Combining Equational Reasoning

Ashish Tiwari

Abstract: Given a theory \(\mathbb{T}\), a set of equations E, and a single equation e, the uniform word problem (UWP) is to determine if \(E\Rightarrow e\) in the theory \(\mathbb{T}\). We recall the classic Nelson-Oppen combination result for solving the UWP over combinations of theories and then present a constructive version of this result for equational theories. We present three applications of this constructive variant. First, we use it on the pure theory of equality (\(\mathbb{T}_{EQ}\)) and arrive at an algorithm for computing congruence closure of a set of ground term equations. Second, we use it on the theory of associativity and commutativity (\(\mathbb{T}_{AC}\)) and obtain a procedure for computing congruence closure modulo AC. Finally, we use it on the combination theory \(\mathbb{T}_{EQ}\cup\mathbb{T}_{AC}\cup\mathbb{T}_{PR}\), where \(\mathbb{T}_{PR}\) is the theory of polynomial rings, to present a decision procedure for solving the UWP for this combination.