Formalizing Timing Diagram Requirements in Discrete Duration Calulus

Several temporal logics have been proposed to formalise timing diagram requirements over hardware and embedded controllers. These include LTL, discrete time MTL and the recent industry standard PSL. However, succintness and visual structure of a timing diagram are not adequately captured by their formulae. Interval temporal logic QDDC is a highly succint and visual notation for specifying patterns of behaviours. In this paper, we propose a practically useful notation called SeCeCntnl which enhances negation free fragment of QDDC with features of nominals and limited liveness. We show that timing diagrams can be naturally (compositionally) and succintly formalized in SeCeCntnl as compared with PSL and MTL. We give a linear time translation from timing diagrams to SeCeCntnl. As our second main result, we propose a linear time translation of SeCeCntnl into QDDC. This allows QDDC tools such as DCVALID and DCSynth to be used for checking consistency of timing diagram requirements as well as for automatic synthesis of property monitors and controllers. We give examples of a minepump controller and a bus arbiter to illustrate our tools. Giving a theoretical analysis, we show that for the proposed SeCeCntnl, the satisfiability and model checking have elementary complexity as compared to the non-elementary complexity for the full logic QDDC

Learning the Boundary of Inductive Invariants

We study the complexity of invariant inference and its connections to exact concept learning. We define a condition on invariants and their geometry, called the fence condition, which permits applying theoretical results from exact concept learning to answer open problems in invariant inference theory. The condition requires the invariant's boundary---the states whose Hamming distance from the invariant is one---to be backwards reachable from the bad states in a small number of steps. Using this condition, we obtain the first polynomial complexity result for an interpolation-based invariant inference algorithm, efficiently inferring monotone DNF invariants with access to a SAT solver as an oracle. We further harness Bshouty's seminal result in concept learning to efficiently infer invariants of a larger syntactic class of invariants beyond monotone DNF. Lastly, we consider the robustness of inference under program transformations. We show that some simple transformations preserve the fence condition, and that it is sensitive to more complex transformations