Fracterm Calculus for Partial Meadows
This work offers a formal foundation for partial meadows, which may be of interest to researchers in algebraic specification and partial algebras, but the results are incremental and theoretical.
The paper develops a fracterm calculus for partial meadows, providing a formalization of fields with a division operator using three-valued short-circuit logic. It shows that the resulting logic cannot express that division by zero must be undefined, and that the ⊥-enlargement of a partial meadow is a common meadow.
Partial algebras and datatypes are discussed with the use of signatures that allow partial functions, and a three-valued short-circuit (sequential) first order logic with a Tarski semantics. The propositional part of this logic is also known as McCarthy calculus and has been studied extensively. Axioms for the fracterm calculus of partial meadows are given. The case is made that in this way a rather natural formalisation of fields with division operator is obtained. It is noticed that the logic thus obtained cannot express that division by zero must be undefined. An interpretation of the three-valued sequential logic into $\bot$-enlargements of partial algebras is given, for which it is concluded that the consequence relation of the former logic is semi-computable, and that the $\bot$-enlargement of a partial meadow is a common meadow.