Applied and Computational Mathematics Multisorted tree algebra (original) (raw)

Imbrication algebras - algebraic structures of nesting order

Comput. Sci. J. Moldova, 2018

This paper is about “imbrication algebras”, universal algebras with one binary operator in their signature, the operator for formation of ordered pairs, called here “pairing operator”, and with the “characteristic property of ordered pairs” as their sole axiom. These algebras have been earlier introduced by the first author as reducts of “aggregate algebras”, universal algebras proposed as models for a set theory convenient for formalization of data structures. The term “aggregate” is used to generalize three fundamental notions of set theory: set, atom and ordered pair. Thus, this paper initiates the research of aggregate algebras by narrowing the focus to one type of their main reducts – the reduct which deals with ordered pairs.

Tree algebras and varieties of tree languages

Theoretical Computer Science, 2007

We consider several aspects of Wilke's [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput. Sci. 154 (1996) 85-106] tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way. A convergent term rewriting system yields normal form representations of binary trees and contexts, as well as a new completeness proof and a computational decision method for the axiomatization of tree algebras. Varieties of binary tree languages are compared with varieties of tree languages studied earlier in the literature. We also prove a variety theorem thus solving a problem noted by several authors. Syntactic tree algebras are studied and compared with ordinary syntactic algebras. The expressive power of the language of tree algebras is demonstrated by giving equational definitions for some well-known varieties of binary tree languages.

Multisorted Tree-Algebras for Hierarchical Resources Allocation

This paper presents a generic abstract model for the study of disparities between goals and results in hierarchical multiresources allocation systems. In an organization, disparities in resource allocation may occur, when, after comparison of a resource allocation decision with an allocation reference goal or property, some agents have surplus resources to accomplish their tasks, while at the same time other agents have deficits of expected resources. In the real world, these situations are frequently encountered in organizations facing scarcity of resources and/or inefficient management. These disparities can be corrected using allocation decisions, by measuring and reducing gradually such disparities and their related costs, without totally canceling the existing resource distribution. While a lot of research has been carried out in the area of resource allocation, this specific class of problems has not yet been formally studied. The paper exposes the results of an exploratory research study of this class of problems. It identifies the commonalities of the family of hierarchical multiresource allocation systems and proposes the concept of multisorted tree-algebra for the modeling of these problems. The research presented here is not yet an in-depth descriptive research study of the mathematical theory of multisorted tree-algebra, but a formal study on modelling hierarchical multiresource allocation problems.

VARIETIES OF REGULAR ALGEBRAS AND UNRANKED TREE LANGUAGES

In this paper we develop a variety theory for unranked tree languages and unranked algebras. In an unranked tree any symbol may label a node with any number of successors. Such trees appear in markup languages such as XML and as syntactic descriptions of natural languages. In the corresponding algebras each operation is defined for any number of arguments, but in the regular algebras used as tree recognizers the operations are finite-state computable. We develop the basic theory of regular algebras for a setting in which algebras over different operator alphabets are considered together. Using syntactic algebras of unranked tree languages we establish a bijection between varieties of unranked tree languages and varieties of regular algebras. As varieties of unranked tree languages are usually defined by means of congruences of term algebras, we introduce also varieties of congruences and a general device for defining such varieties. Finally, we show that the natural unranked counterparts of several varieties of ranked tree languages form varieties in our sense.

Algebraic Backgrounds for Data Structures Models

2019

The main aim of the Algebraic Aggregate Theory is to present uniformly data structures in mathematics and its applications in terms of Universal Algebra. In the given paper are characterized three sub-algebras of the Algebraic Aggregate Theory. These sub-algebras are the sub-algebra of ordered pairs, the successor sub-algebra and the algebra of Semi-Boolean Systems. Some applications of the obtained results for solving mathematical problems and software development are illustrated.

An Algebra of Hierarchical Graphs

2010

We define an algebraic theory of hierarchical graphs, whose axioms characterise graph isomorphism: two terms are equated exactly when they represent the same graph. Our algebra can be understood as a high-level language for describing graphs with a node-sharing, embedding structure, and it is then well suited for defining graphical representations of software models where nesting and linking are key aspects.

Inheritance hierarchies: Semantics and unification

Journal of Symbolic Computation, 1989

Inheritance hierarchies are introduced as a means of representing taxonomically organized data. The hierarchies are built up from so-called feature types that are ordered by subtyping and whose elements are records. Every feature type comes with a set of features prescribing fields of its record elements. So-called feature terms are available to denote subsets of feature types. Feature unification is introduced as an operation that decides whether two feature terms have a nonempty intersection and computes a feature term denoting the intersection.

A framework for order-sorted algebra

Algebraic Methodology and Software Technology, 2002

Order-sorted algebra is a generalization of many-sorted algebra obtained by having a partially ordered set of sorts rather than merely a set. It has numerous applications in computer science. There are several variants of order sorted algebra, and some relationships between these are known. However there seems to be no single conceptual framework within which all the connections between the variants can be understood. This paper proposes a new approach to the understanding of order-sorted algebra. Evidence ...

A Category of Ordered Algebras Equivalent to the Category of Multialgebras

Bulletin of the Section of Logic

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s. We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \({\em bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \({\em ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of...

Some Nuances of Many-sorted Universal Algebra: A Review

Bulletin of the EATCS, 2011

proof of the results. We give some examples of this, indicating how equational calculus, Birkhoff's variety theorem and interpolation results should be adjusted for many-sorted algebras. 2 Basic Definitions and Facts 2.1 Many-sorted Sets Let S be any set; we think of elements of S as sort names, or sorts for short. An S-sorted set is an S-indexed family of sets X = X s s∈S. We say that such an S-sorted set X is empty if X s is empty for all s ∈ S. The empty S-sorted set will be written (ambiguously) as ∅. We say that X is everywhere non-empty if X s ∅ for all s ∈ S ; otherwise we say that X is somewhere empty. Clearly, if S has at least two elements, there are S-sorted sets that are neither empty nor everywhere non-empty. S-sorted set X is finite if X s is finite for all s ∈ S and X s = ∅ for almost all s ∈ S (that is, for all but a finite number of s ∈ S , X s = ∅). Let X = X s s∈S and Y = Y s s∈S be S-sorted sets. Union, intersection, Cartesian product, disjoint union, inclusion (subset) and equality of X and Y are defined component-wise in the obvious manner. An S-sorted function f : X → Y is an S-indexed family of functions f = f s : X s → Y s s∈S ; X is called the domain (or source) of f , and Y is called its codomain (or target). An S-sorted function f : X → Y is an identity (inclusion, surjection, injection, bijection, etc) if for every s ∈ S , the function f s : X s → Y s is an identity (inclusion, surjection, injection, bijection, etc). The identity S-sorted function on X will be written as id X : X → X.