Many apparently divergent approaches to specifying formal semantics of programming languages are applications of initial algebra semantics. In this paper an overview of initial algebra semantics is provided. The major technical feature is an initial continuous algebra which permits unified algebraic treatment of iterative and recursive semantic features in the same framework as more basic operations.

Initial Algebra Semantics and Continuous Algebras

Publisher Summary This chapter presents three mathematical tools that can be used to describe graph-grammars and the sets generated by them. The chapter describes graph properties by logical formulas and presents a comparison of the powers of several logical languages. The chapter discusses first-order logic, second-order logic, and monadic second-order logic, together with a few variants and restrictions of these three languages. It presents algebraic techniques that are useful for defining and studying graph rewriting rules and context-free graph-grammars. The chapter also presents the use of category theory for specifying graph rewriting rules in a precise and concise way, and for properly defining the least or rather the initial solution of a system of graph equations. It presents some links between the context-free graph-grammars and monadic second-order logic. Some applications to the definition of sets of graphs by forbidden configurations and to the theory of NP-completeness are also presented in the chapter.

Graph Rewriting: An Algebraic and Logic Approach.

This paper describes how Coloured Petri Nets (CP-nets) have been developed — from being a promising theoretical model to being a full-fledged language for the design, specification, simulation, validation and implementation of large software systems (and other systems in which human beings and/or computers communicate by means of some more or less formal rules).

Coloured Petri Nets: a high level language for system design and analysis

Fundamental properties of infinite trees

Tree languages

IO and OI. II

A fixed point characterization of the inside-out (IO) and the outside- in (OI) context-free tree languages is given. This characterization is used to obtain a theory of nondeterministic systems of context-free equations with parameters. Several ''Mezei and Wright like'' results are obtained which relate to context-free tree languages, to recognizable tree languages and to nondeterministic recursive program(scheme)s (called by value and called by name). The emptiness problem and closure properties of the context-free tree languages are discussed. Hierarchies of higher level equational subsets of an algebra are considered.

IO and OI

We define a quasidictionary to be a data structure that supports the following operations: check-in(v) inserts a data item v and returns a positive integer tag to be used in future references to v; check-out(x) deletes the data item with tag x; access(x) inspects and/or modifies the data item with tag x. A quasidictionary is similar to a dictionary, the difference being that the names identifying data items are chosen by the data structure rather than by its user. We describe a deterministic quasidictionary that executes the operations check-in and access in constant time and check-out in constant amortized time, works in linear space, and uses only tags bounded by the maximum number of data items stored simultaneously in the quasidictionary since it was last empty.

Erik Meineche Schmidt

Papers

IO and OI. II

IO and OI

Algorithm Theory — SWAT 2002

Algorithm Theory — SWAT '94

Experimental Algorithmics, From Algorithm Design to Robust and Efficient Software [Dagstuhl seminar, September 2000]