Ground confluence of order-sorted conditional specifications modulo axioms

Terminating functional programs should be deterministic, i.e., should evaluate to a unique result, regardless of the evaluation order. For equational functional programs such determinism is exactly captured by the ground confluence property. For operationally terminating conditional equations this is equivalent to ground local confluence, which follows from local confluence. Checking local confluence by computing critical pairs is the standard way to check ground confluence [33]. The problem is that some perfectly reasonable equational programs are not locally confluent and it can be very hard or even impossible to make them so by adding more equations. We propose three methods, called Methods 1–3, that can be synergistically combined to prove an order-sorted conditional specification modulo axioms B ground locally confluent. Method 1 applies the strategy proposed in [14] to use non-joinable critical pairs as completion hints to either achieve local confluence or reduce the number of critical pairs. Method 2 uses the inductive joinability inference system proposed in this paper to try to prove the critical pairs remaining after applying Method 1 ground joinable. It can furthermore show ground local confluence of the original specification. Method 3 is hierarchical in nature: it can be used to prove the ground local confluence of a conditional equational specification whose conditions belong to a subspecification that has already been proved ground confluent and operationally terminating, and that is conservatively extended by the overall specification in an appropriate sense. These methods apply to order-sorted and possibly conditional equational programs modulo axioms such as, e.g., Maude functional modules. We show their effectiveness in proving the ground confluence of non-trivial examples that have eluded previous proof attempts.