Reachability In Parametric Timed Automata With Two Parametric Clocks - PowerPoint PPT Presentation

Introduction Upper Bound Lower Bound Reachability In Parametric Timed Automata With Two Parametric Clocks And One Parameter Is EXPSPACE-complete Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020 June 23, 2020 Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound Outline Introduction 1 Definition Reachability Problem State of the Art Upper Bound 2 Parametric One-Counter Automata Difficulties Unpumping N -runs Depumping Lemma The 5 / 6-Lemma Lower Bound 3 Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound Definition An automaton is a tuple ( Q , R , q 0 , F ) where Q finite set of states R finite set of transitions q 0 initial state F finite set of final states Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound Parametric Timed Automata Automaton A : y = p x > 3 q 0 q 1 q f start x := 0 Definition PTA A parametric timed automaton (PTA) is an automaton extended with finite set of clocks X = { x , y , z , . . . } over N finite set of parameters P = { p , p 1 , p 2 , . . . } transitions with guards x ⊲ ⊳ c or x ⊲ ⊳ p , where ⊲ ⊳ ∈ { <, = , > } resets of the form x := 0 Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound Run in PTAs PTA A : y = p x > 3 q 0 q 1 q f start x := 0 10-run i.e. run with µ ( p ) = 10 configurations: state × clock valuations q 0 (0 , 0) x > 3 → q 1 (4 , 4) x :=0 → q 0 (0 , 5) x > 3 y = p − − − − − − − → q 1 (4 , 9) − − → q f (5 , 10) Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound PTA-Reachability Reachability in Parametric Timed Automata Input: parametric timed automata A with parameters P and two parametric clocks, state q f . Output: Does there exists a valuation µ : P → N of P such that there is a run ending in q f ? Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound PTA-Reachability : State of the Art Central open problem : Is reachability in PTA with two parametric clocks decidable ? parametric clocks lower bound upper bound 1 clock NEXP[BBLS15] NEXP[BBLS15] PSPACE NEXP [BO17] 2 clocks open 3+ clocks undecidable[AHV93] undecidable[AHV93] Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound Our Contribution Main Theorem PTA-Reachability is EXPSPACE-complete for two parametric clocks and one parameter. Improves best previously known bounds from Bundala & Ouaknine [BO17] in 2017 PSPACE NEXP lower bound decidabilty upper bound Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Outline Introduction 1 Definition Reachability Problem State of the Art Upper Bound 2 Parametric One-Counter Automata Difficulties Unpumping N -runs Depumping Lemma The 5 / 6-Lemma Lower Bound 3 Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Parametric One-Counter Automata Automata C : < p mod 3 q 0 q 1 q f start +1 Definition A parametric one-counter automaton (POCA) is an automaton extended with transitions extended with +1/ − 1/+ p / − p ⊲ ⊳ c / ⊲ ⊳ p where ⊲ ⊳ ∈ { <, = , > } and c ∈ N mod c where c ∈ N Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Runs in POCAs POCA C : < p mod 3 q 0 q 1 q f start +1 5-run i.e. run with µ ( p ) = 5 x < p → q 1 (0) +1 → q 0 (1) x < 5 → q 1 (1) +1 → q 0 (2) x < 5 → q 1 (2) +1 q 0 (0) − − − − − − − − − − − − → x < p → q 1 (3) mod 3 q 0 (3) − − − − − − → q f (3) Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma From PTA to POCA revisited Lemma The following is computable in exponential time: INPUT: PTA A over one parameter p and two parametric clocks. OUTPUT: A POCA C such that: 1 all accepting N -runs have values in [0 , 6 · max( N , | C | )] 2 reachability holds for A if, and only if, reachability holds for C Inspired by a construction due to Bundala/Ouaknine [BO17] Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Main Theorem Theorem For a POCA C with parameter p given an accepting N -run with N > 2 poly( | C | ) and countervalues in [0 , 6 · N ] there exists an accepting N ′ -run for N ′ smaller with countervalues in [0 , 6 · N ′ ]. Corollary PTA-Reachability is in EXPSPACE for two parametric clocks and one parameter. Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Outline Introduction 1 Definition Reachability Problem State of the Art Upper Bound 2 Parametric One-Counter Automata Difficulties Unpumping N -runs Depumping Lemma The 5 / 6-Lemma Lower Bound 3 Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Difficulties unpumping N -runs Figure: Parametric update followed by some arbitrary run Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Difficulties unpumping N -runs Figure: Parametric update with a lower value followed by same arbitrary run Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Difficulties unpumping N -runs We need to change the run following the parametric update! Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Outline Introduction 1 Definition Reachability Problem State of the Art Upper Bound 2 Parametric One-Counter Automata Difficulties Unpumping N -runs Depumping Lemma The 5 / 6-Lemma Lower Bound 3 Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Depumping Lemma Lemma For every N-run π of a POCA C with N > 2 poly( | C | ) counter effect larger than 2 poly( | C | ) same number of + p and − p never more than 12 pending + p / − p there exists an N-run with smaller absolute counter effect. Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Application of the Depumping Lemma Figure: Parametric update followed by a run with the right properties Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Application of the Depumping Lemma Figure: Parametric update with a smaller value followed by same run Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020

Introduction Upper Bound Lower Bound The 5 / 6-Lemma Application of the Depumping Lemma Figure: Section where to cut subruns from the well-bracketed subrun Reachability In Parametric Timed Automata With Two Parametric Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨ oller MOVEP 2020