Nondeterministic Polynomial-time Turing Machines Research Articles

The complexity of counting models of linear-time temporal logic

We determine the complexity of counting models of bounded size of specifications expressed in linear-time temporal logic. Counting word-models is #P-complete, if the bound is given in unary, and as hard as counting accepting runs of nondeterministic polynomial space Turing machines, if the bound is given in binary. Counting tree-models is as hard as counting accepting runs of nondeterministic exponential time Turing machines, if the bound is given in unary. For a binary encoding of the bound, the problem is at least as hard as counting accepting runs of nondeterministic exponential space Turing machines, and not harder than counting accepting runs of nondeterministic doubly-exponential time Turing machines. Finally, counting arbitrary transition systems satisfying a formula is #P-hard and not harder than counting accepting runs of nondeterministic polynomial time Turing machines with a PSPACE oracle, if the bound is given in unary. If the bound is given in binary, then counting arbitrary models is as hard as counting accepting runs of nondeterministic exponential time Turing machines.

Hierarchical Unambiguity

We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to generalize known constructs involving relativized unambiguity based complexity classes (UP and Promise-UP) to new constructs involving arbitrary higher levels of the relativized unambiguous polynomial hierarchy (UPH). Our techniques are developed on constraints imposed by hierarchical arrangement of unambiguous nondeterministic polynomial-time Turing machines, and so they differ substantially, in applicability and in nature, from standard methods (such as the switching lemma [J. Håstad, Computational Limitations of Small-Depth Circuits, MIT Press, Cambridge, 1987]), which play roles in carrying out similar generalizations. Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoč in [Complexity Theory, Cambridge University Press, Cambridge, UK, 1993, pp. 101–146], on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP.

Inverting onto functions

We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including: In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept Σ*. Every total multivalued nondeterministic function has a polynomial-time computable refinement. In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments.

Exploration of NP-hard enumeration problems by simulated annealing — the spectrum values of permanents

The Monte Carlo optimisation technique simulated annealing has been used to sample the possible values of the permanent of fully indecomposable (0,1)-matrices. The basic idea in simulated annealing is that a Markov chain explores the space of feasible solutions, each of which has an associated “energy” (here minus the permanent of the matrix). The control parameter “temperature” is gradually reduced in order to focus the search on the low-energy region of state space and eventually reaching the minimum. Although the minimum energy per se (largest permanent) is known, the spectrum above the minimum is not, so this importance sampling technique is quite efficient. Interest in permanents is based on the fact that it is “complete” for the class # P of enumeration problems, that is as hard as counting the number of accepting computations of any nondeterministic polynomial time Turing machine. The spectrum of possible values for the permanent of a (0, 1)-matrix with given topological dimension d is quite interesting but only partially understood. The present work is an experimental study of this spectrum using the handle basis representation of directed graphs and simulated annealing to sample the possible values. The results show that gaps exist in the spectrum of values of the permanent but also show the existence of values for d = 8 and 9 not hitherto observed. The new values in the spectrum of the permanent for d = 8 is 72 and for d = 9 are 119, 132, 136, 138, and 140. Further we found that the density of states grows exponentially with the permanent values. The larger the value of the permanent the larger the number of neighbours and the larger the energy barriers between the permanent values of the neighbours.

Logical Definability of Counting Functions

The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is #P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logicL, #Lis the class of functions on finite structures counting the tuples (T, ) satisfying a given formulaψ(T, ) inL. Saluja, Subrahmanyam, and Thakur showed that on classes of ordered structures #FO=#P (where FO denotes first-order logic) and that every function in #Σ1has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in #Σ1. A consequence is that functions counting the number of cliques, the number of Hamilton cycles, and the number of pairs with distance greater than two in a graph, are not contained in #Σ1. It is shown that on ordered structures #Σ11captures the previously studied class spanP. On unordered structures #FO is a proper subclass of #P and #Σ11is a proper subclass of spanP; in fact, no class #Lcontains all polynomial-time computable functions on unordered structures. However, it is shown that on unordered structures every function in #P is identical almost everywhere with some function #FO, and similarly for #Sgr;11and spanP. Finally, we discuss the closure properties of #FO under arithmetical operations.

THE COMPLEXITY OF FINDING MIDDLE ELEMENTS

Seinosuke Toda introduced the class Mid P of functions that yield the middle element in the set of output values over all paths of nondeterministic polynomial time Turing machines. We define two related classes: Med P consists of those functions that yield the middle element in the ordered sequence of output values of nondeterministic polynomial time Turing machines (i.e. we take into account that elements may occur with multiplicities greater than one). [Formula: see text] P consists of those functions that yield the middle element of all accepting paths (in some resonable encoding) of nondeterministic polynomial time Turing machines. We exhibit similarities and differences between these classes and completely determine the inclusion structure between these classes and some other well-known classes of functions like Valiant’s # P and Köbler, Schöning, and Torán’s span-P, that hold under general accepted complexity theoretic assumptions such as the counting hierarchy does not collapse. Our results help in clarifying the status of Toda’s very important class Mid P in showing that it is closely related to the class PPNP .

Counting classes: thresholds, parity, mods, and fewness

Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomial-time Turing machines. Well-known examples of counting classes are NP, co-NP, ⊕ P, and PP. Every counting class is a subset of P #P[1], the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k i P = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra (1990) by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in Valiant and Vazirani (1986)).

Relativized counting classes: Relations among thresholds, parity, and mods

Well-known complexity classes such as NP, co-NP, ⊕P (PARITY-P), and PP are produced by considering a nondeterministic polynomial time Turing machine N and defining acceptance in terms of the number of accepting paths in N. That is, they are subclasses of P #P[1]. Other interesting classes such as MOD k P and C + P are also subclasses of P #P[1]. Many relations among these classes are unresolved. Of course, these classes coincide if P = PSPACE. However, we develop a simple combinatorial technique for constructing oracles that separate counting classes. Our results suggest that it will be difficult to resolve the unknown relationships among different counting classes. In addition to presenting new oracle separations, we simplify several previous constructions.

Some observations on the connection between counting and recursion

Based on Valiant's class # P of all functions counting the number of accepting computations of nondeterministic polynomial-time Turing machines, the polynomial-time hierarchy of counting functions is introduced. The class PHCF of all functions of this hierarchy and some of its subclasses are characterized by recursion-theoretic means. It turns out that, from the recursion-theoretic point of view, PHCF is an analogue to Kalmár's class E of elementary functions, to the class Pspace of polynomial-space computable functions as well as to the class P of polynomial-time computable functions.