Nondeterministic Branching Research Articles

Circuits on Cylinders

We consider the computational power of constant width polynomial size cylindrical circuits and nondeterministic branching programs. We show that every function computed by a Pi_2 o MOD o AC^0 circuit can also be computed by a constant width polynomial size cylindrical nondeterministic branching program (or cylindrical circuit) and that every function computed by a constant width polynomial size cylindrical circuit belongs to ACC^0.

Comparing the sizes of nondeterministic branching read- k-times programs

We compare the complexities of Boolean functions for nondeterministic syntactic read- k-times branching and branching read- s k -times programs. It is shown that for each natural number k, k⩾2, there exists a sequence of Boolean functions such that the complexity of computation of each function of this sequence by nondeterministic syntactic branching read- k-times programs is exponentially larger (with respect to the number of variables of the Boolean function) than by nondeterministic branching read- (k ln k/ ln 2+C) -times programs, where C is a constant independent of k. Besides, it is shown that for each natural numbers N and k( N), where 4⩽k(N)<C 2 ln N / ln ln N and C 2< 2 is a constant independent of k and N, there exists a Boolean function in N variables such that the complexity of this function for nondeterministic syntactic read- k-times branching programs is exponentially larger (with respect to N) than for nondeterministic syntactic read- (k ln k/ ln 2+C) -times branching programs.

An Improved Hierarchy Result for Partitioned BDDs

One of the great challenges of complexity theory is the problem of analyzing the dependence of the complexity of Boolean functions on the resources nondeterminism and randomness. So far this problem could be solved only for very few models of computation. For so-called partitioned binary decision diagrams , which are a restricted variant of nondeterministic read-once branching programs, Bollig and Wegener have proven an astonishing hierarchy result which shows that the smallest possible decrease of the available amount of nondeterminism may incur an exponential blow-up of the branching program size. They have shown that k -partitioned BDDs which may nondeterministically choose between k alternative subprograms may be exponentially larger than (k+1) -partitioned BDDs for the same function if k = o(( log n / log log n) 1/2 ) , where n is the input size. In this paper an improved hierarchy result is established which still works if the number of nondeterministic decisions is O((n/ log 1+e n) 1/4 ) , where e > 0 is an arbitrary small constant.

Neither reading few bits twice nor reading illegally helps much

We first consider the so-called (1, + s)-branching programs in which along every consistent path at most s variables are tested more than once. We prove that any such program computing a characteristic function of a linear code C has size at least 2 Ω(min/s d 1 , d 2 s ), where d 1 and d 2 are the minimal distances of C and its dual C ⊥. We apply this criterion to explicit linear codes and obtain a super-polynomial lower bound for s = o( n logn ) . Then we introduce a natural generalization of read- k-times and (1, + s)-branching programs that we call semantic branching programs. These programs correspond to corrupting Turing machines which, unlike eraser machines, are allowed to read input bits even illegally, i.e. in excess of their quota on multiple readings, but in that case they receive in response an unpredictably corrupted value. We generalize the above-mentioned bound to the semantic case, and also prove exponential lower bounds for semantic read-once nondeterministic branching programs.

A Linear Metalanguage for Concurrency

A metalanguage for concurrent process languages is introduced.&lt;br /&gt;Within it a range of process languages can be defined, including&lt;br /&gt;higher-order process languages where processes are passed and received as arguments. (The process language has, however, to be linear, in the sense that a process received as an argument can be run at most once, and not include name generation as in the Pi-Calculus.) The metalanguage is provided with two interpretations both of which can be understood as categorical models of a variant of linear logic. One interpretation is in a&lt;br /&gt;simple category of nondeterministic domains; here a process will denote its set of traces. The other interpretation, obtained by direct analogy with the nondeterministic domains, is in a category of presheaf categories; the nondeterministic branching behaviour of a process is captured in its denotation as a presheaf. Every presheaf category possesses a notion of (open-map) bisimulation, preserved by terms of the metalanguage. The&lt;br /&gt;conclusion summarises open problems and lines of future work.

Inductive Counting for Width-Restricted Branching Programs

As an application of the inductive counting technique to a circuit-like model, we prove that complementation on nondeterministic branching programs can be done without increasing the width excessively. A consequence of this result is that the class of languages recognized by a generalization of nonuniform finite automata to nonconstant space is closed under complement.

Formal Characterisation of Immediate Actions in SPA with Nondeterministic Branching

Stochastic Process Algebras (SPA) are process algebras in which the duration of each activity is given by a random variable. If the stochastic aspect is restricted to Markovian, i.e. exponentially distributed durations, nice algebraic foundations are available. They include a formal semantics and an equational theory for Markovian bisimulation, a congruence that can be seen as a stochastic counterpart of strong bisimulation. This paper extends that theory with a stochastic notion of Milner's observational congruence. We enrich a basic SPA with immediate actions that happen instantaneously if enabled. For the enriched calculus we will derive a sound and complete characterisation of Markovian observational congruence, a conservative extension of both Markovian bisimulation and observational congruence. The usefulness of immediate actions together with their equational theory will be illustrated by means of an example. Additionally, we will discuss some implementation and modelling issues arising from our results.

Automatic derivation of test cases for LOTOS expressions with data parameters

AbstractIt is known as generally impossible to generate automatically the test cases for arbitrary LOTOS expression containing data parameters. A subclass of LOTOS, called P‐LOTOS, is defined. In P‐LOTOS, the data type is restricted to be integer and Boolean and the operation on the integer is restricted to be the addition/subtraction and comparison.This paper presents an algorithm for automatic generation of the test cases for this class of LOTOS descriptions. Algorithms also are presented solving the detection problems for the deadlocks, the unexecutable edges, and the nondeterministic branches. The proposed algorithm is applied to a simplified OSI session protocol, and the result is reported in this paper.

Combining Unification Algorithms

An algorithm is presented for solving equations in a combination of arbitrary theories over disjoint sets of function symbols. It is an extension of (Boudet el al. 1989) in which the problem was treated for the combination of an arbitrary and a simple theory. The algorithm consists of a set of transformation rules that simplify a unification problem until a solved form is obtained. Each rule is shown to preserve solutions, and solved problems are unification problems in normal form. The rules terminate for any control that delays replacement until the end. The algorithm is more efficient than Schmidt-Schauß (1990) because nondeterministic branching is performed only when necessary, that is when theory clashes or compound cycles are encountered. The results presented here come from the author's thesis (Boudet (1990a)).

The power of nondeterminism in polynomial- size bounded-width branching programs

Nondeterministic branching programs introduced by Meinel (1986) proved to be an interesting computational tool for describing higher complexity classes (Meinel 1988). Investigation of the power of nondeterminism in the case of bounded-width nondeterministic branching programs yields the following results: 1.1-time-only-nondeterministic, polynomial-size, and bounded-width branching programs are no more powerful than (ordinary) polynomial-size and bounded-width branching programs;2.k-times-only-nondeterministic, polynomial-size, and bounded-width branching programs, (k > 1), are as powerful as nondeterministic, polynomial-size, and unbounded-width branching programs.That is, Pbw−n1BP = NC1 and Pbw−nkBP = NPpoly, k > 1.

Complementing deterministic Büchi automata in polynomial time

For any Buchi automaton Γ with n states which accepts the (ω-regular) language L (Γ), an explicit construction is given for a Büchi automaton Γ with 2n states which, when Γ is deterministic, accepts exactly the complementary language L (Γ)′. It follows that the nonemptiness of complement problem for deterministic Buchi automata ( i.e., whether L (Γ)′ = ⊘) is solvable in polynomial time. The best previously known construction for complementing a deterministic Büchi automaton with n states has O(2 4 n 2) states; for nondeterministic Γ, determining whether L (Γ)′ = ⊘, is known to be PSPACE-complete. Interest in deterministic Büchi automata arises from the suitability of deterministic automata in general to describe properties of physical systems; such properties have been found to be more naturally expressible by deterministic automata than by nondeterministic automata. However, if Γ is nondeterministic, then Γ provides a “poor man's” approximate inverse to Γ in the following sense: L (Γ)′ ⊂ L ( Γ ), and as nondeterministic branches of T are removed, the two languages become closer. Hence, for example, given two nondeterministic Buchi automata Λ and Γ, one may test for containment of their associated languages through use of the corollary that L (Λ ∗ Γ = ⊘ ⇒ L (Λ) ⊂ L (Γ) (where Γ ∗ Γ is one of the standard constructions satisfying L (Λ ∗ Γ ) = L (Λ) ∩ L ( Γ )). The “error term” L = L ( Γ ) ⧹ L (Γ)′ may be deter exactly, and whether L = ⊘ may be determined in time O( e 2), where e is the number of edges of Γ.