Zhenhua Duan

A decision procedure for propositional projection temporal logic with infinite models

This paper investigates the satisfiability of Propositional Projection Temporal Logic (PPTL) with infinite models. A decision procedure for PPTL formulas is given. To this end, Normal Form (NF) and Labeled Normal Form Graph (LNFG) for PPTL formulas are defined, and algorithms for transforming a formula to its normal form and constructing the LNFG for the given formula are presented. Further, the finiteness of LNFGs is proved in details. Moreover, the decision procedure is extended to check the satisfiability of the formulas of Propositional Interval Temporal Logic. In addition, examples are also given to illustrate how the decision procedure works.

A Unified Model Checking Approach with Projection Temporal Logic

This paper presents a unified model checking approach with Projection Temporal Logic (PTL) based on Normal Form Graphs (NFGs). To this end, a Modeling, Simulation and Verification Language (MSVL) is defined based on PTL. Further, normal forms and NFGs for MSVL programs and Propositional PTL (PPTL) formulas are defined. The finiteness for NFGs of MSVL programs is proved in details. Moreover, by modeling a system with an MSVL program p, and specifying the desirable property of the system with a PPTL formula i¾?, whether or not the system satisfies the property (whether or not pi¾?i¾?is valid) can equivalently be checked by evaluating whether or not ¬(pi¾?i¾?) i¾? pi¾? ¬i¾?is unsatisfiable. Finally, the satisfiability of a formula in the form of pi¾? ¬i¾?is checked by constructing the NFG of pi¾? ¬i¾?, and then inspecting whether or not there exist paths in the NFG.

Model checking propositional projection temporal logic based on SPIN

This paper investigates a model checking algorithm for Propositional Projection Temporal Logic (PPTL) with finite models. To this end, a PPTL formula is transformed to a Normal Form Graph (NFG), and then a Nondeterministic Finite Automaton (NFA). The NFA precisely characterizes the finite models satisfying the corresponding formula and can be equivalently represented as a Deterministic Finite Automaton (DFA). When the system to be verified can be modeled as a DFA As, and the property of the system can be specified by a PPTL formula P, then ¬P can be transformed to a DFA Ap. Thus, whether the system satisfies the property or not can be checked by computing the product automaton of As and Ap, and then checking whether or not the product automaton accepts the empty word. Further, this method can be implemented by means of the verification system SPIN.

Complexity of propositional projection temporal logic with star

This paper investigates the complexity of Propositional Projection Temporal Logic with Star (PPTL*). To this end, Propositional Projection Temporal Logic (PPTL) is first extended to include projection star. Then, by reducing the emptiness problem of star-free expressions to the problem of the satisfiability of PPTL* formulas, the lower bound of the complexity for the satisfiability of PPTL* formulas is proved to be non-elementary. Then, to prove the decidability of PPTL*, the normal form, normal form graph (NFG) and labelled normal form graph (LNFG) for PPTL* are defined. Also, algorithms for transforming a formula to its normal form and LNFG are presented. Finally, a decision algorithm for checking the satisfiability of PPTL* formulas is formalised using LNFGs.

Propositional projection temporal logic, Büchi automata and ω-regular expressions

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An improved decision procedure for propositional projection temporal logic

A new decision procedure for Propositional Projection Temporal Logic (PPTL) is proposed which is an improvement to the decision procedure given in [4]. The main contribution of the paper is as follows: (1) the relationship between paths in the NFG of a formula R and its models is established and proved; (2) a new Labeled NFG (LNFG) with a set of labels (propositions) is defined; (3) given a formula R, an LNFG of R can be generated by the new decision algorithm, and all models of R can be found; (4) based on the new decision procedure, an improved model checking algorithm is presented and implemented.

An efficient approach for abstraction-refinement in model checking

Abstraction is one of the most important strategies for dealing with the state space explosion problem in model checking. In an abstract model, the state space is largely reduced, however, a counterexample found in such a model may not be a real counterexample. Accordingly, the abstract model needs to be further refined where an NP-hard state separation problem is often involved. In this paper, a novel approach is presented, in which extra boolean variables are added to the abstract model for the refinement. With this approach, not only the NP-hard state separation problem can be avoided, but also a smaller refined abstract model can be obtained.

A note on stutter-invariant PLTL

Peled and Wilke proved that every stutter-invariant propositional linear temporal property is expressible in Propositional Linear Temporal Logic (PLTL) without @? (next) operators. To eliminate next operators, a translation @t which converts a stutter-invariant PLTL formula @f to an equivalent formula @t(@f) not containing @? operators has been given. By @t, for any formula @?@f, where @f contains no @? operators, a formula with the length of O(4^n|@f|) is always produced, where n is the number of distinct propositions in @f, and |@f| is the number of symbols appearing in @f. Etessami presented an improved translation @t^. By @t^, for @?@f, a formula with the length of O(nx2^n|@f|) is always produced. We further improve Etessamis result in the worst case to O(nx2^n|@f|) by providing a new translation @t^, and we show that the worst case will never occur.

Decidability of propositional projection temporal logic with infinite models

This paper investigates the satisfiability of Propositional Projection Temporal Logic (PPTL) with infinite models. A decision procedure for PPTL formulas is formalized. To this end, Normal Form (NF) and Normal Form Graph (NFG) for PPTL formulas are defined and an algorithm constructing NFG for PPTL formulas is presented. Further, examples are also given to illustrate how the decision algorithm works.

Making abstraction-refinement efficient in model checking

Abstraction is one of the most important strategies for dealing with the state space explosion problem in model checking. In the abstract model, the state space is largely reduced, however, a counterexample found in such a model may not be a real counterexample. Accordingly, the abstract model needs to be further refined where an NP-hard state separation problem is often involved. In this paper, a novel method is presented, in which extra variables are added to the abstract model for the refinement. With this method, not only the NP-hard state separation problem can be avoided, but also a smaller refined abstract model can be obtained.