Enumerators l Show that a language is decidable

Enumerators l Show that a language is decidable iff some enumerator enumerates the language in lexicographic order. l Show that every infinite recognizable language has an infinite decidable language as a subset.

ﺯﺑﺎﻥ ﻫﺎی ﺗﺼﻤیﻢ پﺬیﺮ M is a Turing machine l Does M take more than k steps on input x? l Does M take more than k steps on some input? l Does M take more than k steps on all inputs? l Does M ever move the tape head more than k cells away from the starting position?

ﺯﺑﺎﻥ ﻫﺎی ﺗﺼﻤیﻢ پﺬیﺮ l {M: M is the description of a Turing machine and L(M) is a Turing recognizable language}

ﺯﺑﺎﻥ ﻫﺎی ﺗﺸﺨیﺺ ﻧﺎپﺬیﺮ Consider the following language L: L = { <M> | for every input string w, M will halt within 1000|w|2 steps } Show that this language is not recognizable. (Reduce from ~ATM. ) complement of

Close look to the formal definition of a TM l Exercise 3. 5: ¡ Can a Turing machine ever write the blank symbol on its tape? ¡ Can the tape alphabet be the same as the input alphabet? ¡ Can a Turing machine's read head ever be in the same location in two successive steps? ¡ Can a Turing machine contain just a single state?

Robustness • doubly infinite tape • k-stack PDAs (k>1) • A Turing machine with only RIGHT and RESET moves • Cyclical Turing machine • A queue automaton • 2(k) head Turing machine • Turing machine with k-dimensional tape × A single tape TM not allowed to change the input -> regular language × Only Right and Stay Put moves -> regular language

Clue to the Solution: input-read-only TM At most the last |Q| squares of input on tape can be determining. Myhill-Nerode theorem l if a language L partitions ∑* into a finite number of equivalence classes then L is regular. l See: http: //www. eecs. berkeley. edu/~tah/172/7. pdf http: //en. wikipedia. org/wiki/Myhill-Nerode_theorem