Undecidable Problems (unsolvable problems) Fall 2006 Costas Busch - RPI

Decidable Languages Recall that: A language is decidable, if there is a Turing machine (decider) that accepts the language and halts on every input string Decision On Halt: Turing Machine YES Accept Input string Decider for NO Reject Fall 2006 Costas Busch - RPI

A computational problem is decidable if the corresponding language is decidable We also say that the problem is solvable Fall 2006 Costas Busch - RPI

Corresponding Language: (Decidable) Problem: Does DFA accept the empty language ? Corresponding Language: (Decidable) Description of DFA as a string (For example, we can represent as a binary string, as we did for Turing machines) Fall 2006 Costas Busch - RPI

Determine whether there is a path from Decider for : On input : Determine whether there is a path from the initial state to any accepting state DFA DFA Decision: Accept Reject Fall 2006 Costas Busch - RPI

Corresponding Language: (Decidable) Problem: Does DFA accept a finite language? Corresponding Language: (Decidable) Fall 2006 Costas Busch - RPI

Check if there is a walk with a cycle Decider for : On input : Check if there is a walk with a cycle from the initial state to an accepting state DFA DFA infinite finite Decision: Accept Reject (NO) (YES) Fall 2006 Costas Busch - RPI

Corresponding Language: (Decidable) Problem: Does DFA accept string ? Corresponding Language: (Decidable) Fall 2006 Costas Busch - RPI

Decider for : On input string : Run DFA on input string If accepts Then accept (and halt) Else reject (and halt) Fall 2006 Costas Busch - RPI

accept the same language? Problem: Do DFAs and accept the same language? Corresponding Language: (Decidable) Fall 2006 Costas Busch - RPI

Let be the language of DFA Decider for : On input : Let be the language of DFA Construct DFA such that: (combination of DFAs) Fall 2006 Costas Busch - RPI

and Fall 2006 Costas Busch - RPI

or Fall 2006 Costas Busch - RPI

which is a solvable problem for DFAs: Therefore, we only need to determine whether which is a solvable problem for DFAs: Fall 2006 Costas Busch - RPI