Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.
Table 1. Syntax and Semantics of a Calculus for Propositional Logic
The simplest expression for logical truth is the empty word, typically denoted by or in formal languages, where it is the identity element for concatenation. To make it visible in context, it may be denoted by the equivalent expression or, especially if operating in an algebraic context, by a simple Also when working in an algebraic mode, the plus sign may be used for exclusive disjunction. Thus we have the following translations of algebraic expressions into cactus expressions.
It is important to note the last expressions are not equivalent to the 3-place form