A fundamental result in rewriting is : E |= s ≈ t ⇐⇒1 E ` s = t ⇐⇒2 s↔E t 1. is Birkhoff’s theorem expressing that semantic consequence (|=) coincides with syntactic consequence (`). 2. expresses that equality in equational logic (=) coincides with convertibility in rewriting (↔∗), a property known as logicality. We show that the result can be extended to hold for several sub-equational logics ...

Equational logic programming is an extended programming paradigm of equational programming. Central to the notion of equational logic programming is the problem of solving equations, which is also called unification in equational theories. In this paper, we investigate the problem of solving equations in O’Donnell’s equational language. We define an equality theory for this language which adequ...

We formalize equational propositional logic, prove that it is sound and complete, and compare the equational-proof style with the more traditional Hilbert style.

Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multi-sorted version [26], and also to synthesise a new version of the Nominal Algebra...

Equational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of equations using a simple graphical syntax called !-box notation. While this does greatly increase the proving power of string diagrams, previous attempts to go beyond equational reasoning have been l...