Timed Pattern Matching FORMATS’14 gan Ulus 1 ere 1 Eugene Asarin 2 Oded Maler 1 Do˘ Thomas Ferr` 1 VERIMAG, CNRS and the University of Grenoble-Alpes 2 LIAFA, University of Paris Diderot / CNRS September 10, 2014 Timed Pattern Matching Thomas Ferr` ere 1 / 26
Pattern matching Problem (Pattern matching for regular expressions) Given a word w ∈ Σ ∗ and a regular expression ϕ find subwords v of w that match ϕ . Classical solutions ◮ algorithms: often based on automata ◮ tools: grep , sed , programming languages: perl , python etc. Timed Pattern Matching Thomas Ferr` ere 2 / 26
Timed pattern matching Why do it? ◮ Monitoring of embedded systems ◮ Hardware specification languages ◮ Timed texts (music, speech) ◮ Quantitative pattern matching (texts, DNA sequences) Why is it not straightforward? ◮ Classical algorithms based on (implicit) determinization ⇒ heavy subset construction ◮ Timed automaton does not explicitely provide all matches Timed Pattern Matching Thomas Ferr` ere 3 / 26
Example ◮ Expression: ϕ = � ( p ∧ q ) · ¯ q · q � [4 , 5] · ¯ p ◮ Signals: p q 0 2 4 6 8 10 12 ◮ Matches: p ∧ q ¯ q q p ¯ 1 . 3 2 . 0 6 . 06 . 3 6 . 8 p ∧ q ¯ q q ¯ p 1 . 5 2 . 0 6 . 0 6 . 4 7 . 0 Timed Pattern Matching Thomas Ferr` ere 4 / 26
Outline Problem statement 1 The solution 2 Practical algorithmics 3 Experiments 4 Timed Pattern Matching Problem statement Thomas Ferr` ere 5 / 26
Outline Problem statement 1 The solution 2 Practical algorithmics 3 Experiments 4 Timed Pattern Matching Problem statement Thomas Ferr` ere 6 / 26
Timed Boolean signals Example p q 0 2 4 6 8 10 12 Definition (Boolean Signals) Let T = [0 , d ] (time domain). ◮ a Boolean signal is a function w : T → B m . ◮ can be represented using m Boolean variables p ( t ) . ◮ assumption: the number of discontinuities is finite. Timed Pattern Matching Problem statement Thomas Ferr` ere 7 / 26
Timed regular expressions – to specify sets of signals Definition (Syntax of TRE) ϕ := ǫ | p | p | ϕ · ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | ϕ ∗ | � ϕ � I p propositional variable, I integer bounded interval Definition (Semantics of TRE) ( w, t, t ′ ) t = t ′ ↔ � ǫ t < t ′ and ∀ t ′′ ∈ ( t, t ′ ) , p [ t ′′ ] = 1 ( w, t, t ′ ) � p ↔ ( w, t, t ′ ) ↔ � p . . . ∃ t ′′ ∈ ( t, t ′ ) , ( w, t, t ′′ ) � ϕ and ( w, t ′′ , t ′ ) � ψ ( w, t, t ′ ) � ϕ · ψ ↔ ( w, t, t ′ ) ( w, t, t ′ ) � ϕ or ( w, t, t ′ ) � ψ � ϕ ∨ ψ ↔ ( w, t, t ′ ) � ϕ ∧ ψ ↔ . . . ( w, t, t ′ ) � ϕ ∗ ∃ k ≥ 0 , ( w, t, t ′ ) � ϕ k ↔ t ′ − t ∈ I and ( w, t, t ′ ) � ϕ ( w, t, t ′ ) � � ϕ � I ↔ Timed Pattern Matching Problem statement Thomas Ferr` ere 8 / 26
Timed regular expressions – to specify sets of signals Definition (Syntax of TRE) ϕ := ǫ | p | p | ϕ · ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ | ϕ ∗ | � ϕ � I p propositional variable, I integer bounded interval Definition (Semantics of TRE) ( w, t, t ′ ) t = t ′ ↔ � ǫ t < t ′ and ∀ t ′′ ∈ ( t, t ′ ) , p [ t ′′ ] = 1 ( w, t, t ′ ) � p ↔ ( w, t, t ′ ) ↔ � p . . . ∃ t ′′ ∈ ( t, t ′ ) , ( w, t, t ′′ ) � ϕ and ( w, t ′′ , t ′ ) � ψ ( w, t, t ′ ) � ϕ · ψ ↔ ( w, t, t ′ ) ( w, t, t ′ ) � ϕ or ( w, t, t ′ ) � ψ � ϕ ∨ ψ ↔ ( w, t, t ′ ) � ϕ ∧ ψ ↔ . . . ( w, t, t ′ ) � ϕ ∗ ∃ k ≥ 0 , ( w, t, t ′ ) � ϕ k ↔ t ′ − t ∈ I and ( w, t, t ′ ) � ϕ ( w, t, t ′ ) � � ϕ � I ↔ Timed Pattern Matching Problem statement Thomas Ferr` ere 8 / 26
Problem statement Definition (Match-set) For a signal w and an expression ϕ the match-set is M ( ϕ, w ) := { ( t, t ′ ) ∈ T × T : ( w, t, t ′ ) � ϕ } Problem (Timed pattern matching) Given a signal and an expression compute the match-set. Timed Pattern Matching Problem statement Thomas Ferr` ere 9 / 26
Problem statement Definition (Match-set) For a signal w and an expression ϕ the match-set is M ( ϕ, w ) := { ( t, t ′ ) ∈ T × T : ( w, t, t ′ ) � ϕ } Problem (Timed pattern matching) Given a signal and an expression compute the match-set. Timed Pattern Matching Problem statement Thomas Ferr` ere 9 / 26
Outline Problem statement 1 The solution 2 Practical algorithmics 3 Experiments 4 Timed Pattern Matching The solution Thomas Ferr` ere 10 / 26
Data structure Definition (2d zone) A 2d zone is a subset of R 2 described by inequalities e < x − y < f a < x < b c < y < d with a, b, c, d, e, f real constants. About 2d zones ◮ convex polygons (with up to 6 edges) ◮ only vertical, horizontal and diagonal edges Timed Pattern Matching The solution Thomas Ferr` ere 11 / 26
Data structure Definition (2d zone) A 2d zone is a subset of R 2 described by inequalities e < x − y < f a < x < b c < y < d with a, b, c, d, e, f real constants. About 2d zones ◮ convex polygons (with up to 6 edges) ◮ only vertical, horizontal and diagonal edges Timed Pattern Matching The solution Thomas Ferr` ere 11 / 26
Main result Theorem The match-set M ( ϕ, w ) is computable as a finite union of 2d zones. Proof principle Structural induction over ϕ ⇒ recursive algorithm over the expression syntactic tree Timed Pattern Matching The solution Thomas Ferr` ere 12 / 26
Main result Theorem The match-set M ( ϕ, w ) is computable as a finite union of 2d zones. Proof principle Structural induction over ϕ ⇒ recursive algorithm over the expression syntactic tree Timed Pattern Matching The solution Thomas Ferr` ere 12 / 26
Structural induction: base cases Empty word M ( ǫ, w ) = { ( t, t ′ ) : t = t ′ } and the diagonal is a zone A literal Example M ( p, w ) is a union of triangles over the diagonal Timed Pattern Matching The solution Thomas Ferr` ere 13 / 26
Structural induction: base cases Empty word M ( ǫ, w ) = { ( t, t ′ ) : t = t ′ } and the diagonal is a zone A literal Example M ( p, w ) is a union of triangles 12 over the diagonal 11 10 9 8 7 6 5 4 3 2 1 p 0 p 0 1 2 3 4 5 6 7 8 9 10 11 12 Timed Pattern Matching The solution Thomas Ferr` ere 13 / 26
Structural induction: Boolean closure Intersection Zones are closed under intersection Union Unions of zones are closed under union Time restriction Example M ( � ϕ � I , w ) = M ( ϕ, w ) ∩{ ( t, t ′ ) : t ′ − t ∈ I } is an intersection of zones Timed Pattern Matching The solution Thomas Ferr` ere 14 / 26
Structural induction: Boolean closure Intersection Zones are closed under intersection Union Unions of zones are closed under union Time restriction Example M ( � ϕ � I , w ) = M ( ϕ, w ) ∩{ ( t, t ′ ) : t ′ − t ∈ I } is an intersection of zones Timed Pattern Matching The solution Thomas Ferr` ere 14 / 26
Structural induction: Boolean closure Intersection Zones are closed under intersection Union Unions of zones are closed under union Time restriction Example M ( � ϕ � I , w ) = M ( ϕ, w ) ∩{ ( t, t ′ ) : t ′ − t ∈ I } 12 is an intersection of zones 11 10 9 8 7 6 5 4 3 2 1 � p � [1 , 2] 0 p 0 1 2 3 4 5 6 7 8 9 10 11 12 Timed Pattern Matching The solution Thomas Ferr` ere 14 / 26
Structural induction: concatenation Lemma (Concatenation = composition of binary relations) M ( ϕ · ψ, w ) = M ( ϕ, w ) ◦ M ( ψ, w ) Composition preserves zones ( t, t ′ ) ∈ z 1 ◦ z 2 ↔ ∃ t ′′ : ( t, t ′′ ) ∈ z 1 ∧ ( t ′′ , t ′ ) ∈ z 2 . Can be obtained using standard zones operations. Timed Pattern Matching The solution Thomas Ferr` ere 15 / 26
Structural induction: concatenation Lemma (Concatenation = composition of binary relations) M ( ϕ · ψ, w ) = M ( ϕ, w ) ◦ M ( ψ, w ) Composition preserves zones ( t, t ′ ) ∈ z 1 ◦ z 2 ↔ ∃ t ′′ : ( t, t ′′ ) ∈ z 1 ∧ ( t ′′ , t ′ ) ∈ z 2 . Can be obtained using standard zones operations. Timed Pattern Matching The solution Thomas Ferr` ere 15 / 26
Example Concatenation of ϕ = � p � [1 , ∞ ] with ψ = � q � [0 , 2] 8 7 6 t ′ 5 t ′′ 4 3 2 ϕ ψ 1 ϕ · ψ 0 p p q q 0 1 2 3 4 5 6 7 8 t t ′′ t ′ a match ( t, t ′ ) of ϕ · ψ zones for ϕ , ψ and ϕ · ψ Timed Pattern Matching The solution Thomas Ferr` ere 16 / 26
Structural induction: Kleene star Definition (size of a signal) σ ( w ) – minimal number of intervals that: ◮ cover the time domain of w ◮ are of length < 1 ◮ w is constant in each interval Lemma (Star is bounded) If σ ( w ) = k then M ( ϕ ∗ , w ) = M ( ϕ ≤ 2 k +1 , w ) . Timed Pattern Matching The solution Thomas Ferr` ere 17 / 26
Structural induction: Kleene star Definition (size of a signal) σ ( w ) – minimal number of intervals that: ◮ cover the time domain of w ◮ are of length < 1 ◮ w is constant in each interval Lemma (Star is bounded) If σ ( w ) = k then M ( ϕ ∗ , w ) = M ( ϕ ≤ 2 k +1 , w ) . Timed Pattern Matching The solution Thomas Ferr` ere 17 / 26
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