refer to languages as “recursive” and problems as “decidable” . If a language is not recursive, then we call the problem expressed by that language is “undecidable”.

The Language Ld

• This language is the set of strings of 0’s and 1’s that, when interpreted as a TM , are not in the language of that TM. The language Ld is a good example of a language that is not RE; i.e., no Turing machine accept it.

The Universal Language

• The language Lu consists of strings that are interpreted as a TM followed by an input for that TM. The string is in the Lu if the TM accepts that input. Lu is a good example of a language that is RE but not recursive.

Rice’s Theorem

• Any nontrivial property (not empty or not all RE) of the languages accepted by Turing machines is undecidable. For instance, the set of codes for Turning machines whose language is empty is undecidable by Rice’s theorem. In fact, this language is not RE, although its complement – the set of codes for TM’s that accept at least one string – is RE but not recursive.

Post's Correspondence Problem

• This question asks, given two lists of the same number of strings, whether we can pick a sequence of corresponding string form the two lists and form the same string by concatenation. PCP is an important example of an undecidable problem. PCP is a good choice for reducing to other problems and thereby proving them undecidable.

Undecidable Context-Free-Language Problems

• By reduction from PCP, we can show a number of questions about CFL’s or their grammars to be undecidable. For instance, it is undecidable whether a CFG is ambiguous, whether one CFL is contained in another, or whether the intersection of two CFL’s is empty.

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