Introduction Volume Functional Analysis Discretization Information Theory Conclusion Entropy of Timed Regular Languages eal 3 Aldric Degorre 1 and Eugene Asarin 1 , Nicolas Basset 2 , Marie-Pierre B´ Dominique Perrin 3 1 IRIF — Universit´ e de Paris-Diderot 2 LIP6 — Universit´ e Pierre et Marie Curie 3 LIGM — Universit´ e Paris Est - Marne-la-Vall´ ee May 10 2016 – EQINOCS final Workshop – IRIF – Paris Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 1 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Measuring Size of Timed Languages: Why? Motivations Verification (original motivation): Quality of an over-approximation L ⊃ M (compare # L and # M ) Quantitative model-checking Information theory: Information content Security: timed information flow Timed channel capacity [ABBDP’12] Quasi-uniform random simulation [B’13] And of course: links with symbolic dynamics (entropy of timed subshifts) Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 2 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Reminder: Size of Languages Size and entropy of discrete languages Take a language L ⊂ Σ ∗ . Count its words a of length n (# L n , L n = def Σ n ∩ L ) a we could also count prefixes or factors An automaton: ❛ ❜ ✶ ✷ ❛ ✱ � ❜ ❛ ✸ Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 3 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Reminder: Size of Languages Size and entropy of discrete languages Take a language L ⊂ Σ ∗ . Count its words a of length n (# L n , L n = def Σ n ∩ L ) Typically: exponential growth Growth rate - entropy H ( L ) = lim sup log 2 # L n n a we could also count prefixes or factors ✶ Languages L 0 , . . . , L 4 : ❜ ❛ ∅ ; { b } ; { ab } ; { aab , baa , bac } ; ✶ ✷ { aaab , abaa , abac , babb } ; ❛ ❜ ❛ { aaaab , aabaa , aabac , ababb , babab , ✶ ✷ ✸ baaaa , baaac , bacaa , bacac } . . . ❜ ❝ ❛ ❛ ❛ ❜ Cardinalities: 0,1,1,3,4,9, . . . ✶ ✷ ✸ ✶ ✷ ✷ Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 3 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Computing the entropy of regular languages Entropy for a deterministic automaton = logarithm of the spectral radius of the adjacency matrix. ❛ 1 1 0 ❜ ✶ ✷ M = 0 0 1 1 2 0 ❛ ✱ � ❜ ❛ ✸ Spectral radius: maximal norm of the eigenvalues For this M : ρ ( M ) ≈ 1 . 80194; entropy: H = log ρ ( M ) ≈ 0 . 84955. Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 4 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Context Timed automata A model for verification of real-timed systems Invented by Alur and Dill in early 1990s Precursors: time Petri nets (Berthomieu) Now: an efficient model for verification, supported by tools ( Uppaal ) A popular research topic ( > 8000 citations for papers by Alur and Dill) modeling and verification decidability and algorithmics automata and language theory very recent: dynamics Inspired by TA: hybrid automata, data automata, automata on nominal sets Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 5 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Foreword: timed words and languages A word: u = abbabb represents a sequence of events in some Σ. A timed word: w = 0 . 8 a 2 . 66 b 1 . 5 b 0 a 3 . 14159 b 2 . 71828 b represents a sequence of events and delays. It lives in a timed monoid Σ ∗ ⊕ R + (but nevermind this!). For us it sits in ( R + × Σ) ∗ (words on some infinite alphabet), that is w = (0 . 8 , a ) , (2 . 66 , b ) , (1 . 5 , b ) , (0 , a ) , (3 . 14159 , b ) , (2 . 71828 , b ). Geometrically w is a point in several copies of R n : w = (0 . 8 , 2 . 66 , 1 . 5 , 0 , 3 . 14159 , 2 . 71828) ∈ R 6 abbabb A timed language is a set of timed words – examples below. Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 6 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion So, what is a TA? Recipe for making a timed automaton : take a finite automaton; add some variables x 1 , . . . , x n , called clocks; add guards to transitions (e.g. x 3 < 7); add resets to transitions (e.g. x 2 := 0); make all clocks run at speed ˙ x i = 1 everywhere and interpret behaviors in continuous time; enjoy! Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 7 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion An example of timed automaton ❛ ❀ ① � ❬ ✁ ✂ ✄ ❪ ☎ q q ✶ ✷ Timed automaton A : ❜ ❀ ① ✿ ✆ ✵ A run: ( q 1 , 0) 1 . 83 → ( q 2 , 1 . 83) 4 . 1 1 a b → ( q 1 , 1 . 83) → ( q 2 , 5 . 93) → ( q 1 , 0) → ( q 1 , 1) → . . . Its trace 1 . 83 a 4 . 1 b 1 a is a timed word. The timed language of the TA: set of all traces starting in q 1 , ending in q 1 : { t 1 as 1 bt 2 as 2 b . . . t n a |∀ i . t i ∈ [1; 2] } Observation: clock value of x : time since the last reset of x . Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 8 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Outline Introduction 1 Entropy of regular languages Timed Languages and Timed Automata Volume 2 Measuring timed languages Some simple volume computations Functional Analysis Approach 3 Computing the volume Main Theorem Symbolic method Numerical method Discretization Approach 4 Information Theory 5 Discrete channel coding Time channel coding Conclusion 6 Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 9 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Talking about size Timed languages typically are non-countable sets (continuous choice of delays). How does one describe the “size” of such an object? (and thus translate a nice classical theory to the realm of timed automata / timed shifts → extra-motivation). Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 10 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Talking about size Timed languages typically are non-countable sets (continuous choice of delays). How does one describe the “size” of such an object? (and thus translate a nice classical theory to the realm of timed automata / timed shifts → extra-motivation). The idea: timed regular languages must be seen as unions of polytopes → instead of counting words, we sum up their volumes. Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 10 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Volume and Entropy for Timed Languages u 3 u 1 u 2 t 3 t 1 t 2 Choice of a timed word ( � t , u ) ∈ L n = discrete choice of path u (untiming) + continuous choice of delay vector � t (timing). t , u ) ∈ L n } ⊆ R n is a polytope (e.g. hypercube, simplex...) Given u , L u = { � t | ( � Measure of L n , Vol ( L n ) = � u ∈ Σ n Vol ( L u ) (Rate of volumic) entropy: H = lim 1 n log 2 ( Vol ( L n )) Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 11 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Simple n -volumes hypercubes dimension 1 dimension 2 dimension 3 dimension n ? t 1 ≤ d t 1 , t 2 ≤ d t 1 , t 2 , t 3 ≤ d t 1 , . . . , t n ≤ d Volume d 2 Volume d 3 Volume d n Volume d a , x ≤ d / x := 0 Timed word : ( t 1 , a )( t 2 , a ) . . . ( t n , a ) Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 12 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Simple n -volumes simplices dimension 1 dimension 2 dimension 3 dimension n ? t 1 ≤ 1 t 1 + t 2 ≤ 1 t 1 + t 2 + t 3 ≤ 1 t 1 + · · · + t n ≤ 1 Volume 1 Volume 1 / 2 Volume 1 / 6 Volume 1 / n ! a , x ≤ 1 Timed word : ( t 1 , a )( t 2 , a ) . . . ( t n , a ) Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 13 / 64
Introduction Volume Functional Analysis Discretization Information Theory Conclusion Volume and entropy of timed automata Example 1: rectangles ❛ ❀ ① ✷ ❬ � ✁ ✹ ❪ ❂ ① ✿ ✂ ✵ Language: L 1 = ([2; 4] a + [3; 10] b ) ∗ ♣ ❜ ❀ ① ✷ ❬ ✄ ✁ ✶ ✵ ❪ ❂ ① ✿ ✂ ✵ Entropy of Timed Regular Languages May 10 2016 – EQINOCS final Workshop – IRIF – Paris 14 / 64
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