Defining soft sortedness by abstract interpretation

Sorted languages can improve the expressiveness and efficiency of reasoning. A conventional sorted language typically includes well-sortedness rules amongst the rules for well-formedness. A major disadvantage of this approach is that many intuitively meaningful expressions are ill-sorted and hence not part of the language. To overcome this limitation, soft sorting regards as well-formed, all firstorder expressions of the corresponding unsorted language, and lets the semantics be the basis for defining the significance of the sort syntax. In this paper we show how soft sortedness can be defined by abstract interpretations which characterise semantic properties of softly sorted expressions

Defining Soft Sortedness by Abstract Interpretation

Abstract When a language refers to a universe of discourse which contains more than one sort of object, it is often useful for the language to include notations for describing sort restrictions on functions and predicates so as to achieve more flexible representations and more efficient reasoning. Increased efficiency is typically based on sort restrictions, which may be checked either statically or dynamically. Languages using syntactic sort constraints ("signatures") on function and relation symbols support more static sort checking but are less expressive than languages using semantic sort constraints ("sort predicates"). In this paper, we describe an approach, called soft sorting, in which both static and dynamic sort checking can be performed within an unified framework. In this framework, we aim to do as much static sort checking as possible, relying on dynamic sort checking only when necessary. We describe the basic concepts and results of this approach in the context of first-order languages. 1 Introduction Symbols in a first-order language denote objects of an intended universe of discourse and functions and relations on them. When a universe of discourse contains more than one sort of object, it is often useful to include in the language, notations for describing sort constraints on those functions and predicates, so as to achieve more flexible representations and more efficient reasoning. Hereafter we will use the word "expression " to refer generally to terms and formulas. In this paper, sorting means the use of sort constraints on the symbols of a language and consequently on expressions of the language