In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer[1] and independently by Kozen[2] in 1976, with a joint journal publication in 1981.[3]
The definition of NP uses the existential mode of computation: if any choice leads to an accepting state, then the whole computation accepts. The definition of co-NP uses the universal mode of computation: only if all choices lead to an accepting state does the whole computation accept. An alternating Turing machine (or to be more precise, the definition of acceptance for such a machine) alternates between these modes.
An alternating Turing machine is a non-deterministic Turing machine whose states are divided into two sets: existential states and universal states. An existential state is accepting if some transition leads to an accepting state; a universal state is accepting if every transition leads to an accepting state. (Thus a universal state with no transitions accepts unconditionally; an existential state with no transitions rejects unconditionally). The machine as a whole accepts if the initial state is accepting.
M=(Q,\Gamma,\delta,q0,g)
Q
\Gamma
\delta:Q x \Gamma → l{P}(Q x \Gamma x \{L,R\})
q0\inQ
g:Q → \{\wedge,\vee,accept,reject\}
If M is in a state
q\inQ
g(q)=accept
g(q)=reject
g(q)=\wedge
g(q)=\vee
q0
Note that it is impossible for a configuration to be both accepting and rejecting, however, some configurations may be neither accepting or rejecting, due to the possibility of nonterminating computations.
When deciding if a configuration of an ATM is accepting or rejecting using the above definition, it is not always necessary to examine all configurations reachable from the current configuration. In particular, an existential configuration can be labelled as accepting if any successor configuration is found to be accepting, and a universal configuration can be labelled as rejecting if any successor configuration is found to be rejecting.
An ATM decides a formal language in time
t(n)
t(n)
s(n)
s(n)
A language that is decided by some ATM in time
c ⋅ t(n)
c>0
ATIME(t(n))
c ⋅ s(n)
ASPACE(s(n))
Perhaps the most natural problem for alternating machines to solve is the quantified Boolean formula problem, which is a generalization of the Boolean satisfiability problem in which each variable can be bound by either an existential or a universal quantifier. The alternating machine branches existentially to try all possible values of an existentially quantified variable and universally to try all possible values of a universally quantified variable, in the left-to-right order in which they are bound. After deciding a value for all quantified variables, the machine accepts if the resulting Boolean formula evaluates to true, and rejects if it evaluates to false. Thus at an existentially quantified variable the machine is accepting if a value can be substituted for the variable that renders the remaining problem satisfiable, and at a universally quantified variable the machine is accepting if any value can be substituted and the remaining problem is satisfiable.
Such a machine decides quantified Boolean formulas in time
n2
n
The Boolean satisfiability problem can be viewed as the special case where all variables are existentially quantified, allowing ordinary nondeterminism, which uses only existential branching, to solve it efficiently.
The following complexity classes are useful to define for ATMs:
AP=cupk>0ATIME(nk)
APSPACE=cupk>0ASPACE(nk)
AEXPTIME=cupk>0
| nk | |
| ATIME(2 |
)
These are similar to the definitions of P, PSPACE, and EXPTIME, considering the resources used by an ATM rather than a deterministic Turing machine. Chandra, Kozen, and Stockmeyer proved the theorems
ASPACE(f(n))=cupc>0DTIME(2cf(n))=DTIME(2O(f(n)))
ATIME(g(n))\subseteqDSPACE(g(n))
NSPACE(g(n))\subseteqcupc>0ATIME(c x g(n)2),
f(n)\gelog(n)
g(n)\gelog(n)
A more general form of these relationships is expressed by the parallel computation thesis.
An alternating Turing machine with k alternations is an alternating Turing machine that switches from an existential to a universal state or vice versa no more than k−1 times. (It is an alternating Turing machine whose states are divided into k sets. The states in even-numbered sets are universal and the states in odd-numbered sets are existential (or vice versa). The machine has no transitions between a state in set i and a state in set j < i.)
ATIME(C,j)=\SigmajTIME(C)
f\inC
j-1
TIME(C)
coATIME(C,j)=\PijTIME(C)
ATIME(f,j)
ASPACE(C,j)=\SigmajSPACE(C)
Consider the circuit minimization problem: given a circuit A computing a Boolean function f and a number n, determine if there is a circuit with at most n gates that computes the same function f. An alternating Turing machine, with one alternation, starting in an existential state, can solve this problem in polynomial time (by guessing a circuit B with at most n gates, then switching to a universal state, guessing an input, and checking that the output of B on that input matches the output of A on that input).
It is said that a hierarchy collapses to level if every language in level
k\gej
As a corollary of the Immerman–Szelepcsényi theorem, the logarithmic space hierarchy collapses to its first level.[4] As a corollary the
SPACE(f)
f=\Omega(log)
An alternating Turing machine in polynomial time with k alternations, starting in an existential (respectively, universal) state can decide all the problems in the class
| p | |
| \Sigma | |
| k |
| p | |
| \Pi | |
| k |
\Sigmak\rm{P}
\Pik\rm{P}
Another special case of time hierarchies is the logarithmic hierarchy.