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LIACS - Second Course . . .I
Chapter 1
Review of
Formal Languages and
Automata Theory
I 1Chomsky hierarchy
grammar automaton3regular
right-linear finite stateA→aB
a2context-free
A→α
pushdown(+lifo stack)1context-sensitive
(β?,A,βr)→α linear bounded monotone0recursively enumerable
turing machineI 2review of formal languages and automata theory
1.1 Sets
1.2 Symbols, strings, and languages1.3 Regular expressions and regular languages1.4 Finite automata1.5 Context-free grammars and languages1.6 Turing machines1.7 Unsolvability1.8 Complexity theory
1.4 Finite automata
MOVED to Chapter 3
1.5 Context-free grammars and languages
MOVED to Chapter 4
1.6 Turing machines
I 3adding one
B B 1 1 0 1 B control unitinput tape reading head r present state begin: r end: ∅symbol state 0 1b r r,0,R r,1,R ?,b,L ∅,1,N ?,0,L ∅,1,L B B 1 1 0 1 B r???? B B 1 1 0 1 B r???? B B 1 1 0 1 B r???? B B 1 1 0 1 B r???? B B 1 1 0 1 B r???? B B 1 1 0 1 B B B 1 1 0 0 B B B 1 1 1 0 BI 4Turing machine
M= (Q,Σ,Γ,δ,q0,h)
Qstates
q0?Qinitial state
h?Qhalting stateΓ tape alphabet
Σ?Γ input alphabetB?Γ-Σ
transition function (partial)δ:Q×Γ→Q×Γ× {L,R,S}
(nondet)δ?Q×Γ×Q×Γ× {L,R,S} wqxconfigurationw,x?Γ?,q?QαZpXβ?αqZY βifδ(p,X) = (q,Y,L)
αpXβ?αY qβifδ(p,X) = (q,Y,R)
αpXβ?αqY βifδ(p,X) = (q,Y,S)
L(M) =
{x?Σ?|q0Bx??αhβfor someα,β?Γ?}I 5TM models
a input tape··· ···δ p finitecontrol working tapesDSPACE(f) NSPACE(f)
space complexity online Turing machine multiple working tapes single sidedδ p finitecontrol··· B BB···
B a B working tapesDTIME(f) NTIME(f)
time complexity input on tape multiple working tapes double sidedI 6Turing computations
- blocks: no move defined'no" - move off tape (one-sided)'no" - infinite computation 'loop"'no"(but we cannot tell) - halt in stateh'yes"RErecursively enumerable
enumerate REC recursive - TM always stopsdecideREC closed complementation, RE is not
equivalent variants: multiple tapes one sided, two sided two-dimensional tape nondeterminismI 7the universal machineTheorem 1.6.1
"It is possible to invent a single machine whichcan be used to compute any computable se-quence. If this machine
U is supplied with a tape on the beginning of which is written the S.D. [=description] of some computing machine M , then U will compute the same sequence as M A.M. Turing,On Computable Numbers, with an Application to the Entscheidungsproblem.Proc. London Math. Soc. Ser. 2