Programme Reduction - applications Prove non-decidable properties - PowerPoint PPT Presentation

Programme Reduction - applications • Prove non-decidable properties of TMs • Theorem 11.7 Acc- Λ = {e(T) | Λ ∈ L(T) } is not recursive – Reduction technique Λ • Describe non-decidable properties of other universal Proof: Show Acc Acc- ≤ formalisms – Chomsky grammars • Theorem – Java Let T U denote the universal Turing Machine, then • Prove non-decidable properties of non-universal Uni-Acc = {e(w) | w ∈ L(T U ) } is not recursive formalisms – Games – Context-free grammars Proof: Show Acc Uni-Acc ≤ dBerLog 2007 1 dBerLog 2007 2 Reduction - applications Rice’s Theorem - definition • Theorem 11.8 • Definition AccSome = {e(T) | L(T) is nonempty} is not recursive Proof: Show Acc- Λ AccSome A property of languages is said to be nontrivial iff it is ≤ satisfied by some but not all recursively enumerable AccEver = {e(T) | L(T)’ is empty} is not recursive languages Proof: Show Acc- Λ AccEver ≤ Subset = {e(T 1 )e(T 2 ) | L(T 1 ) ⊆ L(T 2 )} is not recursive Proof: Show Acc-Ever Subset ≤ dBerLog 2007 3 dBerLog 2007 4

Nontrivial language properties Rice’s Theorem ∀ Λ ∈ L • Theorem 11.9 • L = Ø Let R be any nontrivial property of languages, then • L = Σ * P R = {e(T) | L(T) has property R} is not recursive! Proof: Show Acc- Λ P • L is finite ≤ R • L is regular • All strings in L have even length dBerLog 2007 5 dBerLog 2007 6 Reduction: Acc- Λ P Reduction: Acc- Λ P R ≤ ≤ R • Construct TM accepting Acc- Λ • Assume you had TM accepting P R Acc- Λ Y Y Y Λ ∈ L(T) L(T) sat R L(T’) sat R P R P R e(T) e(T) e(T’) N N N Λ ∉ L(T) L(T) viol R L(T’) viol R Assume Ø viol R. Then T R exists s.t. L(T R ) sat R Construct T’ s.t. if Λ ∉ L(T) then L(T’)=Ø else L(T’)= L(T R ) dBerLog 2007 7 dBerLog 2007 8

Programme Harel diagonalization • Prove non-decidable properties of TMs Assume halting problem solvable in JAVA – Reduction technique • Describe non-decidable properties of other universal prog in formalisms – Chomsky grammars P – Java • Prove non-decidable properties of non-universal formalisms FINE LOOP if prog(in) ↓ if prog(in) ↑ – Context-free grammars dBerLog 2007 9 dBerLog 2007 10 Harel diagonalization Harel diagonalization prog Q Construct program Q Run Q with input Q COPY COPY prog prog Q Q P P Q Q FINE LOOP FINE LOOP LOOP LOOP dBerLog 2007 11 dBerLog 2007 12

Busy Beaver Chomsky grammars • A Chomsky grammar is a tuple G = (V, Σ , S, P), where • Definition BB( n ):= the maximal number of 1’s printed by a Turing V and Σ are finite disjoint sets of variables and terminals resp. machine starting with n 1s on the tape and with n states S is the start variable , an element of V P is a set of productions of the form • Exercise Type 3 : A → a or A → aB, where A, B ∈ V and a ∈ Σ BB is not computable! Type 2 : A → β , where A ∈ V and β ∈ (V ∪ Σ )* Type 0 : α → β , where α ∈ (V ∪ Σ )*V (V ∪ Σ )* and β ∈ (V ∪ Σ )* dBerLog 2007 13 dBerLog 2007 14 Chomsky type 0 languages Chomsky type 0 example • Given a Chomsky type 0 grammar G = (V, Σ , S, P), define • Let G = (V, Σ , S, P), where V = {S, A, B, C} Σ = {a, b, c} if α → β ∈ P, then for all α ’, α ’’, β ’, β ’’ ∈ (V ∪ Σ )* P: S → FT α ’ α α ’’ ⇒ β ’ β β ’’ T → ABCT T → ABC BA → AB CA → AC CB → BC L(G) = {w ∈ Σ *  S ⇒ * w} FA → a aA → aa aB → ab bB → bb bC → bc cC → cc L(G) = {a i b i c i  i > 0} dBerLog 2007 15 dBerLog 2007 16

Turing and Chomsky Programme • Prove non-decidable properties of TMs • Theorems 10.8 and 10.9 – Reduction technique For any language L ⊆ Σ *, • Describe non-decidable properties of other universal L is generated by a Chomsky type 0 grammar formalisms iff (constructively!!) – Chomsky grammars L is accepted by a Turing machine – Java • Prove non-decidable properties of non-universal formalisms • Corollary – Games All nontrivial properties of languages for Chomsky type 0 – Context-free grammars grammars are undecidable! dBerLog 2007 17 dBerLog 2007 18 Post’s correspondence problem - example Post’s correspondence problem - example • List A: List B: • List A: List B: α 1 = b β 1 = bbb α 1 = b β 1 = bbb α 2 = babbb β 2 = ba α 2 = babbb β 2 = ba α 3 = ba β 3 = a α 3 = ba β 3 = a Does there exist a sequence of indices i1, i2,..., im ∈ {1,2,3} such that Solution? α i1 α i2 ...... α im = β i1 β i2 ...... β im dBerLog 2007 19 dBerLog 2007 20

Post’s correspondence problem - example Post’s correspondence problem - example • List A: • List A: List B: List B: α 1 = b β 1 = bbb α 1 = ba β 1 = bab α 2 = babbb β 2 = ba α 2 = abb β 2 = bb α 3 = ba β 3 = a α 3 = bab β 3 = abb Solution? YES: 2 1 1 3 Solution? α 2 α 1 α 1 α 3 = babbbbbba = β 2 β 1 β 1 β 3 dBerLog 2007 21 dBerLog 2007 22 Post’s correspondence problem - example Post’s correspondence problem - formally • Given two finite lists of strings over some alphabet Γ • List A: List B: α 1 = ba β 1 = bab List A: α 1 , α 2 ,.., α k α 2 = abb β 2 = bb List B: β 1 , β 2 ,...., β k α 3 = bab β 3 = abb • Does there exist a sequence of indices Solution? NO! i1, i2,...,im ∈ {1,2,..,k} such that α i1 α i2 ...... α im = β i1 β i2 ...... β im ? dBerLog 2007 23 dBerLog 2007 24

Modified Post’s correspondence problem Post’s correspondence problem • Given two finite lists of strings over some alphabet Γ • Theorem List A: α 1 , α 2 ,.., α k List B: β 1 , β 2 ,...., β k Post’s correspondence problem is undecidable! • Does there exist a sequence of indices i2, i3,...,im ∈ {1,2,..,k} such that α 1 α i2 ...... α im = β 1 β i2 ...... β im ? dBerLog 2007 25 dBerLog 2007 26 Post’s correspondence problem - reductions Modified Post’s correspondence problem • Theorem 11.11 Modified Post’s correspondence problem is undecidable! Acc MPCP PCP Reduction I Reduction II dBerLog 2007 27 dBerLog 2007 28

Reduction I A Turing Machine • Given Turing machine T and input w, construct algoritmically MPCP T,w such that 0/X,R 1/Y, L p q r T accepts w iff p010 |- Xq10 |- rXY0........ MPCP T,w has a solution # p010 # Xq10 # rXY0 #........ dBerLog 2007 29 dBerLog 2007 30 Reduction I - lists of MPCP T,w Reduction I • List A: List B: • Given T = (Q, Σ , Π , δ , q 0 ,) and w ∈ Σ * α 1 = # β 1 = #q 0 ∆ w# (assume w.l.g. that T has no Stay-moves!) • Alphabet of MPCP T,w Γ := (Q ∪ {h a , h r }) ∪ ( Π ∪ { ∆ }) ∪ {#} dBerLog 2007 31 dBerLog 2007 32

Reduction I - lists of MPCP T,w Reduction I - lists of MPCP T,w • List A: • List A: List B: List B: α 1 = # β 1 = #q 0 ∆ w# α 1 = # β 1 = #q 0 ∆ w# α d = X β d = X for all X ∈ Γ α d = X β d = X for all X ∈ Γ α q,X = qX β q,X = Yp if δ (q,X) = (p, Y, R) α q,X = qX β q,X = Yp if δ (q,X) = (p, Y, R) α q,X = ZqX β q,X = pZY if δ (q,X) = (p, Y, L) α q,X = ZqX β q,X = pZY if δ (q,X) = (p, Y, L) α q,B = q# β q,B = Yp# if δ (q, ∆ ) = (p, Y, R) α q,B = q# β q,B = Yp# if δ (q, ∆ ) = (p, Y, R) α q,B = Zq# β q,B = pZY#if δ (q, ∆ ) = (p, Y, L) α q,B = Zq# β q,B = pZY#if δ (q, ∆ ) = (p, Y, L) α a1 = Xh a Y β a1 = h a for all X,Y ∈ Γ α a2 = Xh a β a2 = h a for all X,Y ∈ Γ α a3 = h a Y β a3 = h a for all X,Y ∈ Γ dBerLog 2007 33 dBerLog 2007 34 Reduction I - lists of MPCP T,w Reduction II • List A: List B: • Given MPCP over alphabet Γ α 1 = # β 1 = #q 0 ∆ w# α d = X β d = X for all X ∈ Γ • Construct PCP over alphabet Γ ’ such that α q,X = qX β q,X = Yp if δ (q,X) = (p, Y, R) α q,X = ZqX β q,X = pZY if δ (q,X) = (p, Y, L) α q,B = q# β q,B = Yp# if δ (q, ∆ ) = (p, Y, R) MPCP has solution α q,B = Zq# β q,B = pZY#if δ (q, ∆ ) = (p, Y, L) iff α a1 = Xh a Y β a1 = h a for all X,Y ∈ Γ PCP has solution α a2 = Xh a β a2 = h a for all X,Y ∈ Γ α a3 = h a Y β a3 = h a for all X,Y ∈ Γ α s = h a ## β s = # dBerLog 2007 35 dBerLog 2007 36

Definitions Reduction II ir, il : Γ * → ( Γ ∪ {#})* • Given MPCP with k lists over alphabet Γ A : α 1 , α 2 ,.... α k B : β 1 , β 2 ,..... β k ir ( ε ) = ε ir (ax) = a# • ir (x) a ∈ Γ , x ∈ Γ * il ( ε ) = ε il (ax) = #a • il (x) a ∈ Γ , x ∈ Γ * • Construct PCP with k+2 lists over Γ ∪ {#, $} A’: B’: Examples: α ’ 0 = # ir ( α 1 ) β ’ 0 = il ( β 1 ) ir (bob) = b#o#b# α ’ i = ir ( α 1 ) β ’ i = il ( β i ) for i = 1,2,..k il (bob) = #b#o#b α ’ k+1 = $ β ’ k+1 = #$ dBerLog 2007 37 dBerLog 2007 38 Context-free Grammar for expressions Ambiguity - example • G = ({E, I}, {1,2,+, ∗ ,(,)}, P, S) E E P: E → I | E + E | E ∗ E | (E) E E ∗ E E I → 1 | 2 + I E E I E E ∗ + 2 2 I I I I 1 2 2 1 dBerLog 2007 39 dBerLog 2007 40