TR-2013005: Realization Implemented
Melvin Fitting · CUNY Academic Works (City University of New York) · 2013
Justification logics are connected to modal logics via realization theorems. These have both constructive and non-constructive proofs. In this report we do two things. First we provide a new path to constructive realization proofs, going through an intermediate quasi-realization stage. Quasirealizers are easier to produce than realizers, though like them they are constructed from cut-free proofs. Quasi-realizers in turn constructively convert to realizers, and this conversion is independent of the justification logic in question. The construction depends only on the structure of the formula involved. The second thing we do is provide a Prolog implementation of quasi-realization, and quasirealization to realization conversion, for the logic LP. Many other justification logics can obviously be treated similarly. Our quasi-realization algorithm, and its implementation, assumes the underlying modal proof system (for S4) is based on tableaus. Since these may not be familiar to everybody, we provide a sketch of how tableaus work. Then we present our algorithms, our implementation, and a discussion of implementation behavior and design decisions. We believe our algorithms are simple and straightforward. The original realization algorithm, for instance, needed the entire cut-free proof as input. Our quasi-realization algorithm works one formal proof step at a time. There is, in the literature, another realization construction that works one step at a time, but it requires extensive use of substitution while our quasi-realization algorithm does not. The conversion algorithm is, as noted above, independent of particular justification logics and so only needs to be understood once. It is only here that substitution is needed. The reason for the length of this report has less to do with algorithm complexity than with the desire to supply background and discussion. We hope we have been able to make the ideas clear. A text version of the Prolog program discussed here can be obtained from my web site: comet.lehman.cuny.edu/fitting