From MTL to Deterministic Timed Automata Dejan Nickovic Nir Piterman IST Austria Imperial College London (University of Leicester)
Introduction Property-based analysis and synthesis of digital systems Specification Temporal Logic LTL Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of digital systems Specification Temporal Logic LTL Non−Deterministic Automaton Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of digital systems Specification Temporal Logic LTL Non−Deterministic Automaton On−the−fly Determinization Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of digital systems Specification Temporal Logic LTL Subset Construction Deterministic Non−Deterministic Finite Automaton Automaton On−the−fly Determinization Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of digital systems Specification Temporal Logic LTL Subset Safra’s Construction Construction Deterministic Non−Deterministic Deterministic Finite Automaton Automaton ω -Automaton On−the−fly Determinization Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of real-time systems Real−time Specification MITL Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of real-time systems Real−time Specification MITL Non−Deterministic Timed Automaton On−the−fly Determinization Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of real-time systems Real−time Specification MITL ?? ?? Deterministic Timed Non−Deterministic Deterministic Finite Automaton Timed Automaton Timed ω -Automaton On−the−fly Determinization Model Controller Monitoring Checking Synthesis From MTL to Deterministic Timed Automata
Introduction Property-based analysis and synthesis of real-time systems Real−time Specification MITL ?? ?? Deterministic Timed Non−Deterministic Deterministic Finite Automaton Timed Automaton Timed ω -Automaton On−the−fly Determinization Model Controller Monitoring Checking Synthesis Timed automata are non-determinizable in general!! From MTL to Deterministic Timed Automata
Metric Temporal Logic - MTL • AP - set of atomic propositions Signal over AP - w : R ≥ 0 → 2 AP • • w p - projection of w to proposition p ∈ AP Syntax: ϕ :== p | ¬ ϕ 1 | ϕ 1 ∨ ϕ 2 | ϕ 1 U I ϕ 2 where p belongs to the set AP of atomic propositions and I is an interval of the form [ b, b ] , [ a, b ] , [ a, b ) , ( a, b ] , ( a, b ) , [ a, ∞ ) , ( a, ∞ ) where 0 ≤ a < b . • Derived operators: ✸ I ϕ = T U I ϕ and ✷ I ϕ = ¬ ✸ I ¬ ϕ • MITL - restricion of MTL to non-singular modalities From MTL to Deterministic Timed Automata
MTL - Metric Temporal Logic Semantics: ( w, t ) | ↔ = p w p [ t ] = 1 ( w, t ) | = ¬ ϕ ↔ ( w, t ) �| = ϕ ( w, t ) | = ϕ 1 ∨ ϕ 2 ↔ ( w, t ) | = ϕ 1 or ( w, t ) | = ϕ 2 ∃ t ′ ∈ t + I st ( w, t ) | ( w, t ) | = ϕ 1 U I ϕ 2 ↔ = ϕ 2 ∧ ∀ t ′′ ∈ ( t, t ′ ) ( w, t ′′ ) | = ϕ 1 Formula ϕ satisfied by w if ( w, 0) | = ϕ From MTL to Deterministic Timed Automata
MTL and Non-Determinism 1. Unbounded variability p → ✸ ( a,b ) q memorize changes p q t t + a t + b 2. Acausality p U ( a,b ) q p q t ′ t t + a t + b From MTL to Deterministic Timed Automata
Signals with Bounded Variability • Signal w is of bounded variability k if for every proposition p , it changes its value at most k times in every interval of length 1 k − 1 1 2 3 k t t + 1 • Reasonable assumption for many applications • Almost all systems have a bound on the frequency they operate • From now on, we assume that every input signal is of bounded variability From MTL to Deterministic Timed Automata
From MTL to Deterministic Timed Automata - Overview • Translation from MTL to deterministic TA assuming bounded variability of input signals MTL Specification From MTL to Deterministic Timed Automata
From MTL to Deterministic Timed Automata - Overview • Translation from MTL to deterministic TA assuming bounded variability of input signals MTL Specification Translation Non−Deterministic TA Proposition Prediction Monitor Generator Non−Deterministic Deterministic TA Dependent TA From MTL to Deterministic Timed Automata
From MTL to Deterministic Timed Automata - Overview • Translation from MTL to deterministic TA assuming bounded variability of input signals MTL Specification Translation Non−Deterministic TA Proposition Prediction memorizes events passive use of clocks Monitor Generator deterministic by Non−Deterministic discrete predictions Deterministic TA construction Dependent TA From MTL to Deterministic Timed Automata
From MTL to Deterministic Timed Automata - Overview • Translation from MTL to deterministic TA assuming bounded variability of input signals MTL Specification Translation Non−Deterministic TA Proposition Prediction memorizes events passive use of clocks Monitor Generator deterministic by Non−Deterministic discrete predictions Deterministic TA construction Dependent TA Determinization Deterministic TA Proposition Prediction Monitor Generator Deterministic Deterministic TA Dependent TA From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - Overview • Computation of the truth value of a formula ϕ at time t with a delay at time t + f where f is a bound p U ( a,b ) q memorize evaluate p p p ( w, t ) �| = p U ( a,b ) q q t t + a t + b From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - Overview • Computation of the truth value of a formula ϕ at time t with a delay at time t + f where f is a bound p U ( a,b ) q memorize evaluate p ( w, t ) �| = p U ( a,b ) q q q t t + a t + b From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - Overview • Computation of the truth value of a formula ϕ at time t with a delay at time t + f where f is a bound p U ( a,b ) q memorize evaluate p ( w, t ) | = p U ( a,b ) q q q t t + a t + b From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - Overview • Computation of the truth value of a formula ϕ at time t with a delay at time t + f where f is a bound p U ( a,b ) q memorize evaluate p ( w, t ) | = p U ( a,b ) q q q t t + a t + b p U ( a, ∞ ) q memorize evaluate p p p ( w, t ) �| = p U ( a, ∞ ) q q t t + a From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - Overview • Computation of the truth value of a formula ϕ at time t with a delay at time t + f where f is a bound p U ( a,b ) q memorize evaluate p ( w, t ) | = p U ( a,b ) q q q t t + a t + b p U ( a, ∞ ) q memorize evaluate p ??? q q t t + a From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - Overview • Computation of the truth value of a formula ϕ at time t with a delay at time t + f where f is a bound p U ( a,b ) q memorize evaluate p ( w, t ) | = p U ( a,b ) q q q t t + a t + b p U ( a, ∞ ) q predict p U q memorize evaluate p q q t t + a From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - future Function • Computation of the truth value of a formula ϕ at time t by looking in the interval [ t, t + future ( ϕ )) future ( p ) = p future ( ¬ ϕ 1 ) = future ( ϕ 1 ) future ( ϕ 1 ∨ ϕ 2 ) = max ( future ( ϕ 1 ) , future ( ϕ 2 )) future ( ϕ 1 U ( a,b ) ϕ 2 ) = b + max ( future ( ϕ 1 ) , future ( ϕ 2 )) future ( ϕ 1 U ( a, ∞ ) ϕ 2 ) = 2 + a + max ( future ( ϕ 1 ) , future ( ϕ 2 )) • Why 2 additional lookaheads for future ( ϕ 1 U ( a, ∞ ) ϕ 2 ) ? [ t, t + a ) never sufficient to determine whether p U ( a, ∞ ) holds at t p q q t t + a From MTL to Deterministic Timed Automata
Evaluating MTL Formulas - future Function • Computation of the truth value of a formula ϕ at time t by looking in the interval [ t, t + future ( ϕ )) future ( p ) = p future ( ¬ ϕ 1 ) = future ( ϕ 1 ) future ( ϕ 1 ∨ ϕ 2 ) = max ( future ( ϕ 1 ) , future ( ϕ 2 )) future ( ϕ 1 U ( a,b ) ϕ 2 ) = b + max ( future ( ϕ 1 ) , future ( ϕ 2 )) future ( ϕ 1 U ( a, ∞ ) ϕ 2 ) = 2 + a + max ( future ( ϕ 1 ) , future ( ϕ 2 )) • Why 2 additional lookaheads for future ( ϕ 1 U ( a, ∞ ) ϕ 2 ) ? [ t, t + a ) never sufficient to determine whether p U ( a, ∞ ) holds at t p q q t t + a From MTL to Deterministic Timed Automata
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