‫ماشین های تورینگ‪ ،‬تشخیص پذیری و تصمیم پذیری‬
‫زبان ها‬
‫جلسات حل تمرین نظریه زبان ها و ماشین ها‬
‫دانشگاه صنعتی شریف‬
‫بهار ‪87‬‬
Enumerators
 Show that a language is decidable iff some
enumerator enumerates the language in
lexicographic order.
 Show that every infinite recognizable language
has an infinite decidable language as a subset.
‫طراحی تصمیم گیر‬
‫زبان های مکمل‪-‬تشخیص پذیر(‪)co-recognizable‬‬
‫زبان های تصمیم پذیر‬
‫زبان های تصمیم پذیر‬
M is a Turing machine
 Does M take more than k steps on input x?
 Does M take more than k steps on some input?
 Does M take more than k steps on all inputs?
 Does M ever move the tape head more than k
cells away from the starting position?
‫زبان های تصمیم پذیر‬
 {M: M is the description of a Turing machine and L(M) is a Turing
recognizable language}
‫زبان های تصمیم ناپذیر‬
‫زبان های تشخیص ناپذیر‬
‫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
‫طراحی تشخیص دهنده‬
Close look to the formal definition of a TM
 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?
‫خواص بسته بودن‬
‫• زبان های تشخیص پذیر‪:‬‬
‫• اجتماع‬
‫• اشتراک‬
‫• تکرار(*)‬
‫• الحاق‬
‫• زبان های تصمیم پذیر‬
‫• اجتماع‬
‫• اشتراک‬
‫• مکمل گیری‬
‫• تکرار(*)‬
‫• الحاق‬
Robustness
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•
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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
 if a language L partitions ∑* into a finite number of equivalence
classes then L is regular.
 See:
http://www.eecs.berkeley.edu/~tah/172/7.pdf
http://en.wikipedia.org/wiki/Myhill-Nerode_theorem