[Paper Review] An Emptiness Algorithm for Regular Types with Set Operators

Lunjin Lu, John G. Cleary|ArXiv.org|Nov 11, 1998

Logic, programming, and type systems32 references3 citations

TL;DR

This paper presents a decision algorithm for determining the emptiness of regular type expressions extended with set operators (union, intersection, complement), given parameterized type definitions. It generalizes prior work by not assuming tuple distributivity and supports full set-theoretic operations, enabling equivalence and inclusion checking between regular type expressions through emptiness checks on their set differences.

ABSTRACT

An algorithm to decide the emptiness of a regular type expression with set operators given a set of parameterised type definitions is presented. The algorithm can also be used to decide the equivalence of two regular type expressions and the inclusion of one regular type expression in another. The algorithm strictly generalises previous work in that tuple distributivity is not assumed and set operators are permitted in type expressions.

Motivation & Objective

• To develop a decision procedure for determining whether a regular type expression with set operators is empty.
• To extend existing emptiness algorithms to support full set-theoretic operations (union, intersection, complement) in type expressions.
• To eliminate the need for tuple distributivity assumptions, which restrict prior approaches.
• To enable equivalence and inclusion checking between regular type expressions via emptiness testing of set differences.
• To provide a formal and general framework for reasoning about regular types with complex set operations.

Proposed method

• The algorithm operates on regular type expressions defined over a set of parameterized type definitions.
• It uses a closure-based construction to compute the set of all possible values representable by a type expression.
• Set operations (union, intersection, complement) are handled through algebraic manipulation of regular expressions and their associated automata.
• Emptiness is determined by checking whether the closure of a type expression contains any valid value.
• Equivalence and inclusion are reduced to emptiness checks on the symmetric difference and relative complement, respectively.
• The method avoids tuple distributivity by directly manipulating type expressions and their semantics using formal language theory.

Experimental results

Research questions

• RQ1 Can we decide the emptiness of a regular type expression that includes set operators like union, intersection, and complement?
• RQ2 Does the algorithm remain sound and complete without assuming tuple distributivity in type expressions?
• RQ3 Can the emptiness decision procedure be extended to check for equivalence and inclusion between regular type expressions?
• RQ4 How does the algorithm handle parameterized type definitions within complex set operations?
• RQ5 What is the theoretical foundation that allows the generalization beyond prior work relying on tuple distributivity?

Key findings

• The algorithm correctly determines the emptiness of any regular type expression with set operators, even without tuple distributivity.
• The approach supports equivalence and inclusion checking by reducing them to emptiness checks on derived expressions.
• The method generalizes previous work by allowing arbitrary set operators in type expressions, not just union.
• The algorithm is sound and complete for the given formalism, as proven through closure and automata-theoretic reasoning.
• The framework is extensible to parameterized type definitions, enabling modular reasoning over complex type structures.
• The solution is formally grounded in logic and automata theory, ensuring correctness and decidability.