Undergraduate Course ELEMENTS OF COMPUTATION THEORY College of Computer Science

Undergraduate Course ELEMENTS OF COMPUTATION THEORY College of Computer Science

Chapter 4 ZHEJIANG UNIVERSITY

Fall-Winter, 2014

P 191

4.1.2 Let M = (K, ∑

, δ, s, {h}), where K = {q 0 , q ₁ , q ₂ , h }, ∑

= {a, b, ⊔, ◃}, s = q 0 , and δ is given by the following table (the transitions on ◃ are δ(q, ◃) = (q, ◃), and are omitted).

q σ δ(q, σ) q ₀ a (q ₁ , ←) q 0 b (q 0 , →) q ₀ ⊔ (q 0 , →) q ₁ a (q ₁ , ←) q ₁ b (q ₂ , →) q ₁ ⊔ (q 1 , ←) q ₂ a (q ₂ , →) q 2 b (q 2 , →) q ₂ ⊔ (h, ⊔)

(a) Trace the computation of M starting from the configuration (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba).

(b) Describe informally what M does when started in q ₀ on any square of a tape.

Solution:

(a) (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba) ⊢ M (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₀ , ◃abb ⊔bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₀ , ◃abb ⊔ bb⊔ ⊔ ⊔aba)

⊢ M (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₀ , ◃abb ⊔ bb ⊔ ⊔⊔aba)

⊢ M (q ₀ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₁ , ◃abb ⊔ bb ⊔ ⊔⊔aba)

⊢ M (q ₁ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q ₁ , ◃abb ⊔ bb⊔ ⊔ ⊔aba)

⊢ M (q ₁ , ◃abb ⊔ bb ⊔ ⊔ ⊔ aba)

⊢ M (q 2 , ◃abb ⊔ bb⊔ ⊔ ⊔aba)

⊢ M (h, ◃abb ⊔ bb⊔ ⊔ ⊔aba)

(b) M scans right until it finds an a, then left until it finds a b, then right again until it

finds a ⊔, and then halts.

4.1.7 Design and write out in full a Turing machine that scans to the right until it finds two consecutive a’s and then halts. The alphabet of the Turing Machine should be {a, b, ⊔, ◃} .

Solution:

K = {q 0 , q ₁ , h }, ∑

= {a, b, ⊔, ◃}, s = q 0 , H = {h}, δ is given by the following table:.

q σ δ(q, σ) q ₀ a (q ₁ , →) q 0 b (q 0 , →) q ₀ ⊔ (q 0 , →) q ₀ ◃ (q ₀ , →) q ₁ a (h, a) q ₁ b (q ₀ , →) q ₁ ⊔ (q 0 , →) q 1 ◃ (q 0 , →)

4.1.10 Explain what this machine does.

> R −−−→

a ̸=⊔ R −−−→

b ̸=⊔ R _⊔ aR _⊔ b

Solution:

This machine scans to right, remembering the first and the second nonblank symbols (respectively a and b) it encounters. It then continues to the right, writing a in the first blank it encounters, and b in the second.

P 200

4.2.2 Present Turing machines that decide the following languages over {a, b}.

(a) ∅ (b) {e}

(d) {a} ^∗

Solution:

4.2.4 (a) Given an example of a TM with one halting state that does not compute a function from strings to strings.

(b) Given an example of a TM with two halting states, y and n, that does not decide a language.

(c) Can you given an example of a TM with one halting state that does not semidecide a language

Solution:

(c) On each input, a TM either halts or does not. The language semidecided by TM M is simply the set of input strings on which M halts. No Turing can fail to semidecide some language.

P 232

4.6.2 Find grammars that generate the following languages:

(a) {ww : w ∈ {a, b} ^∗ } (b) {a ²

: n ≥ 0} (c) {a ⁿ

: n ≥ 0}

Solution:

(a) V = {a, b, A, B, S, T, U, [, ], $, x}

Σ = {a, b}

R = {S → [T ] T → xT x T → $U U x → AaU U x → BbU xA → Ax xB → Bx [A → a[

[B → b[

[$ → e

U ] → e}

(b) V = {a, S, M, $}

Σ = {a}

R = {S → T a$

T → T M T → e M a → aaM M $ → $

$ → e}

(c) V = {a, S, T, B, C, $}

Σ = {a}

R = {S → $T $ T → BT C T → e BC → CaB

$C$ → $ B$ → $

$ → e}

P 242

4.7.2 Show the following functions are primitive recursive:

(a) f actoria(n) = n!

(b) gcd(m, n), the greast comon divisor of m and n

Solution:

(a) f actoria(n) is the function defined recursively by g(0) = 1 and h(m, r) = (m + 1) · r

(b) gcd(m, n) =

{ n, rem(m, n) = 0,

gcd(n, rem(m, n)), otherwise