12.3 Decidability in formal languages and automata theory
3 min read•july 22, 2024
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
Top images from around the web for Concept of decidability
CS 340: Lecture 8: Decidability and the Halting Problem View original
Is this image relevant?
Turing machine - Wikipedia, the free encyclopedia View original
Is this image relevant?
CS 340: Lecture 8: Decidability and the Halting Problem View original
Is this image relevant?
CS 340: Lecture 8: Decidability and the Halting Problem View original
Is this image relevant?
Turing machine - Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 3
Top images from around the web for Concept of decidability
CS 340: Lecture 8: Decidability and the Halting Problem View original
Is this image relevant?
Turing machine - Wikipedia, the free encyclopedia View original
Is this image relevant?
CS 340: Lecture 8: Decidability and the Halting Problem View original
Is this image relevant?
CS 340: Lecture 8: Decidability and the Halting Problem View original
Is this image relevant?
Turing machine - Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 3
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 () or if two context-free grammars generate the same language ()
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
for Turing machines determines whether a given Turing machine accepts any input string at all
Proved undecidable using a reduction from the , 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:
Type 0: (generated by unrestricted grammars, recognized by Turing machines)
Type 1: Context-sensitive languages (generated by context-sensitive grammars, recognized by linear-bounded automata)
Type 2: (generated by context-free grammars, recognized by )
Type 3: Regular languages (generated by regular grammars, recognized by )
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