My second major interest is in modeling changing information, i.e. learning new facts and accommodating incompatible information.
My thesis contained several results on circumscription. An extended version of the second chapter of my thesis is to appear as [Cos98e]. The major extension beyond my thesis is a result showing that circumscription cannot model a certain default, (the default of temporal projection mentioned earlier). This default has been widely considered by Lin and Reiter, Lifschitz and Giunchiglia and others, and thus it is notable that circumscription is incapable of modeling it.
[Cos98c] contains an extension of my thesis results on allowing varying domains in circumscription. The notable extension is a set of results on when this new form of circumscription can be reduced to a first order formula.
The idea of Cofinal entailment, suggested in my thesis, has been extensively reconsidered and a version [Cos98a] has been accepted by CSLI press for a forth-coming volume. This work contains examples that highlight the difference between minimal entailment, Lewis's idea of truth in sufficiently close possible worlds, and my notion of cofinal entailment, the only form of entailment that preserves satisfiability for arbitrary orders. (Lewis's form preserves satisfiability only for total pre-orders.)
As well as looking at circumscription I have examined counterfactuals, another approach to modeling how things might be if things were different. John McCarthy and I [CM98b] have suggested ``Cartesian counterfactuals'', which differ from classical counterfactuals. They add a new dimension to the regular notion of counterfactual, as they explicitly prescribe a Cartesian product structure over the space of possibilities. Changing to a world where the premise is the case is then done relative to this frame. This allows more information to be stated, and allows different views of the world to be countenanced explicitly, explaining certain earlier problems, (In Korea, Caesar would have used nuclear weapons and catapults). The formalism also allows inference from counterfactuals. That is, a counterfactual can imply new facts about the real world.
A major problem in belief revision was how a believer could represent how his mind should change. Certain triviality results were proved that showed that any believer, (satisfying certain minimal postulates) that could believe sentences of the form ``If I was told A I would believe B'', (A > B) was trivial. In [Cos96] I showed that this triviality result could be overcome by appealing to counterfactuals. I showed that non-trivial agents could believe propositions of the form ``If, in contradiction to my current beliefs I discovered A, I would believe B'' (). The old conditional is definable in terms of as if and only if and , else if . As unprovability is not representable in the object language this definability does not lead to triviality. The paper goes on to show that the standard Lewis axiomatization of counterfactuals exactly corresponds to the postulates on belief revision that Alchourrón, Gárdenfors and Makinson proposed. This solved the problem, first considered Lewis in 1976 and more explicitly by Gárdenfors in 1986, of how an agent could non-trivially have knowledge of how to change his mind.
In addition to circumscription and counterfactuals I have looked, with Anna Patterson, at relations between linear logic and , and how exponentials are modeled [PC97a, PC97b]. We have also looked at operations and quantifiers on modalities. In [CP98a, CP98b] we give a set of operations on contexts/modalities that can generate all first order definable functions on modalities. This gives rise to a formal system that can quantify over modalities, with a closure condition (first order definability) on the universe of possible accessibility relations.
My work with Anna Patterson aims towards a logical spreadsheet that will allow logically represented plans and relations to be manipulated. This will use the notion of Cartesian counterfactuals to allow update of propositions, the operations on modalities to combine information sources, and our work on linear logic [PC97c] to capture how the inputs or assumptions of a proof need to be modified in order to yield a certain output.