Abstract
We extend universal algebra and its equational logic from first to second order as follows. 1 We consider second-order equational presentations as specified by identities between second-order terms, with both variables and parameterised metavariables over signatures of variable-binding operators. 1 We develop an algebraic model theory for second-order equational presentations, generalising the semantics of (first-order) algebraic theories and of (untyped and simply-typed) lambda calculi. 1 We introduce a deductive system, Second-Order Equational Logic, for reasoning about the equality of second-order terms. Our development is novel in that this equational logic is synthesised from the model theory. Hence it is necessarily sound. 1 Second-Order Equational Logic is shown to be a conservative extension of Birkhoff’s (First-Order) Equational Logic. 1 Two completeness results are established: the semantic completeness of equational derivability, and the derivability completeness of (bidirectional) Second-Order Term Rewriting.
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