Section: New Results

Subexponentials

Participants : Dale Miller, Vivek Nigam.

The inference rules of a logic define a logical connective in a canonical fashion if the following test is passed: assume that there are two copies of a logical connective, say a red and blue copy, and assume that both of these connectives have the same introduction rules. If it is possible to prove that the red and blue versions are equivalent within the extended proof system, then we say that the connective is defined canonically. In linear logic, all connectives are canonical in this sense except for the exponentials (!, ?). That is, it is possible to have many exponentials and they do not need to all support weakening and contraction but some subsets of these structural rules. Since they do not need to provide all structural rules, we have called these not exponentials but subexponentials [19] .

Proof theory does not provide canonical solutions for many things in computational logic: for example, the domain of first-order quantification is seldom addressed by proof theory as well as the exact nature of worlds within, say, Kripke models. These non-canonical aspects of logic provide, however, important opportunities for computer scientists to attach structures that they need to logic. Since the exponentials are not canonical, maybe there are possible exploitations of such non-canonical exponentials in computer science. Nigam and Miller have provided a partial answer to this question. In particular, they have shown that rich forms of multiset computation can be supported using subexponentials [19] [12] . In particular, it is possible to specify various multisets with different locations (identified with different exponentials) and to test them for emptiness. In this way, linear logic with an array of subexponential can then be used to declaratively and faithfully specify a wide range of deterministic and non-deterministic algorithms.