Equivalence and difference between institutions: simulating Horn Clause Logic with based algebras

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In this paper, we investigate several logical frameworks whose expressiveness lies between Conditional Equational Logic and Horn Clause Logic. The main result deals with thePART-construction, which interprets total based algebras as partial algebras. This construction can be viewed as a simulation of Horn Clause Theories by means of Conditional Equational Theories. Other constructions in other frameworks are extendable to simulations in a similar way. The notion of categorical retractive simulation captures some essential properties of these, which allows us to measure the equivalence and difference between institutions.