Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey his contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort’s is not widely known, and much of it remains unpublished. A complete list of van Heijenoort’s unpublished writings on tableaux methods and related work in proof theory is appended.

The paper describes tableaux based proof methods for temporal logics of knowledge allowing non-trivial interaction axioms between the modal and temporal components, namely those of synchrony and no learning and synchrony and perfect recall. The interaction axioms allow the description of how knowledge evolves over time and makes reasoning in such logics theoretically more complex. Such logics can be used to specify systems that involve the knowledge of processes or agents and which change over time, for example agent based systems, (...) knowledge games or security protocols. Soundness, completeness and termination results for the tableaux are proved. (shrink)

ABSTRACT In this paper we define two logics, KLn and BLn, and present tableau-based decision procedures for both. KLn is a temporal logic of knowledge. Thus, in addition to the usual connectives of linear discrete temporal logic, it contains a set of unary modal connectives for representing the knowledge possessed by agents. The logic BLn is somewhat similar; it is a temporal logic that contains connectives for representing the beliefs of agents. In addition to a complete formal definition of the (...) two logics and their decision procedures, the paper includes a brief review of their applications in AI and mainstream computer science, correctness proofs for the decision procedures, a number of worked examples illustrating the decision procedures, and some pointers to further work. (shrink)