Theory of Computer Science
C7. Context-Sensitive and Type-0 Languages: Turing Machines
Gabriele R¨ oger
University of Basel
April 3, 2019
Theory of Computer Science
April 3, 2019 — C7. Context-Sensitive and Type-0 Languages: Turing Machines
C7.1 Context-Sensitive and General Grammars C7.2 Turing Machines
C7.3 Summary
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 2 / 22
Overview
Automata &
Formal Languages
Languages
& Grammars
Regular Languages
Context-free Languages
Context-sensitive &
Type-0 Languages
Turing machines
Closure properties
& decidability
C7.1 Context-Sensitive and General Grammars
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 4 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Context-Sensitive and General Grammars
Repetition: (Context-Sensitive) Grammars
Definition (Grammar)
A grammar is a 4-tuple hΣ, V , P , S i with:
I Σ finite alphabet of terminal symbols
I V finite set of variables (with V ∩ Σ = ∅)
I P ⊆ (V ∪ Σ) + × (V ∪ Σ) ∗ finite set of rules
I S ∈ V start variable Type 0 and type 1:
I Every grammar is type 0.
I A grammar is context-sensitive (type-1) if all rules w 1 → w 2 satisfy |w 1 | ≤ |w 2 |.
I
only exception: S → ε is allowed for the start symbol S
if S never occurs on a right-hand side.
C7.2 Turing Machines
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 6 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Overview
Automata &
Formal Languages
Languages
& Grammars
Regular Languages
Context-free Languages
Context-sensitive &
Type-0 Languages
Turing machines
Closure properties
& decidability
Automata for Type-1 and Type-0 Languages?
Finite automata
accept exactly the regular languages, push-down automata exactly the context-free languages. Are there automata models for context-sensitive
and type-0 languages?
Yes! Turing machines German: Turingmaschinen
Picture courtesy of imagerymajestic / FreeDigitalPhotos.net Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 8 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Alan Turing (1912–1954)
Picture courtesy of Jon Callas / wikimedia commons
I British logician, mathematician, cryptanalyst and computer scientist
I most important work (for us):
On Computable Numbers, with an Application to the Entscheidungsproblem
Turing machines
I collaboration on Enigma decryption
I conviction due to homosexuality;
pardoned by Elizabeth II in Dec. 2013
I Turing award most important
science award in computer science
Turing Machines: Conceptually
. . . b a c a c a c a . . .
infinite tape
read-write head
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 10 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Nondeterministic Turing Machine: Definition
Definition (Nondeterministic Turing Machine)
A nondeterministic Turing machine (NTM) is given by a 7-tuple M = hQ , Σ, Γ, δ, q 0 , , E i with:
I Q finite non-empty set of states
I Σ 6= ∅ finite input alphabet
I Γ ⊃ Σ finite tape alphabet
I δ : (Q \ E ) × Γ → P (Q × Γ × {L, R, N}) transition function
I q 0 ∈ Q start state
I ∈ Γ \ Σ blank symbol
I E ⊆ Q end states
German: Turingmaschine, Zust¨ ande, Eingabealphabet, Bandalphabet,
German: Ubergangsfunktion, Startzustand, Blank-Symbol, Endzust¨ ¨ ande
Turing Machine: Transition Function
Let M = hQ, Σ, Γ, δ, q 0 , , Ei be an NTM.
What is the Intuitive Meaning of the Transition Function δ?
hq 0 , b, Di ∈ δ(q , a):
I If M is in state q and reads a, then
I M can transition to state q 0 in the next step,
I replacing a with b,
I and moving the head in direction D ∈ {L, R, N}, where:
I
L: one step to the left,
I
R: one step to the right,
I
N: neutral (no) movement.
q a → b, D q
0Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 12 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Nondeterministic Turing Machine: Example
M = h{q 0 , q 1 , . . . , q 6 }, {1, +, =}, {1, +, =, X, }, δ, q 0 , , {q 6 }i
q
0q
1q
2q
3q
4q
5q
6+ → +, R
1 → X, R 1 → 1, R
+ → +, R
= → =, R X → X, R
1 → X, L
X → X, L
= → =, L
1 → 1, L + → +, L
X → X , R
X → X, R
= → =, R
→ , N
Turing Machine: Configuration
Definition (Configuration of a Turing Machine)
A configuration of a Turing machine M = hQ, Σ, Γ, δ, q 0 , , Ei is given by a triple c ∈ Γ ∗ × Q × Γ + .
German: Konfiguration
Configuration hw 1 , q, w 2 i intuitively means that
I the non-empty or already visited part of the tape contains the word w 1 w 2 ,
I the read-write head is on the first symbol of w 2 , and
I the TM is in state q .
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 14 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Turing Machine Configurations: Example
Example
configuration h BEFORE, q, AFTER i.
. . . B E F O R E A F T E R . . .
q
Turing Machine: Step
Definition (Transition/Step of a Turing Machine) An NTM M = hQ, Σ, Γ, δ, q 0 , , Ei can transition
from configuration c to configuration c 0 in one step (c ` M c 0 ) according to the following rules:
I ha 1 . . . a m , q, b 1 . . . b n i ` M ha 1 . . . a m , q 0 , cb 2 . . . b n i if hq 0 , c, Ni ∈ δ(q, b 1 ), m ≥ 0, n ≥ 1
I ha 1 . . . a m , q, b 1 . . . b n i ` M ha 1 . . . a m−1 , q 0 , a m cb 2 . . . b n i if hq 0 , c, Li ∈ δ(q, b 1 ), m ≥ 1, n ≥ 1
I hε, q, b 1 . . . b n i ` M hε, q 0 , cb 2 . . . b n i if hq 0 , c, Li ∈ δ(q, b 1 ), n ≥ 1
I ha 1 . . . a m , q, b 1 . . . b n i ` M ha 1 . . . a m c, q 0 , b 2 . . . b n i if hq 0 , c, Ri ∈ δ(q, b 1 ), m ≥ 0, n ≥ 2
I ha 1 . . . a m , q, b 1 i ` M ha 1 . . . a m c , q 0 , i if hq 0 , c, Ri ∈ δ(q, b 1 ), m ≥ 0
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 16 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Turing Machines: Reachability of Configurations
Definition (Reachable Configuration)
Configuration c 0 is reachable from configuration c in NTM M (c ` ∗ M c 0 ) if there are configurations c 0 , . . . , c n (n ≥ 0) where
I c 0 = c ,
I c i ` M c i+1 for all i ∈ {0, . . . , n − 1}, and
I c n = c 0 .
German: c
0ist in M von c erreichbar
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Turing Machines: Recognized Words
Definition (Recognized Word of a Turing Machine)
NTM M = hQ, Σ, Γ, δ, q 0 , , Ei recognizes the word w = a 1 . . . a n
iff an accepting configuration (where M is in an end state) is reachable from the start configuration:
M recognizes w iff hε, q 0 , w i ` ∗ M hw 1 , q, w 2 i for some q ∈ E, w 1 ∈ Γ ∗ , w 2 ∈ Γ +
special case: for w = ε the start configuration is hε, q 0 , i
special case:
rather than hε, q 0 , εi
German: M erkennt w , akzeptierende Konfiguration, Startkonfiguration example: blackboard
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 18 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Turing Machines
Turing Machines: Accepted Language
Definition (Accepted Language of an NTM) Let M be an NTM with input alphabet Σ.
The language accepted by M is defined as
L(M ) = {w ∈ Σ ∗ | M recognizes w }.
German: erkannte Sprache
example: blackboard
Exercise
Specify the state diagram of an NTM that accepts language L = { a n b n c n | n ≥ 1}.
Feel free to solve this together with your neighbour.
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 20 / 22
C7. Context-Sensitive and Type-0 Languages: Turing Machines Summary
C7.3 Summary
Summary
I Turing machines only have finitely many states but an unbounded tape as “memory”.
I Alan Turing proposed them as a mathematical model for arbitrary algorithmic computations.
I In this role, we will revisit them in the parts on computability and complexity theory.
Gabriele R¨oger (University of Basel) Theory of Computer Science April 3, 2019 22 / 22
