Alternating Turing machine

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 and independently by Kozen in 1976, with a joint journal publication in 1981.
Definitions.
Informal description.
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.
Formal definition.
Formally, a (one-tape) alternating Turing machine is a 5-tuple formula_1 where
If "M" is in a state formula_7 with formula_8 then that configuration is said to be "accepting", and if formula_9 the configuration is said to be "rejecting". A configuration with formula_10 is said to be accepting if all configurations reachable in one step are accepting, and rejecting if some configuration reachable in one step is rejecting. A configuration with formula_11 is said to be accepting when there exists some configuration reachable in one step which is accepting and rejecting when all configurations reachable in one step are rejecting (this is the type of all states in an NTM). "M" is said to accept an input string "w" if the initial configuration of "M" (the state of "M" is formula_12, the head is at the left end of the tape, and the tape contains "w") is accepting, and to reject if the initial configuration is rejecting.
Resource bounds.
When deciding if a configuration of an ATM is accepting or rejecting using the above definition, it is not 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 formula_13 if, on any input of length , examining configurations only up to formula_13 steps is sufficient to label the initial configuration as accepting or rejecting. An ATM decides a language in space formula_15 if examining configurations which do not modify tape cells beyond the formula_15 cell from the left is sufficient.
A language which is decided by some ATM in time formula_17 for some constant formula_18 is said to be in the class formula_19, and a language decided in space formula_20 is said to be in the class formula_21.
Example.
Perhaps the simplest 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 which 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 formula_22 and space formula_23.
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.
Complexity classes and comparison to deterministic Turing machines.
The following complexity classes are useful to define for ATMs:
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
when formula_30 and formula_31.
A more general form of these relationships is expressed by the parallel computation thesis.
Bounded alternation.
Definition.
An alternating Turing machine with "k" alternations is an alternating Turing machine which 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".)
formula_32 is the class of function in time formula_33 beginning by existential state and alternating at most formula_34 times. It is called the th level of the formula_35 hierarchy.
formula_36 is the same classes, but beginning by a universal state, it is the complement of the language of formula_37.
formula_38 is defined similarly for space bounded computation.
Example.
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).
Collapsing classes.
It is said that a hierarchy "collapses" to level if every language in level formula_39 of the hierarchy is in its level .
As a corollary of the Immerman–Szelepcsényi theorem, the logarithmic space hierarchy collapses to its first level. As a corollary the formula_40 hierarchy collapses to its first level when formula_41 is space constructible.
Special cases.
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 formula_42 (respectively, formula_43).
These classes are sometimes denoted formula_44 and formula_45, respectively.
See the polynomial hierarchy article for details.
Another special case of time hierarchies is the logarithmic hierarchy.