Temporal Veri cation Diagrams ?

Most formal approaches to the veriication of temporal properties of reactive programs infer temporal conclusions from veriication conditions that are state formulas, i.e., contain no temporal operators. These proofs can often be eeectively presented by the use of veriica-tion diagrams. In this paper, we present a self-contained presentation of veriication diagrams for proving various temporal properties. Beginning with safety properties, we present WAIT-FOR and INVARIANCE diagrams for proving wait-for (precedence) and invariance formulas. Proceeding to liveness properties, we present veriication diagrams for response properties that require a bounded number of helpful steps (CHAIN diagrams) and response properties that require an unbounded number of helpful steps (RANK diagrams). Additional types of diagrams are proposed for handling response properties for parameterized programs (e.g., PRANK diagrams) and response properties that rely on the full spectrum of fairness requirements, including compassionate helpful transitions (e.g., F-CHAIN diagrams).