Operationelle und funktionale Semantik von Σ-Graphen mit Anwendungen auf LISP

Dieser Bericht ist im Wortlaut identisch mit: Peter Padawitz, Church-Rosser-Eigenschaften von Graph-Grammatiken und Anwendungen auf die Semantik von LISP, Diplomarbeit 1978.Previous studies of operational versus functional semantics of symbolic expressions mostly have been confined to treelike expressions and evaluation by "simplification" and substitution of recursive definitions for function symbols. In order to drop these restrictions we introduce Σ-graphs and Σ-grammars to represent expressions and evaluation rules, respectively. Functional semantics of Σ-graphs is defined as an extension of Scott's fixed point semantics of flow diagrams. We prove that derivations via a Σ-grammar P preserve the functional semantics of Σ-graphs if the underlying "semantic algebra" satisfies the equations given by P. To get an operational semantics of a Σ-graph G relative to a Σ-grammar P derivations of G via P must yield a unique normal form. Therefore sufficient conditions for a weak Church-Rosser property of Σ-grammars are formulated and proved for some classes of such grammars. Applying these results to the programming language LISP we show that the evaluation rules of a LISP interpreter are compatible with the semantics of LISP and weak Church-Rosser where garbage collection is included

Implicit complexity for coinductive data: a characterization of corecurrence

We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using productivity (fairness) as the fundamental assertion, rather than bi-simulation. The latter is expressible in terms of the former. As an application to this framework, we give an implicit characterization of corecurrence: a function is definable using corecurrence iff its productivity is provable using coinduction for formulas in which data-predicates do not occur negatively. This is an analog, albeit in weaker form, of a characterization of recurrence (i.e. primitive recursion) in [Leivant, Unipolar induction, TCS 318, 2004].Comment: In Proceedings DICE 2011, arXiv:1201.034

Parameter-preserving data type specifications

AbstractTerm rewriting methods are used for solving the persistency problem of parametrized data type specifications. Such a specification is called persistent if the parameter part of its algebraic symantics agrees with the semantics of the parameter specification. Since persistency mostly cannot be guaranteed for the whole equational variety of the parameter specification, the persistency criteria developed here mainly concern classes of parameter algebras with “built-in” logic