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On Synthesizing Controllers from Bounded-Response Properties Dejan Niˇ ckovi´ Oded Maler c Amir Pnueli Verimag Verimag Weizmann Institute NYU On Synthesizing Controllers from Bounded-Response Properties 1 / 23

Overview Introduction ● Property-based Synthesis ● ✦ Bounded-response Properties M TL -B ● Syntax and Semantics ✦ ✦ Non-Determinism From M TL -B to Deterministic Temporal Testers ● ✦ Pastification of M TL -B formulae Bounded-variability assumption ✦ Application to Synthesis: Arbiter Example ● ✦ Specification in M TL -B ✦ Experimental Results Conclusion ● On Synthesizing Controllers from Bounded-Response Properties 2 / 23

Introduction l 0 r 1 g 2 r 1 g 1 r 1 g 1 r 2 g 2 Controller r 3 g 1 · · · · · · . . . l 1 l 2 r m g n r 2 g 1 r 2 g 2 Environment Controller variables variables . . . BAD Automatic controller synthesis from high-level specifications ● ✦ Problem posed in [Chu63] Theoretically solved in [BL69,TB73] ✦ On Synthesizing Controllers from Bounded-Response Properties 3 / 23

Introduction 0 0 1 0 1 l 0 r 1 g 2 r 1 g 1 ( r 1 → r 1 S g 1 ) r 3 g 1 ( g 1 → [0 , 1] r 1 ) . . . l 1 l 2 ( g 1 → [0 , 2] r 1 ) r 2 g 1 r 2 g 2 . . . . . . BAD Synthesizing controllers from temporal logic formulae [PR89] ● ✦ Recent improvements [PPS06,PP06] Property-based synthesis problem: ● Given a temporal property ϕ defined over two distinct alphabets A and B , build a finite-state transducer (controller) from A ω to B ω such that all of its behaviors satisfy ϕ . We are interested in controller synthesis from real-time temporal logic specifications ● On Synthesizing Controllers from Bounded-Response Properties 4 / 23

Introduction 0 0 1 0 1 l 0 r 1 g 2 r 1 g 1 ( r 1 → r 1 S g 1 ) r 3 g 1 ( g 1 → [0 , 1] r 1 ) . . . l 1 l 2 ( g 1 → [0 , 2] r 1 ) r 2 g 1 r 2 g 2 . . . . . . BAD Synthesizing controllers from temporal logic formulae [PR89] ● ✦ Recent improvements [PPS06,PP06] Property-based synthesis problem: ● Given a temporal property ϕ defined over two distinct alphabets A and B , build a finite-state transducer (controller) from A ω to B ω such that all of its behaviors satisfy ϕ . We are interested in controller synthesis from real-time temporal logic specifications ● On Synthesizing Controllers from Bounded-Response Properties 4 / 23

Introduction 0 0 1 0 1 l 0 r 1 g 2 r 1 g 1 ( r 1 → r 1 S g 1 ) r 3 g 1 ( g 1 → [0 , 1] r 1 ) . . . l 1 l 2 ( g 1 → [0 , 2] r 1 ) r 2 g 1 r 2 g 2 . . . . . . BAD Synthesizing controllers from temporal logic formulae [PR89] ● ✦ Recent improvements [PPS06,PP06] Property-based synthesis problem: ● Given a temporal property ϕ defined over two distinct alphabets A and B , build a finite-state transducer (controller) from A ω to B ω such that all of its behaviors satisfy ϕ . We are interested in controller synthesis from real-time temporal logic specifications ● On Synthesizing Controllers from Bounded-Response Properties 4 / 23

Temporal Logic and Controller Synthesis translation Non−Deterministic Temporal Logic Game Automaton Specification determinization controller synthesis alg. Deterministic Controller Game Automaton On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Temporal Logic and Controller Synthesis translation Non−Deterministic Temporal Logic Game Automaton acceptance conditions Specification non−determinism determinization timed automata controller synthesis alg. Deterministic Controller Game Automaton On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Temporal Logic and Controller Synthesis Past Non−Deterministic Temporal Logic Game Automaton Specification translation safety deterministic controller synthesis alg. Deterministic Controller Game Automaton On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Temporal Logic and Controller Synthesis Bounded Response translation Non−Deterministic Temporal Logic Game Automaton safety Specification non−determinism determinization timed automata controller synthesis alg. Deterministic Controller Game Automaton On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Temporal Logic and Controller Synthesis Bounded Response Non−Deterministic Temporal Logic Game Automaton Specification Eliminate sources of non−determinism translation safety deterministic controller synthesis alg. Deterministic Controller Game Automaton On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Motivation for Bounded-Response Properties Bounded-response correspond to safety properties ● ✦ → Limited scope wrt more general liveness properties Liveness properties abstract away the upper bound requirement of occurrence of ● events But many applications require specifying explicitly such upper bound: ✦ Hard real-time systems ■ Scheduling problems ■ . . . ■ We choose Bounded Response Metric Temporal Logic - M TL -B as the specification ● formalism 0 ✦ M TL [Koy90] without unbounded until ✦ Punctual operators (unlike M ITL [AFH96]) ✦ Allows specifying non-trivial properties Can be interpreted both in discrete and dense time ✦ We consider specifications of type ϕ where ϕ is an M TL -B formula ✦ On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Motivation for Bounded-Response Properties Bounded-response correspond to safety properties ● ✦ → Limited scope wrt more general liveness properties Liveness properties abstract away the upper bound requirement of occurrence of ● events But many applications require specifying explicitly such upper bound: ✦ Hard real-time systems ■ Scheduling problems ■ . . . ■ We choose Bounded Response Metric Temporal Logic - M TL -B as the specification ● formalism 0 ✦ M TL [Koy90] without unbounded until ✦ Punctual operators (unlike M ITL [AFH96]) ✦ Allows specifying non-trivial properties Can be interpreted both in discrete and dense time ✦ We consider specifications of type ϕ where ϕ is an M TL -B formula ✦ On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Motivation for Bounded-Response Properties Bounded-response correspond to safety properties ● ✦ → Limited scope wrt more general liveness properties Liveness properties abstract away the upper bound requirement of occurrence of ● events But many applications require specifying explicitly such upper bound: ✦ Hard real-time systems ■ Scheduling problems ■ . . . ■ We choose Bounded Response Metric Temporal Logic - M TL -B as the specification ● formalism 0 ✦ M TL [Koy90] without unbounded until ✦ Punctual operators (unlike M ITL [AFH96]) ✦ Allows specifying non-trivial properties Can be interpreted both in discrete and dense time ✦ We consider specifications of type ϕ where ϕ is an M TL -B formula ✦ On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Motivation for Bounded-Response Properties Bounded-response correspond to safety properties ● ✦ → Limited scope wrt more general liveness properties Liveness properties abstract away the upper bound requirement of occurrence of ● events But many applications require specifying explicitly such upper bound: ✦ Hard real-time systems ■ Scheduling problems ■ . . . ■ We choose Bounded Response Metric Temporal Logic - M TL -B as the specification ● formalism 0 ✦ M TL [Koy90] without unbounded until ✦ Punctual operators (unlike M ITL [AFH96]) ✦ Allows specifying non-trivial properties Can be interpreted both in discrete and dense time ✦ We consider specifications of type ϕ where ϕ is an M TL -B formula ✦ On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Motivation for Bounded-Response Properties Bounded-response correspond to safety properties ● ✦ → Limited scope wrt more general liveness properties Liveness properties abstract away the upper bound requirement of occurrence of ● events But many applications require specifying explicitly such upper bound: ✦ Hard real-time systems ■ Scheduling problems ■ . . . ■ We choose Bounded Response Metric Temporal Logic - M TL -B as the specification ● formalism 0 ✦ M TL [Koy90] without unbounded until ✦ Punctual operators (unlike M ITL [AFH96]) ✦ Allows specifying non-trivial properties Can be interpreted both in discrete and dense time ✦ We consider specifications of type ϕ where ϕ is an M TL -B formula ✦ On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Motivation for Bounded-Response Properties Bounded-response correspond to safety properties ● ✦ → Limited scope wrt more general liveness properties Liveness properties abstract away the upper bound requirement of occurrence of ● events But many applications require specifying explicitly such upper bound: ✦ Hard real-time systems ■ Scheduling problems ■ . . . ■ We choose Bounded Response Metric Temporal Logic - M TL -B as the specification ● formalism 0 ✦ M TL [Koy90] without unbounded until ✦ Punctual operators (unlike M ITL [AFH96]) ✦ Allows specifying non-trivial properties Can be interpreted both in discrete and dense time ✦ We consider specifications of type ϕ where ϕ is an M TL -B formula ✦ On Synthesizing Controllers from Bounded-Response Properties 6 / 23