Entropy and temporal specifications Eugene Asarin 1 , Michel Bockelet 2 , Aldric Degorre 1 , alin Dima 2 and Chunyan Mu 3 C˘ at˘ 1 LIAFA – Universit´ e de Paris-Diderot 2 LACL – Universit´ e de Paris-Est Cr´ eteil 3 University of Birmingham EQINOCS final workshop, May 9th, 2016 C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 1 / 40
Entropy and quantitative model-checking 1 Quantitative model-checking in very few slides Entropy used as a measure Some experiments 2 Entropy and asymptotics Parametric linear temporal logic (PLTL) Convergence problems for PLTL formulas Main result and techniques 3 Discrete timed automata with parameters (GTBAC) Producing entropy in GTBAC Translating from PLTL to GTBAC Computing limit entropies 4 “Positive” case “Negative” case 5 Conclusions C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 2 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides On qualitative and quantitative model-checking Qualiltative model-checking Given a system S and a property φ decide if S ⊧ φ (answer: YES/NO). S : language of ( ω -) words, automaton, Kripke structure, etc. ϕ : language of ( ω -) words, automaton, formula in some logic (LTL, µ -calculus), etc. ⊧ : language inclusion, model satisfaction, etc. C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 3 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides On qualitative and quantitative model-checking Qualiltative model-checking Given a system S and a property φ decide if S ⊧ φ (answer: YES/NO). S : language of ( ω -) words, automaton, Kripke structure, etc. ϕ : language of ( ω -) words, automaton, formula in some logic (LTL, µ -calculus), etc. ⊧ : language inclusion, model satisfaction, etc. Quantitative model-checking Given a system S and a property φ , measure how much S ⊧ φ (answer: a real number). Approaches: probability (PRISM/UppAal people, etc.) “reward/penalty” models (quantitative languages, simulation distances, etc.). C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 3 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides On qualitative and quantitative model-checking Qualiltative model-checking Given a system S and a property φ decide if S ⊧ φ (answer: YES/NO). S : language of ( ω -) words, automaton, Kripke structure, etc. ϕ : language of ( ω -) words, automaton, formula in some logic (LTL, µ -calculus), etc. ⊧ : language inclusion, model satisfaction, etc. Quantitative model-checking Given a system S and a property φ , measure how much S ⊧ φ (answer: a real number). Approaches: probability (PRISM/UppAal people, etc.) “reward/penalty” models (quantitative languages, simulation distances, etc.). source of this work: entropy. C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 3 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides Why we are not happy with probability Example System S (state-labeled, note Σ = 2 { p , q } ): pq p ¯ q Specifications: φ 1 = always p . 1 pq ¯ p ¯ ¯ q φ 2 = never 100 times in a row p . 2 In Linear Temporal Logic (LTL), φ 1 = ◻ p , φ 2 = ◻ ◇ < 100 p . C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 4 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides Why we are not happy with probability Example System S (state-labeled, note Σ = 2 { p , q } ): pq p ¯ q Specifications: φ 1 = always p . 1 pq ¯ p ¯ ¯ q φ 2 = never 100 times in a row p . 2 In Linear Temporal Logic (LTL), φ 1 = ◻ p , φ 2 = ◻ ◇ < 100 p . Naive analysis Certain effort required to satisfy φ 1 (never go below) A different (smaller?) effort required to satisfy φ 2 (go above at least every 100 units) C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 4 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides Why we are not happy with probability Example System S (state-labeled, note Σ = 2 { p , q } ): pq p ¯ q Specifications: φ 1 = always p . 1 pq ¯ p ¯ ¯ q φ 2 = never 100 times in a row p . 2 In Linear Temporal Logic (LTL), φ 1 = ◻ p , φ 2 = ◻ ◇ < 100 p . Naive analysis Certain effort required to satisfy φ 1 (never go below) A different (smaller?) effort required to satisfy φ 2 (go above at least every 100 units) Probabilistic analysis P ( S ⊧ φ 1 ) = 0 and P ( S ⊧ φ 2 ) = 0 . C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 4 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides Why we are not happy with probability Example System S (state-labeled, note Σ = 2 { p , q } ): pq p ¯ q Specifications: φ 1 = always p . 1 pq ¯ p ¯ ¯ q φ 2 = never 100 times in a row p . 2 In Linear Temporal Logic (LTL), φ 1 = ◻ p , φ 2 = ◻ ◇ < 100 p . Naive analysis Certain effort required to satisfy φ 1 (never go below) A different (smaller?) effort required to satisfy φ 2 (go above at least every 100 units) Probabilistic analysis P ( S ⊧ φ 1 ) = 0 and P ( S ⊧ φ 2 ) = 0 . Mismatch between the two analyses C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 4 / 40
Entropy and quantitative model-checking Quantitative model-checking in very few slides Our approach — entropy Example System S : pq p ¯ q Specifications: φ 1 = always p . 1 pq ¯ p ¯ ¯ q φ 2 = never 100 times in a row p . 2 In Linear Temporal Logic (LTL), φ 1 = ◻ p , φ 2 = ◻ ◇ < 100 p . Entropy analysis We associate a number (entropy) H to everything, Entropy of the system: H( S ) = 2. Entropy of runs satisfying φ 1 is H( S ∩ φ 1 ) = 1 < 2 Entropy of runs satisfying φ 2 is H( S ∩ φ 2 ) > 1 . 99 (close to 2). Matches the intuition! C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 5 / 40
Entropy and quantitative model-checking Entropy used as a measure What is entropy Entropy of a finite word language (Chomsky, Miller) For a language L ⊂ Σ ∗ , with L n = L ∩ Σ n 1 H( L ) = lim sup n log # L n n →∞ Entropy of an ω -language (Staiger) 1 H( L ) = H( pref ( L )) = lim sup n log # pref ( L , n ) n →∞ What does it mean Growth rate of the language: # L n ≈ 2 H n “average log(number of choices for a symbol)” Quantity of information (in bits/symbol) in words of L Related to compression, Kolmogorov complexity, topological entropy, Hausdorff dimension etc. C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 6 / 40
Entropy and quantitative model-checking Entropy used as a measure Entropy — examples Example a 1 b H(L( A )) = log 2 = 1 C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 7 / 40
Entropy and quantitative model-checking Entropy used as a measure Entropy — examples Example a 1 b H(L( A )) = log 2 = 1 a b √ 1 2 H(L( A )) = log 1 + 5 a 2 C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 7 / 40
Entropy and quantitative model-checking Entropy used as a measure Entropy — examples Example a 1 b H(L( A )) = log 2 = 1 a b √ 1 2 H(L( A )) = log 1 + 5 a 2 H( Σ ω ) = log ∣ Σ ∣ ; Infinitely many times p : H ([ [ ◻ ◇ p ] ]) = log ∣ Σ ∣ (no constraint most of the time); Eventually only p : H ([ [ ◇ ◻ p ] ]) = log ∣ Σ ∣ (for any prefix, it is always possible to append p ). C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 7 / 40
Entropy and quantitative model-checking Entropy used as a measure Entropy model-checking The setting A system S — automaton/Kripke structure A specification φ — LTL formula C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 8 / 40
Entropy and quantitative model-checking Entropy used as a measure Entropy model-checking The setting A system S — automaton/Kripke structure A specification φ — LTL formula The metrics With ω -languages L S and L φ consider the numbers: Entropy of the system H S = H ( L S ) . Entropy of its good runs H G = H( L S ∩ L φ ) and default d = H S − H G . Maybe entropy of bad runs H B = H( L S ∖ L φ ) . C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 8 / 40
Entropy and quantitative model-checking Entropy used as a measure Entropy model-checking The setting A system S — automaton/Kripke structure A specification φ — LTL formula The metrics With ω -languages L S and L φ consider the numbers: Entropy of the system H S = H ( L S ) . Entropy of its good runs H G = H( L S ∩ L φ ) and default d = H S − H G . Maybe entropy of bad runs H B = H( L S ∖ L φ ) . An interpretation(???) d : how difficult is it to steer S into φ C. Dima (LIAFA, Univ. Paris-Direrot) Entropy and temporal specifications EQINOCS, 9/05/2016 8 / 40
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