# Locally definable acceptance types for polynomial time machines

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## Abstract

*We introduce m-valued locally definable acceptance types, a new model generalizing the idea of alternating machines and their acceptance behaviour. Roughly, a locally definable acceptance type consists of a set F of functions from* {0,..., *m*−1}^{r}*into* {0,..., *m*−1}, *which can appear as labels in a computation tree of a nondeterministic polynomial time machine. The computation tree then uses these functions to evaluate a tree value, and accepts or rejects depending on that value. The case m* = 2 *was (in some different context) investigated by Goldschlager and Parberry [GP86]. In [He91b] a complete classification of the classes (F)P is given, when F consists of only one binary 3- valued function. In the current paper we justify the restriction to the case of one binary function by proving a normal form theorem stating that for every finite acceptance type there exists a finite acceptance type that characterizes the same class, but consists only of one binary function*.

*Further we use the normal form theorem to show that the system of characterizable classes is closed under operators like ∃, ∀, ⊕, and others. In a similar fashion we show that all levels of boolean hierarchies over characterizable classes are characterizable. As corollaries from these results we obtain characterizations of all levels of the polynomial time hierarchy and the boolean hierarchy over* NP, *or more generally σ* _{ k } ^{ p } .

## Keywords

Polynomial Time Computation Tree Binary Function Counting Class Left Subtree## Preview

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## References

- [BGH90]R. Beigel, J. Gill, U. Hertrampf,
*Counting Classes: Thresholds, Parity, Mods, and Fewness*, Proc. 7th Symposium on Theoretical Aspects of Computer Science, LNCS 415 (1990), pp. 49–57.Google Scholar - [CKS81]A.K. Chandra, D.C. Kozen, L.J. Stockmeyer,
*Alternation*, J. ACM 28 (1981), pp. 114–133.Google Scholar - [GNW90]T. Gundermann, N.A. Nasser, G. Wechsung,
*A Survey on Counting Classes*, 5th Structure in Complexity Theory (IEEE) (1990), pp. 140–153.Google Scholar - [GP86]Leslie M. Goldschlager, Ian Parberry,
*On the Construction of Parallel Computers from various Bases of Boolean Functions*, Theoretical Computer Science 43 (1986), pp. 43–58.Google Scholar - [GW87]T. Gundermann, G. Wechsung,
*Counting Classes of Finite Acceptance Types*, Computers and Artificial Intelligence 6 (1987), pp. 395–409.Google Scholar - [He91a]Ulrich Hertrampf,
*Locally Definable Acceptance Types for Polynomial Time Machines*, Technical Report No. 28, Universität Würzburg, 1991.Google Scholar - [He91b]Ulrich Hertrampf,
*Locally Definable Acceptance Types — The Three-Valued Case*, Technical Report No. 29, Universität Würzburg, 1991.Google Scholar - [Wa88]Klaus W. Wagner,
*Bounded Query Computations*, Proc. 3rd Structure in Complexity Theory (IEEE) (1988), pp. 260–277.Google Scholar - [Wa90]Klaus W. Wagner,
*Bounded Query Classes*, SIAM J. Comput. 19 (1990), pp. 833–846.Google Scholar