Alternating Turing machine
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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 in 1976 (see References).
DefinitionsInformal DescriptionThe 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: if all choices lead to an accepting state, then the whole computation accepts. 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 DefinitionFormally, a (one-tape) alternating Turing machine is a 5-tuple Failed to parse (Missing texvc executable; please see math/README to configure.): M=(Q,\Gamma,\delta,q_0,g) where
is the finite set of states
is the finite tape alphabet
is called the transition function (L shifts the head left and R shifts the head right)
is the initial state
specifies the type of each state If M is in a state Failed to parse (Missing texvc executable; please see math/README to configure.): q\in Q with Failed to parse (Missing texvc executable; please see math/README to configure.): g(q)=accept then that configuration is said to be accepting, and if Failed to parse (Missing texvc executable; please see math/README to configure.): g(q)=reject the configuration is said to be rejecting. A configuration with Failed to parse (Missing texvc executable; please see math/README to configure.): g(q)=\wedge 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 Failed to parse (Missing texvc executable; please see math/README to configure.): g(q)=\vee 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 Failed to parse (Missing texvc executable; please see math/README to configure.): q_0 , 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. Machine with k alternationsAn 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.) For 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). An alternating Turing machine, with k alternations, starting in an existential (respectively, universal) state can decide all the problems in the class Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma_k\rm{P}
(respectively, Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi_k\rm{P}
) in polynomial time. See the polynomial hierarchy article. Resource boundsWhen 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 future configuration is found to be accepting, and a universal configuration can be labelled as rejecting if any future configuration is found to be rejecting. An ATM decides a Formal language in time Failed to parse (Missing texvc executable; please see math/README to configure.): t(n) if, on any input of length Failed to parse (Missing texvc executable; please see math/README to configure.): n , examining configurations only up to Failed to parse (Missing texvc executable; please see math/README to configure.): t(n) steps is sufficient to label the initial configuration as accepting or rejecting. An ATM decides a language in space Failed to parse (Missing texvc executable; please see math/README to configure.): s(n) if examining configurations which do not modify tape cells beyond the Failed to parse (Missing texvc executable; please see math/README to configure.): s(n) cell from the left is sufficient. A language which is decided by some ATM in time Failed to parse (Missing texvc executable; please see math/README to configure.): c\cdot t(n)
for some constant Failed to parse (Missing texvc executable; please see math/README to configure.): c>0
is said to be in the class Failed to parse (Missing texvc executable; please see math/README to configure.): {\rm ATIME}(t(n))
, and a language decided in space Failed to parse (Missing texvc executable; please see math/README to configure.): c\cdot s(n)
is said to be in the class Failed to parse (Missing texvc executable; please see math/README to configure.): {\rm ASPACE}(s(n))
. ExamplePerhaps 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 or rejects according as the resulting boolean formula evaluates to true or 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 Failed to parse (Missing texvc executable; please see math/README to configure.): n^2 and space Failed to parse (Missing texvc executable; please see math/README to configure.): 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. Complexity classes and comparison to deterministic Turing machinesThe following complexity classes are useful to define for ATMs:
are the languages decidable in polynomial time
are the languages decidable in polynomial space
are the languages decidable in exponential time 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
This is expressed by the Parallel Computation Thesis. References
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