Combining Logic Systems: Why, How, What For?

The practical significance of the problem of combining logics is widely recognized, namely in knowledge representation (within artificial intelligence) and in formal specification and verification of algorithms and protocols (within software engineering and security). In these fields, the need for working with several calculi at the same time is the rule rather than the exception. For instance, in a knowledge representation problem it may be necessary to work with temporal, spatial, deontic and probabilistic aspects (e.g., for reasoning with mixed assertions like “with probability greater than 0.99, sometime in the future smoking will be forbidden everywhere”). And in a software verification problem it may be necessary to mix equational, epistemic and dynamic logic features. That is, one needs, at least, to be able to develop theories with components in different logic systems, or, even better, to work with theories defined in the combination of those logic systems (where such mixed assertions are allowed). Motivated by these applications that require the joint use of several deduction formalisms, the interest in combination of logic systems has recently been growing (as reflected in the series [9, 20, 2, 18, 26, 1]), but the topic is also of interest on purely theoretical grounds. For instance, one might be tempted to look at predicate temporal logic as resulting from the combination of first-order logic and propositional temporal logic. However, the approach will be significant only if general preservation results are available about the combination mechanism at hand. For example, if it has been established that completeness is preserved by a combination mechanism • and it is known that logic system L is given by L′ • L′′, then the completeness of L follows from the completeness of L′ and L′′. No wonder that much theoretical effort has been dedicated to establishing preservation results and/or finding preservation counterexamples about different combination mechanisms. For an early overview of the practical and theoretical issues see also [4]. Several forms of combination have been studied, like product [30, 21, 22, 23], fusion [38, 28, 29, 40, 19], temporalization [12, 13, 41, 14], parameterization [6], synchronization [33] and fibring [15, 16, 3, 17, 34, 42]. Fusion is the best understood combination mechanism. In short, the fusion of two modal systems leads to a bimodal system including the two original modal operators and common propositional connectives. Several interesting properties of logic systems (like soundness, weak completeness, Craig interpolation property and decidability) were shown to be preserved when fusing modal systems (see [28, 27]). More recently, research has been directed at fibring, a more general combination mechanism proposed by Gabbay [15, 16]. Fibring can be applied beyond the universe of modal systems and captures fusion as a special case. Although well