12.3 Decidability in formal languages and automata theory

Decidability in formal languages is all about whether we can create algorithms to solve certain problems. Some problems are decidable, meaning we can always find an answer. Others are undecidable, stumping even the most powerful computers.

Undecidability has big implications. It shows there are limits to what computers can do, affecting how we approach problem-solving. It also connects to the Chomsky hierarchy, helping us understand the complexity of different language types and their related problems.

Decidability and Undecidability in Formal Languages

Concept of decidability

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• Decidability refers to the existence of an algorithm that can always correctly determine whether a problem has a "yes" or "no" answer in a finite amount of time
  • Decidable problems can be solved by Turing machines that halt on all inputs (e.g., determining if a string belongs to a regular language)
• Undecidability occurs when no such algorithm exists, meaning the problem cannot be solved by Turing machines that halt on all inputs
  • Examples of undecidable problems in formal languages include determining if a context-free grammar generates all possible strings (universality problem) or if two context-free grammars generate the same language (equivalence problem)
• Formal languages provide the context in which decidability and undecidability are studied, allowing for the classification and analysis of problems based on their computational complexity

Undecidable problems in automata

• Emptiness problem for Turing machines determines whether a given Turing machine accepts any input string at all
  • Proved undecidable using a reduction from the halting problem, which itself is undecidable
• Halting problem determines whether a given Turing machine halts on a given input
  • Undecidability proved by contradiction, showing that assuming a decider for the halting problem leads to a paradox
• Other undecidable problems in automata theory include:
  • Determining if a Turing machine accepts a specific input string
  • Determining if two Turing machines are equivalent and accept the same language

Implications and Relationships of Undecidability

Implications of undecidability

• Theoretical limitations: Undecidability demonstrates that there are problems no computer can solve, regardless of its power, proving inherent limitations to what can be computed
• Practical implications involve identifying likely intractable or impossible problems, guiding research towards problems with computable solutions
• Philosophical implications raise questions about the nature of computation and the limits of human knowledge, highlighting differences between human intelligence and machine computation

Undecidability vs Chomsky hierarchy

• Chomsky hierarchy classifies formal languages into four types based on the complexity of their generating grammars:
  1. Type 0: Recursively enumerable languages (generated by unrestricted grammars, recognized by Turing machines)
  2. Type 1: Context-sensitive languages (generated by context-sensitive grammars, recognized by linear-bounded automata)
  3. Type 2: Context-free languages (generated by context-free grammars, recognized by pushdown automata)
  4. Type 3: Regular languages (generated by regular grammars, recognized by finite automata)
• Most undecidable problems are found in Type 0 languages, with some in Type 1
  • Type 2 and Type 3 languages have more decidable problems, but some undecidable problems still exist (e.g., the emptiness problem is decidable for context-free languages but undecidable for Type 0)
• Understanding decidability within the Chomsky hierarchy helps in designing and analyzing formal languages and their associated automata, balancing expressiveness and decidability