Locally definable acceptance types — The three-valued case

Abstract

We consider polynomial time nondeterministic machines, and augment the nodes of the tree associated with a computation by functions from {0,...,m−1}^r into {0,...,m−1}, where r is the number of successor nodes and m is some fixed positive integer. The leaves of the tree are labelled with constants (less than m), deterministic nodes take the unchanged value from their successor nodes. Thus from the leaves to the root we can compute a value for every node in the tree, and we say the tree accepts, if and only if the root value is 1. If all the functions appearing in nondeterministic nodes of some computation tree are from some set of functions F, then F is called an m-valued locally definable acceptance type. The class of languages recognizable by such machines with acceptance type F is denoted by (F)P.

The case m = 2 was (in some different context) investigated by Goldschlager and Parberry [GP86]. The only classes appearing as (F)P for 2-valued acceptance types F are P, NP, coNP, ⊕P, and PSPACE. m-valued acceptance types for m>2 were introduced in [He91a]. There it is proved that for all finite m-valued acceptance types F there exists an m′-valued acceptance type G (with possibly different constant m′), such that (F)P = (G)P, and G consists of only one binary function.

In the current paper we completely classify all classes (G)P where G is some 3-valued acceptance type consisting of only one binary function. It turns out that 20 classes are characterized by such acceptance types, trivially including the five classes appearing in the two-valued case, further the complete second level of the polynomial time hierarchy (Σ ^p₂ , Π ^p₂ , Δ ^p₂ , Θ ^p₂ ), some interesting very new classes (like ∇P and co-∇P). and some classes that are well known, but did not have such a closed machine characterization before (like P^NP[1]). The proof technique for the main result is the following: first some criteria are given that a function refers to a trivial class (i.e. a class already appearing in the two-valued case), then a computer program is designed to pick out only the relevant cases, and finally the remaining 104 functions are investigated in detail.

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