We discuss an algebraic approach to propositional logic with side effects. To this end, we use Hoare’s conditional [1985], which is a ternary connective comparable to if-then-else. Starting from McCarthy’s notion of sequential evaluation [1963] we discuss a number of valuation congruences and we introduce Hoare-McCarthy algebras as the structures that characterize these congruences.
Further results from @cite_4 concern binary connectives: we prove that the conditional connective cannot be expressed modulo @math (or any finer congruence) if only binary connectives are allowed, but that it can be expressed modulo @math (and @math ); for @math we leave this question open. In the papers @cite_4 @cite_1 we use the notation @math (taken from @cite_6 ) for left-sequential conjunction, defined by [x y=y x , ] and elaborate on the connection between sequential binary connectives, the conditional and negation, defined by [ x= x . ] @cite_1 we define various : the fragments of proposition algebra that remain if only @math and @math can be used. These logics (various choices can be made) are put forward for modeling conditions as used in programming. Typical laws that are valid with respect to each valuation congruence are the associativity of @math , the double negation shift, and @math (and, as explained in the Introduction, a typical non-validity is @math ).
@cite_1 @cite_4 @cite_6
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We propose a combination of Kleene’s three-valued logic and ACP process algebra via the guarded commandconstruct. We present an operational semantics in SOS-style, and a completeness result. © 1998 Elsevier Science B.V. All rights reserved. ABSTRACT In this paper, we survey 3-valued logics and their complete axiomatizations, one of which may be new. We then propose a 4-valued, functionally complete logic that incorporates these 3-valued systems and provide notations for interesting operators and subsystems.