Quasiequational theory

A \emph{quasiequation} is a universal formula of the form $\phi_1 \mbox{ and } \phi_2 \mbox{ and }\cdots\mbox{ and } \phi_m\ \Longrightarrow\ \phi_0$, where the $\phi_i$ are atomic formulas. Note that for an algebraic language, the $\phi_i$ are simply equations. For $m=0$, a quasiequation is just a universal atomic formula.

The \emph{quasiequational theory} of a class of structures is the set of quasiequations that hold in all members of the class.

The \emph{decision problem} for the quasiequational theory of a class of structures is the problem with input: a quasiequation of length $n$ (as a string) and output: “true” if the quasiequation holds in all members of the class, and “false” otherwise.

The quasiequational theory is \emph{decidable} if there is an algorithm that solves the decision problem, otherwise it is \emph{undecidable}.

The complexity of the decision problem (if known) is one of PTIME, NPTIME, PSPACE, EXPTIME, …

A complete deductive system for quasiequations is given in [A. Selman, \emph{Completeness of calculi for axiomatically defined classes of algebras}, Algebra Universalis, \textbf{2}, 1972, 20–32 MRreview. Additional information on quasiequations can be found e.g. in Stanley N. Burris and H.P. Sankappanavar, A Course in Universal Algebra.

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