Simple context-free languages and free monadic recursion schemes

A context-free language is said to be simple if it is accepted by a single-state deterministic pushdown store acceptor that operates in real-time and accepts by empty store. While the problem remains open of deciding whether or not the language accepted by a deterministic pushdown store acceptor is simple, it is shown that this problem is equivalent to another problem in schemata theory. This question is that of determining whether or not a monadic recursion scheme has a strongly equivalent free scheme.This research was supported in part by the National Science Foundation, Grant No. NSF GJ-803 and DCR74-15091.     

Monadic recursion schemes: The effect of constants

When the class of monadic recursion schemes is augmented by individual constants, some of the properties change. It becomes undecidable whether “S diverges” or “S is strongly equivalent to T” for S, T schemes with individual constants. The family of value languages generated by this class of schemes is the family of recursively enumerable languages. The subclass of free schemes with constants is also investigated. It remains decidable whether “S halts” or “S diverges” for S a free scheme with individual constants, but it becomes undecidable whether “T has a strongly equivalent free scheme” for T an arbitrary scheme with individual constants.     

Refining complexity analyses in planning by exploiting the exponential time hypothesis

The use of computational complexity in planning, and in AI in general, has always been a disputed topic. A major problem with ordinary worst-case analyses is that they do not provide any quantitative information: they do not tell us much about the running time of concrete algorithms, nor do they tell us much about the running time of optimal algorithms. We address problems like this by presenting results based on the exponential time hypothesis (ETH), which is a widely accepted hypothesis concerning the time complexity of 3-SAT. By using this approach, we provide, for instance, almost matching upper and lower bounds onthe time complexity of propositional planning.     

Propositional lower bounds: Algorithms and complexity

Propositional greatest lower bounds (GLBs) are logically‐defined approximations of a knowledge base. They were defined in the context of Knowledge Compilation, a technique developed for addressing high computational cost of logical inference. A GLB allows for polynomial‐time complete on‐line reasoning, although soundness is not guaranteed. In this paper we propose new algorithms for the generation of a GLB. Furthermore, we give precise characterization of the computational complexity of the problem of generating such lower bounds, thus addressing in a formal way the question “how many queries are needed to amortize the overhead of compilation?”     

The Polynomially Exponential Time Restrained Analytical Hierarchy

A polynomially exponential time restrained analytical hierarchy is introduced with the basic properties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to any level of the p-arithmetical hierarchies.Then we shall prove that there are recursive sets A and B such that the different levels of the analytical hierarchy relative to A are different and for some n every level higher than n of the analytical hierarchy relative to B is the same as the n-th level.And whether the higher levels are collapsed into some lower level is neither provable nor disprovable in set theory and several other results.     

Some bounds on multiparty communication complexity of pointer jumping

We introduce the model of conservative one-way multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping.? The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one node from the next layer. The output is the node reached by following k edges from the starting node. In a conservative protocol, the ith player can see only the node reached by following the first i–1 edges and the edges on the jth layer for each j > i. This is a restriction of the general model where the ith player sees edges of all layers except for the ith one. In a one-way protocol, each player communicates only once in a prescribed order: first Player 1 writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total number of bits written on the blackboard.?Our main results are the following bounds on k-party conservative one-way communication complexity of pointer jumping with k layers:? (1) A lower bound of order for any .?(2) Matching upper and lower bounds of order for . received March 22, 1996     

The time complexity of typechecking tree-walking tree transducers

Tree-walking tree transducers can be typechecked in double exponential time. More generally, compositions of k tree-walking tree transducers can be typechecked in (k + 1)-fold exponential time. Consequently k-pebble tree transducers, which form a model of XML transformations and XML queries, can be typechecked in (k + 2)-fold exponential time. The results hold for both ranked and unranked trees.     

Tight bounds on the complexity index of one-point iterations

We study the minimal complexity index of one-point iterations without memory for the solution of a system of N nonlinear equations F(x)=0. We present an iteration ^* with maximal order of convergence and with linear combinatory complexity. We show the complexity index of ^* is close to the lower bound on the minimal complexity index.     

Lower bounds for the multiplicative complexity of matrix multiplication

We prove a lower bound of km + mn + k—m + n— 3 for the multiplicative complexity of the multiplication of -matrices with -matrices using the substitution method. Received: July 8, 1997.     

Enhancing prototype reduction schemes with recursion: a method applicable for "large" data sets

Most of the prototype reduction schemes (PRS), which have been reported in the literature, process the data in its entirety to yield a subset of prototypes that are useful in nearest-neighbor-like classification. Foremost among these are the prototypes for nearest neighbor classifiers, the vector quantization technique, and the support vector machines. These methods suffer from a major disadvantage, namely, that of the excessive computational burden encountered by processing all the data. In this paper, we suggest a recursive and computationally superior mechanism referred to as adaptive recursive partitioning (ARP)_PRS. Rather than process all the data using a PRS, we propose that the data be recursively subdivided into smaller subsets. This recursive subdivision can be arbitrary, and need not utilize any underlying clustering philosophy. The advantage of ARP_PRS is that the PRS processes subsets of data points that effectively sample the entire space to yield smaller subsets of prototypes. These prototypes are then, in turn, gathered and processed by the PRS to yield more refined prototypes. In this manner, prototypes which are in the interior of the Voronoi spaces, and thus ineffective in the classification, are eliminated at the subsequent invocations of the PRS. We are unaware of any PRS that employs such a recursive philosophy. Although we marginally forfeit accuracy in return for computational efficiency, our experimental results demonstrate that the proposed recursive mechanism yields classification comparable to the best reported prototype condensation schemes reported to-date. Indeed, this is true for both artificial data sets and for samples involving real-life data sets. The results especially demonstrate that a fair computational advantage can be obtained by using such a recursive strategy for "large" data sets, such as those involved in data mining and text categorization applications.     

On lower bounds of the closeness between complexity classes

We show that if an NP-m-hard set is the union of a set in P_btt(Sparse) and the setA, then NP P_dtt(A). Since co-NP, R, and FewP are closed under _dtt ^P -reductions, so no NP-m-hard set is the union of a set in P_btt(Sparse) and a set in co-NP (resp. R, FewP) unless NP = co-NP (resp. NP = R, NP = FewP). We also study the distance between many important complexity classes. LetA,B be two sets. The distance function dist _A,B is defined as in [S1] such that dist _A,B (n) is the number of strings of length n inA B = (A – B) (B – A). We show that there exists a setA in p_{(k + 1)–tt}(Sparse) such that, for everyB in P _k–tt(Sparse), dist _A,B has an exponential lower bound.This research was supported in part by HTP 863.     

Lower bounds for the bilinear complexity of associative algebras

Let R(A) denote the bilinear complexity (also called rank) of a finite dimensional associative algebra A.?We prove that if the decomposition of into simple algebras contains only noncommutative factors, that is, the division algebra is noncommutative or . In particular, -matrix multiplication requires at least essential bilinear multiplications. We also derive lower bounds of the form essential bilinear multiplications. We also derive lower bounds of the form for the algebra of upper triangular -matrices and the algebra of truncated bivariate polynomials in the indeterminates X,Y over some field k.?A class of algebras that has received wide attention in this context con-sists of those algebras A for which the Alder—Strassen Bound is sharp, i.e., R(A) = 2dim A—t is the number of maximal twosided ideals in A. These algebras are called algebras of minimal rank. We determine all semisimple algebras of minimal rank over arbitrary fields and all algebras of minimal rank over algebraically closed fields. Received: January 12, 2000.     

The monadic second-order logic of graphs X: Linear orderings

Graphs are finite and handled as relational structures. We give some answers to the following general questions:

1. (1) For which classes of graphs is it possible to specify a linear ordering of the set of vertices of each graph of by fixed monadic second-order formulas?

2. (2) For which classes of graphs does there exist an extension of monadic second-order logic such that a subclass L of is recognizable if and only if it is the class of graphs in that satisfy a formula of ? (In this paper, recognizability is understood in an algebraic sense, relative to a finite set of graph operations and basic graphs that generate all graphs of .)

3. (3) For which classes of graphs is it possible to construct, in every graph of the class, and by fixed formulas of a suitable extension of monadic second-order logic, its hierarchichal structure, i.e., a finite term written with the operations and basic graphs of (2), that defines the considered graph?

Applications concern dependency graphs of partially commutative words, partial k-paths, cographs, and graphs, the modular decomposition of which uses prime graphs of bounded size.