Der Satz von Immerman-Szelepcsényi

Transcription

1 Der Satz von Immerman-Szelepcsényi Sommerakademie Rot an der Rot AG 1 Wieviel Platz brauchen Algorithmen wirklich? Martin Seybold Fakultät für Informatik TU München 9. August 2010 Martin Seybold: NSPACE=coNSPACE 1/ 15

2 Introduction Motivation Motivation 1964 Kuroda: ContextSensitive = NLINSPACE = NSPACE(O(n)) 1. LBA Problem: DSPACE(n) = NSPACE(n)? still open, Theorem of Savitch 2. LBA Problem: L := Σ \ L also context sensitive? 1988 Immermann: Nondeterministic space is closed under complementation Szelepcsényi: The Method of Forced Enumeration for Nondeterministic Automata. Martin Seybold: NSPACE=coNSPACE 2/ 15

3 Introduction Problem Description 2. LBA Problem: Description w / L Deterministic vs. nondeterministic calculation tree f (n)-space bounded TM: Graph with at most c f (n) nodes How to insure that there exists no yes -node along the reachables from S(w) without loosing the space bound? Idea: Time consumption is totally irrelevant Save space through ad-hoc recalculation Martin Seybold: NSPACE=coNSPACE 3/ 15

4 Introduction Problem Description Outline 1 Introduction Motivation Problem Description Notation 2 Theorem Proof 3 Consequences 4 Efficient k-tape TM simulation Martin Seybold: NSPACE=coNSPACE 4/ 15

5 Introduction Notation Notation TM M = (Q, Σ, Γ, δ, q 0, F, ) Q finite state set, q 0 Q initial state, F Q set of final states, Σ input alphabet, Γ tape alphabet, Γ \ Σ blank symbol, δ Q (Σ { }) Γ k Q Γ k {L, R, N} k+1 Configuration / snapshot of a TM: α, β, α = (i, q, w 1,, w k ) k-tape TM α = a 1 a i 1 qa i a n 1-tape TM start configuration S(w) := q 0 w α 1 M ᾱ, α k M ᾱ Length of a configuration: α = a 1 a i 1 + a i a n NSPACE(f (n)) = {L L accepted by nondet. TM, longest configuration f (n)} conspace(f (n)) = { L := Σ \ L L NSPACE(f (n))} Martin Seybold: NSPACE=coNSPACE 5/ 15

6 Theorem Theorem (Immermann Szelepcsényi) Let M TM with L(M) NSPACE(f ), f Ω(log(n)) There exists a nondet. TM M with L(M ) = L and L(M ) NSPACE(O(f )). NSPACE is closed unter complement: NSPACE(f ) = conspace(f ) Martin Seybold: NSPACE=coNSPACE 6/ 15

7 Theorem Preobservations M is a 1-tape TM (sec. 4) Configurations are strings, easy enumerable Unreach(Graph, S(w), yes )? Nondeterminism allows guess & check Need of the count of different strings to be guessed Check if yes -string is contained. Martin Seybold: NSPACE=coNSPACE 7/ 15

8 Theorem Proof General simplifications order on configurations (e.g. length lexicographic) yes = α 0 is shortest and only accepting configuration R(k) = {α S(w) k M α} reachables r(k) = R(k) r( ) = r( ) c f ( w ) Martin Seybold: NSPACE=coNSPACE 8/ 15

9 Theorem Proof M : Unreach Data: w Σ Result: w / L or stop α := α 0 ; for r( ) do Guess: S(w) M ᾱ; if ᾱ α then stop; else α := ᾱ; return w / L ; Algorithm 1: Unreach w / L: R( ) = {α 1, α 2,, α r( ) } and α i α i+1. w L: α 0 R( ) => stop. Martin Seybold: NSPACE=coNSPACE 9/ 15

10 Theorem Proof Data: S(w) Result: r( ) or stop r(0) := 1; m(0) := S(w) ; k := 0; while r(k) r(k + 1) do k := k + 1; return r(k); Algorithm 2: r( ) Data: r(k), m(k) N Result: r(k + 1), m(k + 1) or stop α := α 0 ; r := 0; m := 0; for β : β m(k) + 1 do if β R(k + 1) then r := r + 1; m := max{m, β }; return r, m; Algorithm 3: Inductive counting Martin Seybold: NSPACE=coNSPACE 10/ 15

11 Theorem Proof Data: β: Konfiguration, k N Result: β R(k + 1) or stop α := α 0 ; b := false; for r(k) do Guess: S(w) k M ᾱ R(k); if ᾱ α then stop; else α := ᾱ; if ᾱ 1 M β then b := true; return b; Algorithm 4: β? R(k + 1) Martin Seybold: NSPACE=coNSPACE 11/ 15

12 Theorem Proof Space consumption of M check α β needs at most 2 f (n) 6 configurations as local variables k, m log(c f (n) ) O(f (n)) increment is easy with TM check α 1 M β in 2 f (n) L(M ) NSPACE(O(f (n))) Martin Seybold: NSPACE=coNSPACE 12/ 15

13 Consequences Consequences 2. LBA Problem: is Σ \ L 1 of type 1? Yes Context sensitive languages are closed under complement. Martin Seybold: NSPACE=coNSPACE 13/ 15

14 Efficient k-tape TM simulation How to Simulate a k-tape TM with 1 tape? Aim: Simulation of k-tape TM M on a 1-tape TM M Idea: Alphabet extention to tuples Let M = (Q, Σ, Γ, δ, q 0, F, ) k-tape TM M := (Q, Σ, (Γ ˆΓ) k, δ, q 0, F, ) verbal description of δ copy input w = a 1 a n to tupels (a 1, ˆ,, ˆ )(a 2,,, ) (a n,,, ) for q at a i search sequentially for head positions save head letters in state perform replacement of δ Martin Seybold: NSPACE=coNSPACE 14/ 15

15 Efficient k-tape TM simulation Thank you for paying attention! Do you have questions? [Die95] V. Diekert, 8.4 Komplexitätstheorie - Teubner-Taschenbuch der Mathematik Teil II, B. G. Teubner, Stuttgart-Leipzig, 7. edition, [Pap94] C. Papadimitriou, Computational Complexity, Addison-Wesley Publishing Company, Reading, Mass., [Tan10] T. Tantau, Theoretische Informatik, Februar 2010 Martin Seybold: NSPACE=coNSPACE 15/ 15