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Mathematical systems theory

, Volume 13, Issue 1, pp 1–27 | Cite as

An order-algebraic definition of knuthian semantics

  • Laurian M. Chirica
  • David F. Martin
Article

Abstract

This paper presents a formulation, within the framework of initial algebra semantics, of Knuthian semantic systems (K-systems) which contain both synthesized and inherited attributes. The approach is based on viewing the semantics of a derivation tree of a context-free grammar as a set of values, called an attribute valuation, assigned to the attributes of all its nodes. Any tree's attribute valuation which is consistent with the semantic rules of the K-system may be chosen as the semantics of that derivation tree.

The set of attribute valuations of a given tree is organized as a complete partially ordered set such that the semantic rules define a continuous transformation on this set. The least fixpoint of this transformation is chosen as the semantics of a given derivation tree. The mapping from derivation trees to their least fixpoint semantics is a homomorphism between certain algebras. Thus, the semantics of a K-system is an application of the Initial Algebra Semantics Principle of Goguen and Thatcher. This formulation permits a precise definition of K-systems, and generalizes Knuth's original formulation by defining the meaning of recursive (circular) semantic specifications.

The algebraic formulation of K-systems is applied to proving the equivalence of K-systems having the same underlying grammar. Such proofs may require verifying that a K-system possesses certain properties and to this end, a principle of structural induction on many-sorted algebras is formulated, justified, and applied.

Keywords

Computational Mathematic Original Formulation Precise Definition Algebraic Formulation Attribute Valuation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Laurian M. Chirica
    • 1
  • David F. Martin
    • 2
  1. 1.Computer Science DepartmentUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Computer Science DepartmentUniversity of CaliforniaLos Angeles90024

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