Lazy Approximation for Dense Real-Time Systems

ing TCTL Formulas Abstraction predicates are extracted from the timed-bounded modalities of the TCTL formula A new clock variable zi is introduced for every time-bounded operator Example: φ = EG<2 p ∧ A[qU≤4 r] Set of predicates: Ψφ = {z1 < 2 } {{ } ψ1 , z2 ≤ 4 } {{ } ψ2 } Abstract CTL formula: φ̂ = EG(p∧ψ1) ∧ A[qU (r∧ψ2)] Lazy Approximation – p.6 Soundness and Completeness Basis predicates: Set of abstraction predicates expressive enough for distinguishing between any two clock regions Example: M , C = {x, y}, c̃ = 1 Ψ = {x = 0, y = 0, x < 1, x > 1, y < 1, y > 1, x > y, x < y} Lazy Approximation – p.7 Soundness and Completeness Basis predicates: Set of abstraction predicates expressive enough for distinguishing between any two clock regions Example: M , C = {x, y}, c̃ = 1 Ψ = {x = 0, y = 0, x < 1, x > 1, y < 1, y > 1, x > y, x < y} Given: M timed automaton, φ TCTL formula Ψ(M,φ) basis predicates w.r.t. M and φ M̂Ψ abstraction of M, φ̂Ψ abstraction of φ Lazy Approximation – p.7 Soundness and Completeness Basis predicates: Set of abstraction predicates expressive enough for distinguishing between any two clock regions Example: M , C = {x, y}, c̃ = 1 Ψ = {x = 0, y = 0, x < 1, x > 1, y < 1, y > 1, x > y, x < y} Given: M timed automaton, φ TCTL formula Ψ(M,φ) basis predicates w.r.t. M and φ M̂Ψ abstraction of M, φ̂Ψ abstraction of φ Theorem: Abstraction is strongly property preserving M |= φ ⇔ M̂Ψ |= φ̂Ψ Lazy Approximation – p.7 Lazy Approximation The computation of the exact abstraction using the entire basis of predicates Ψ(M,φ) is expensive In many cases it suffices to compute a coarser approximation for proving or refuting the desired property Typically only a few predicates required in the practice Problem: How to guess the necessary predicates for computing good approximations Lazy approximation Computes a sequence of approximations on demand and refines these incrementally New predicates are chosen based on the analysis of failed model-checking attempts on the current approximation (i.e. symbolic counterexamples) Lazy Approximation – p.8 Lazy Approximation (Cont.) M , φ initial approximation M̂, φ̂ M̂ |= φ̂M |= φ yes counterex no concrete counterex M 6|= φ yesrefining approximation no Predicate abstraction Finite-state model checking Theorem proving Symbolic counterexamples Lazy Approximation – p.9 Lazy Approximation (Cont.)