Tsutomu Kamimura

Parallel and two-way automata on directed ordered acyclic graphs

In this paper we study automata which work on directed ordered acyclic graphs, in particular those graphs, called derivation dags ( d -dags), which model derivations of phrase-structure grammars. A rather complete characterization of the relative power of the following features of automata on d -dags is obtained: parallel versus sequential, deterministic versus nondeterministic and finite state versus a (restricted type of) pushdown store. New results concerning trees follows as special cases. Closure properties of classes of d -dag languages definable by various automata are studied for some basic operations. Characterization of general directed ordered acyclic graphs by these automata is also given. Contents. Abstract. 1. Introduction. 2. Definitions of the graphs. 3. Parallel dag automata. 4. Finite state relabeling. 5. Two-way dag-walking automata. 6. Comparison. 7. Closure properties of d -dag languages. 8. Recognition of doags. Acknowledgment. References.

Total objects of domains

Abstract The main result of this paper is a characterization of the space induced by Scott topology on the set of total objects of a bounded-complete cpo. This characterization is simple with bounded-complete algebraic cpos where a family of clopen sets plays an essential role in the characterization. In the more general case of continuity, one can only produce distinct families of open sets and closed sets to axiomatize the essential properties of a total space. Besides the main characterization, we also discuss its relation to continuous spaces, studied in our earlier paper, and the relation between compact T spaces and total spaces under the lower topology.

Tree automata and attribute grammars

The translational mechanism of attribute grammars using tree automata are investigated. The pushdown tree-to-string transducer with a certain synchronization facility as a model to realize transformations by attribute grammars is proposed and its basic properties using tree-walking finite state automata are studied. To demonstrate the utility of this model, it is shown that noncircular attribute grammars are equally powerful as arbitrary attribute grammars, and a method is provided to show that a certain type of transformations is impossible by attribute grammars.The translational mechanism of attribute grammars using tree automata are investigated. The pushdown tree-to-string transducer with a certain synchronization facility as a model to realize transformations by attribute grammars is proposed and its basic properties using tree-walking finite state automata are studied. To demonstrate the utility of this model, ~t is shown that noncircular attribute grammars are equally powerful as arbitrary attribute grammars, and a method is provided to show that a certain type of transformations is impossible by attribute grammars.

Transductions of dags and trees

Directed acyclic graphs (dags) model derivations of phrasestructure grammars analogously to the way that trees model derivations of context-free grammars.In this paper we introduce translations of such dags which naturally extend the bottom-up tree translations. Composition results of these dag-to-tree transformations are studied. It is shown that every “recursively enumerable tree language” can be obtained from a recognizable dag language by such a transduction. Tree languages obtained from some subsets of recognizable dag languages by these transductions are investigated.

Effectively given spaces

Abstract The theory of domains has been presented in various formalisms. The purpose of this paper is to show how to use the ideas in effectively given domains to formalize a theory of effectively given T 0 spaces. In an effectively given space, there are two equivalent formulations of the notion of a computable object, an operational one and a topological one. The operational notion may be viewed as an implementation of the topological notion.

Algebraic relations and presentations

Abstract In this paper the authors examine the mathematical foundations of taking quotients in the category of Scotts continuous lattices (Scott, 1977). They study two questions: (1) What are the limitations of taking quotients in the category of algebraic lattices? (2) What is the meaning of an effectively given congruence relation?

An effectively given initial semigroup

SummaryThe notion of effectively given domains is extended to semigroups. We show that the initial algebra for the semigroups with finite generators is effectively given. The initial algebra we describe uses expressions and trees of symbols of generators and the partial order in the algebra is defined using a formal system on expressions. The main body of the paper lies in the demonstration of the decidability of this formal system.

Kleene chain completeness and fixedpoint properties

Fixedpoint theorems play a fundamental role in denotational semantics of programming languages. In 1955, Tarski [6] and Davis [l] showed that a lattice L is complete if and only if every monotonic function f: L + L has a fixedpoint. Since then, other fixedpoint properties are considered, by imposing either the function to be o-continuous or the fixedpoint to be the least fixedpoint. The key question is whether such fixedpoint properties can as well be characterized by some completeness properties of the given partially ordered set. In this direction, Markowsky [2] showed that a partially ordered set is chain-complete if and only if every monotonic function f: D + D has a least fixedpoint. Suppose we replace monotonic functions by o-continuous functions which play a prominent role in Scott’s theory of computation [4,5]. A slight modification of Tarski’s proof shows that if a partially ordered set D with a least element 1 is o-chain complete, then every w-continuous function f: D + D has a least fixedpoint given by UnEWfn (I). The latter result is the wellknown Tarski-Kleene-Knaster theorem. In 1978, Plotkin asked the validity of the converse of Tarski-Kleene-Knaster theorem beta use an affirmative answer would give us a characterization of w-chain complete pin-tially ordered sets in terms of the least fixedpoint property for o-continuous functions. Throughout the paper, D stands for a partially .3rdered set with a least element 1. Let us consider Plotkin’s puzzle in the followirlg version: Given a D, if every o-continuous function f: D + D has a least fixedpoint given by UIltWf’*(l), is D o-chain complete? In this paper, we answer the problem negatively (Mashburn obtained the same result independently in [3j). Despite this negative answer, we show a rather astonishing result, namely: If D is either countable or countably algebraic, then the converse of Tarski-Kleene-Knaster theorem is true. This positive answer is rather pleasing because most of the D’s used in denotational semantics are either countable or countably algebraic.