CPSC

CPSC 313, Fall 2012
Handout 5 — Computability x∈L x∈L

L ∈ REC halts & accepts halts & rejects Recursive
Def 11.2 p 278 §11.1
L ∈ RE halts & accepts —
Recursive enumerable Def 11.1 p 278 §11.1
L ∈ co-RE — halts & accepts
L ∈ RE ⇔ L ∈ co-RE
By def.
Thm 11.4 p 283 §11.1
L ∈ REC ⇔ L ∈ RE and L ∈ RE
L ∈ RE ⇔ ∃ unrestricted grammar ⇐ Thm 11.6 p 285 §11.2 ⇒ Thm 11.7 p 289 §11.2
L Recursive
= L Decidable
= χL Computable
L Not recursive = L Undecidable = χL Noncomputable
REG CF REC RE

RE

REC

Member

CF

Halt

co-RE
Diag
Empty

Equal

Diag
Member
Halt
Empty
Finite
Equal

L1 ∪ L2
L1 ∩ L2
L
L1 \ L2
L1 L2
L∗
LR prefix(L) h(L) λ-free h(L)

Finite

=
=
=
=
=
=

M
M, w
M, w
M
M
M1 , M2

| M ∈ L(M )
& M is a TM
| w ∈ L(M )
& M is a TM
| M halts on input w & M is a TM
| L(M ) = ∅
& M is a TM
| L(M ) is finite
& M is a TM
| L(M1 ) = L(M2 )
& M1 and M2 are TMs

(1)
(2)
(3)

(4)
(5)

(6)

Thm 11.3 p 281 §11.1
Sec 10.4 p 269 §10.4
Def 12.1 p 301 §12.1
Thm 12.3 p 309 §12.2
Thm 12.4 p 309 §12.2
Finite

Diag
Member
Halt
Halt
Member
Member
Empty

≤m
≤m
≤m
≤m
≤m
≤m
≤m

Member
Empty
Thm 12.3 p 309 §12.2
Finite
Thm 12.4 p 309 §12.2
Finite
Equal
Halt
Equal

Halt
Diag

Member

Finite

Empty

Equal

Equal

Language A ⊆ Σ∗ reduces to language B ⊆ Σ∗ if there is a computable function f : Σ∗ → Σ∗
A
B
A
B such that w ∈ A ⇔ f (w) ∈ B for all strings w ∈ Σ∗ . p 304 §12.1
A
If A reduces to B, denoted A ≤m B, then problem A is at most as difficult as problem B.
(A ≤m B and A ∈ RE) ⇒ B ∈ RE (If problem A is hard, so is B.)
(A ≤m B and B ∈ RE) ⇒ A ∈ RE (If problem B is easy, so is A.)
(A ≤m B) ⇔ (A ≤m B)
Transitivity: A ≤m B and B ≤m C ⇒ A ≤m C