5 Alternating Complexity Classes

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Complexity Theory WS 2009/10

Prof. Dr. Erich Grädel

Mathematische Grundlagen der Informatik RWTH Aachen

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Dieses Werk ist lizensiert uter:

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http://www.logic.rwth-aachen.de

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5 Alternating Complexity Classes

Alternating algorithms are a generalization of non-deterministic algorithms based on two-player games. Indeed, one can view non- deterministic algorithms as the restriction of alternating algorithms to solitaire (i.e., one-player) games. Since complexity classes are mostly defined in terms of Turing machines, we focus on the model of alternating Turing machines. But note that alternating algorithms can be defined in terms of other computational models, also.

Definition 5.1. An alternating Turing machineis a non-deterministic Turing machine whose state set Q is divided into four classes Q_∃, Q_∀, Q_acc, andQ_rej. This means that there are existential, universal, accepting and rejecting states. States inQ_acc∪Q_rejare final states. A configuration ofMis called existential, universal, accepting, or rejecting according to its state.

The computation graphG_M,xof an alternating Turing machineM for an inputxis defined in the same way as for a non-deterministic Turing machine. Nodes are configurations (instantaneous descriptions) ofM, there is a distinguished starting nodeC₀(x)which is the input configuration ofMfor inputx, and there is an edge from configuration C to configurationC^′ if, and only if,C^′ is a successor configuration ofC. Recall that fornon-deterministicTuring machines, the acceptance condition is given by the reachability problem: Maccepts xif, and only if, in the graphG_M,xsome accepting configurationCais reachable fromC₀(x). ForalternatingTuring machines, acceptance is defined by thegameproblem (see Sect. 3.3): the players here are called∃and∀, where∃moves from existential configurations and∀from universal ones. Further,∃wins at accepting configurations and loses at rejecting ones. By definition,Macceptsxif, and only if, Player∃has a winning strategy fromC₀(x)for the game onG_M,x.

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5.1 Complexity Classes

When considering the computation treeT_M,x, which corresponds to the unraveling of the configuration graph from C₀(x), we call a subtreeT_Caccepting if∃has a winning strategy fromC.

5.1 Complexity Classes

Time and space complexity are defined as for nondeterministic Turing machines. For a functionF:N→R, we say that an alternating Turing machineMisF-time-bounded if for all inputsx, all computation paths fromC₀(x)terminate after at mostF(|x|)steps. Similarly,MisF-space- bounded if no configuration of Mthat is reachable fromC₀(x) uses more thanF(|x|)cells of work space. The complexity classes Atime(F) and Aspace(F)contain all problems that are decidable by, respectively, F-time bounded andF-space bounded alternating Turing machines.

The following classes are of particular interest:

• ALogspace=Aspace(O(logn)),

• APtime=^S_d∈NAtime(n^d),

• APspace=^S_d∈NAspace(n^d).

Example5.2. qbf ∈ Atime(O(n)). We assume that, without loss of generality, negations appear only in front of variables. An alternating version of Eval(ψ,I)is the following:

Algorithm 5.1.Alternating Eval(ψ,I)

Input:(ψ,I) whereψ∈qbfundI: free(ψ)→ {0, 1} ifψ=Ythen

ifI(Y) =1thenacceptelsereject endif

ifψ=ϕ₁∨ϕ₂then “∃” guessesi∈ {1, 2};returnEval(ϕ_i,I) ifψ=ϕ₁∧ϕ₂then “∀” choosesi∈ {1, 2};returnEval(ϕ_i,I) ifψ=∃Xϕ then “∃” guessesj∈ {0, 1};returnEval(ϕ,I[X=j]) ifψ=∀Xϕ then “∀” choosesj∈ {0, 1};returnEval(ϕ,I[X=j])

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5 Alternating Complexity Classes

5.2 Alternating Versus Deterministic Complexity

There is a general slogan that parallel time complexity coincides with sequential space complexity.

Theorem 5.3. LetS(n)be space-constructible withS(n)≥n. Then, Nspace(S)⊆Atime(S²).

Proof. We use the same trick as in the proof of Savitch’s Theorem: Let L be decided by a nondeterministic Turing machine Mwith space bounded by S(n) and in time 2^c·S(n). Let Conf[S(n)] be the set of configurations of M with space≤ S(n). The alternating algorithm Reach(C₁,C₂,t)(Algorithm 5.2) decides whether the configurationC₂∈ Conf[S(n)] can be reached from configurationC₁ ∈ Conf[S(n)] in at most 2^tsteps. The algorithm is correct becauseC₂is reachable fromC₁ in at most 2^tsteps if there is someCsuch that Reach(C₁,C,t−1)and Reach(C,C₂,t−1)accept.

Let f(t) =max_C₁_,C₂∈Conf[S(n)]timeReach(C₁,C₂,t). Furthermore, f(0) =O(S(n))and for allt>0, f(t) =O(S(n)) +f(t−1). Hence,

f(t) = (t+1)·O(S(n)).

Lcan then be decided as follows: At first, for an input x, the input configurationC₀ofMonxis constructed. Then, some accepting final configurationC_aofMis guessed. We will accept if Reach(C₀,C_a,S(n))

Algorithm 5.2.Reach(C₁,C₂,t) Input:C₁,C₂,t

ift=0then

ifC₁=C₂or C₂∈Next(C₁)thenacceptelsereject else /* t>0 */

existentially guessC∈Conf[S(n)]

universally choose(D₁,D₂) = (C₁,C)and(D₁,D₂) = (C,C₂) Reach(D₁,D₂,t−1)

endif

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accepts. This algorithm needs (c·S(n) +1)O(S(n)) =O(S²(n))

steps. By the linear Speed-Up Theorem, which also applies to alternating Turing machines,L∈Atime(S²). q.e.d.

Theorem 5.4. Let T be space-constructible and T(n) ≥ n. Then, Atime(T)⊆Dspace(T²).

Proof. LetL∈ Atime(T)andMbe some alternating Turing machine accepting Lin time bounded byT(n). Then, there is somerso that for all configurations C of M: |Next(C)| ≤ r. Algorithm 5.3, A_T, computes whether or not the subtree T_C is accepting (output 1) or rejecting (output 0) for every configurationCinT_M,x.

Obviously, this algorithm is working correctly.A_T(C₀(x))decides whetherMacceptsxand, hence, is a deterministic decision procedure forL.

Algorithm 5.3.A_T, deterministic evaluation ofT_C Input:C

ifC accepting then output1 ifC rejecting then output0 ifC existentialthen

fori=1, . . . ,rdo

computei-th successor configurationC_iofC ifF(C_i) =1then output1

endfor output0 endif

ifC universal then fori=1, . . . ,rdo

computei-th successor configurationC_iofC ifF(C_i) =0then output0

endfor output1 endif

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5 Alternating Complexity Classes How much space does this algorithm need? LetCbe some node of heighttinT_M,x, i.e., all computations ofMrooted atCneed at most tsteps. Then:

space_A_T(C) =





0 ift=0

max_C_i_∈Next(C)(|C_i|+space_A_T(C_i)) ift>0 . SinceC_i∈Next(C)is of heightt−1, we obtain space_A_T(C)≤t·T(n) and therefore space_A_T(Co)≤T²(n). q.e.d.

In particular, we obtain

Theorem 5.5(Parallel time complexity = sequential space complexity).

• APtime=Pspace.

• AExptime=Expspace.

Proof.

• Atime(n^d)⊆Dspace(n^2d)⊆Pspace,

Dspace(n^d)⊆Nspace(n^d)⊆Atime(n^2d)⊆APtime.

• Atime(2ⁿ^d)⊆Dspace(2²ⁿ^d)⊆Expspace,

Dspace(2ⁿ^d)⊆Atime(2²ⁿ^d)⊆AExptime. q.e.d.

On the other hand, alternating space complexity corresponds to exponential deterministic time complexity.

Theorem 5.6. For any space-constructible functionS(n) ≥logn, we have that Aspace(S) =Dtime(2^O(S)).

Proof. The proof is closely associated with thegameproblem. For any S-space-bounded alternating Turing machine M, one can, given an input x, construct the computation graphG_M,x in time 2^O(S(|x|) and then solve thegameproblem in order to decide the acceptance of x byM.

For the converse, we shall show that for anyT(n) ≥n and any constantc, Dtime(T)⊆Aspace(c·logT).

LetL∈Dtime(T). Then there is a deterministic one-tape Turing machineMthat decidesLin timeT². LetΓ=Σ∪(Q×Σ)∪ {∗}and

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t=G²(n). Every configurationC= (q,i,w)(in a computation on some input of lengthn) can be described by a word

c=∗w₀. . .w_i−1(qw_i)w_i+1. . .w_t∗ ∈Γ^t+2.

Theith symbol of the successor configuration depends only on the symbols at positionsi−1,i, andi+1. Hence, there is a function f_M: Γ³ →Γsuch that, whenever symbolsa₋₁,a₀, anda₁ are at positions i−1,iandi+1 of some configurationc, the symbolf_M(a₋₁,a₀,a₁)will be at positioniof the successor configurationc^′.

The alternating algorithmA(Algorithm 5.4) decidesLusing space O(logT(n)). IfMaccepts the inputx, then Player∃has the following winning strategy for the game onC_A,x: the value chosen forsis the time at which Macceptsx, and(q⁺a),iare chosen so that the configuration ofMat timesis of the form∗w₀. . .w_i−1(q⁺a)w_i+1. . .w_t∗. At thejth iteration of the loop (that is, at configuration s−j), the symbols at positionsi−1,i,i+1 of the configuration ofMat times−jare chosen fora₋₁,a₀,a₁.

Conversely, if Mdoes not accept the inputx, theith symbol of the configuration at time sis not(q⁺a). The following holds for all j: if, in thejth iteration of the loop, Player∃choosesa₋₁,a₀,a₁, then

Algorithm 5.4.Alternating simulation of a determinisitc computation existentially guesss≤T²(n) =t

existentially guessi∈ {0, . . . ,s} existentially guess(q⁺a)∈Q⁺_acc×Σ b:= (q⁺a)

forj=1, . . . ,sdo

existentially guess(a₋₁,a₀,a₁)∈Γ³ if f_M(a₋₁,a₀,a₁)̸=bthenreject universally choosek∈ {−1, 0, 1} b:=a_k

i:=i+k endfor

ifthe i-th symbol of the input configuration of M on x equals bthenaccept elsereject

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5 Alternating Complexity Classes either f(a₋₁,a₀,a₁)̸=b, in which case Player∃loses immediately, or there is at least onek∈ {−1, 0, 1}such that the(i+k)th symbol of the configuration at times−jdiffers froma_k. Player∀then chooses exactly thisk. At the end,a_kwill then be different from theith symbol of the input configuration, so Player∀wins.

HenceAacceptsxif, and only if,Mdoes so. q.e.d.

In particular, it follows that

• ALogspace=Ptime;

• APspace=Exptime.

The relationship between the major deterministic and alternating complexity classes is summarised in Fig. 5.1.

Logspace ⊆ Ptime ⊆ Pspace ⊆ Exptime ⊆ Expspace

|| || || ||

ALogspace ⊆ APtime ⊆ APspace ⊆ AExptime Figure 5.1.Relationship between deterministic and alternating complexity

classes

5.3 Alternating Logarithmic Time

For time boundsT(n)<n, the standard model of alternating Turing machines needs to be modified a little by an indirect access mecha- nism. The machine writes down, in binary, an addression an sep- arate index tape to access the ith symbol of the input. Using this model, it makes sense to define, for instance, the complexity class ALogtime=Atime(O(logn)).

Important examples of problems in ALogtimeare

• the model-checking problem for propositional logic;

• the data complexity of first-order logic.

The results mentioned above relating alternating time and sequential space hold also for logarithmic time and space bounds. Note, however, that these do not imply that ALogtime=Logspace, owing to

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5.3 Alternating Logarithmic Time

the quadratic overheads. It is known that ALogtime⊆Logspace, but the converse inclusion is an open problem.

Exercise 5.1. Construct an ALogtimealgorithm for the set of palin- dromes (i.e., words that are same when read from right to left and from left to right).

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5 Alternating Complexity Classes

Complexity Theory WS 2009/10

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5 Alternating Complexity Classes