Synthesizing Probabilistic Invariants via Doobs decomposition G. - PowerPoint PPT Presentation

Synthesizing Probabilistic Invariants via Doob’s decomposition G. Barthe, T. Espitau , L.M.F Fioriti, J. Hsu CAV, 2016

Introduction 1

Introduction Probabilistic 1 2 Computations Ubiquitous in many fields (ML, Crypto, Privacy,…) But… Difficult to prove Termination ? (Certainly, almost sure, non terminating)

Introduction Probabilistic Doob’s 1 2 3 Martingales? Computations Decomposition Ubiquitous in many fields Difficulty to transfer local to (ML, Crypto, Privacy,…) end of program But… Difficult to prove Reason on average values Termination ? Martingales have the required (Certainly, almost sure, non transfer property terminating)

Introduction Probabilistic Doob’s 1 2 3 Martingales? Computations Decomposition Ubiquitous in many fields Difficulty to transfer local to But ( again )… Difficult to find (ML, Crypto, Privacy,…) end of program good ones But… Difficult to prove Reason on average values Automated generation? Doob’s decomposition Termination ? Martingales have the required Formal method to generate (Certainly, almost sure, non transfer property martingales from a seed . terminating)

Martingale theory 101 (I)

Martingale theory 101 (I) Step 1: Some probabilities ● Ω set of outcomes. Sigma algebra: Probability space ● Set F of subsets of Ω Closed under complements, countable unions, countable intersections. ● Probability measure: Countably additive mapping P : F → [0, 1] P (Ω) = 1.

Martingale theory 101 (II) Step 2: Stochastic process ● Random variable: X : Ω → R measurable ( X -1 ( (a,b] ) ∈ F ) Filtration: ( F i ) ⊂ F s.t: ● F i-1 ⊂ F i ● Process wrt filtration F i : Sequence (X i ) s.t: X i is F i measurable

Martingale theory 101 (II) Interlude: PL setting i = 0 Ω: Element = Possible outcome of samples While b do z[i] ← $ Samplings... F i : Events sampled at iteration i or before x[i] ← f(x[i-1], ... , f[0], z[i], … ,z[0]) i++ end Process (X i ) is adapted to the filtration iff: X i is defined in term of elements sampled at step i or before

Martingale theory 101 (III) Step 3: Expectations & Moments ● Expectation: E [X] = ∑ u ∈ Ω X(u) P (u) Conditional expectation wrt G ⊂ F: E [X|G] ● Y G-mesurable st E [X. 1 A ] = E [Y. 1 A ] for A ∈ G

Martingale theory 101 (IV) Step 4 ( Final! ): Martingales Martingale: ● E [ X i | F i-1 ] = X i-1 Average value of the current step is equal to the value of the previous step

Playing with martingales Doob’s decomposition (X i ) stochastic process → (M i ) martingale M 0 = X 0 M i = X 0 + ∑ i j=1 X j - E [X j | F j-1 ]

Black Magic of martingales Optional Stopping theorem (M i ) martingale → Expectations are invariants E [M j ] = E [M 0 ]

Black Magic of martingales Optional Stopping theorem E [M j ] = E [M 0 ]

Black Magic of martingales Optional Stopping theorem E [M T ] = E [M 0 ] For T a stopping time : T : Ω → R { w ∈ Ω | T(w) ≤ i } ⊂ F i

Black Magic of martingales Optional Stopping theorem E [M T ] = E [M 0 ] For T a stopping time : T : Ω → R { w ∈ Ω | T(w) ≤ i } ⊂ F i and... |M i - M i-1 | ≤ C E [T] < ∞

Let’s play with a program...

Geometric distribution x[0] ← 0; while (z ̸ = 0) do z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution Stopping time? (on average) x[0] ← 0; while (z ̸ = 0) do z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution Stopping time? (on average) x[0] ← 0; while (z ̸ = 0) do 1/(1-p) z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution Equation for x ? X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution Equation for x ? X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do z ← $ Bern(p, {1, 0}); Polynomial extraction x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = X 0 M i = X 0 + ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j | F j-1 ] x ← x[-1] + z; end Doob

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = X 0 + ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j | F j-1 ] x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j | F j-1 ] x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j-1 + Z i | F j-1 ] x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j-1 | F j-1 ] + E [Z i | F j-1 ] x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j-1 | F j-1 ] + E [Z i ] x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - E [X j-1 | F j-1 ] + p x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = ∑ i z ← $ Bern(p, {1, 0}); j=1 X j - X j-1 + p x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = X i - X 0 + i p z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do M 0 = 0 M i = X i + i p z ← $ Bern(p, {1, 0}); Simplify... x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do E [ M 0 ] = E [ M T ] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end Optional Stopping

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do 0 = E [ M T ] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do 0 = E [ X T - Tp] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do 0 = E [ X T ] - E [ Tp] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do Simplify... 0 = E [ X T ] - p E [T] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do Hint 0 = E [T-1] - p E [T] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end X T = T-1

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do 0 = E [T] - 1 - p E [T] z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Geometric distribution M 0 = 0 M i = X i + i p X i = X i-1 + Z i x[0] ← 0; while (z ̸ = 0) do Simplify... E [T] = 1 /(1-p) z ← $ Bern(p, {1, 0}); x ← x[-1] + z; end

Automatization Inputs

Automatization Extract Poly. Inputs

Automatization Doob decomp. Extract Poly. Inputs

Automatization Doob decomp. Extract Simplify Poly. Inputs

Automatization Doob decomp. Extract Simplify OST Poly. Inputs

Automatization Doob decomp. Extract Simplify OST Poly. Inputs Verify Hints.

Automatization Doob decomp. Extract Simplify OST Poly. Inputs Verify Simplify Hints.

Gambler’s ruin

Gambler’s ruin X x[0] ← a; while ( 0 < x < b ) do z ← $ Bern(1/2, {-1, 1}); x ← x + z; end

Automatization Extract Poly. Inputs

Gambler’s ruin X i = X i-1 + Z i x[0] ← a; while ( 0 < x < b ) do z ← $ Bern(1/2, {-1, 1}); x ← x + z; end

Automatization Doob decomp. Extract Poly. Inputs

Gambler’s ruin X i = X i-1 + Z i M 0 = X 0 M i = X i x[0] ← a; while ( 0 < x < b ) do z ← $ Bern(1/2, {-1, 1}); x ← x + z; end

Automatization Doob decomp. Extract Simplify OST Poly. Inputs

Gambler’s ruin X i = X i-1 + Z i M 0 = X 0 M i = X i x[0] ← a; while ( 0 < x < b ) do z ← $ Bern(1/2, {-1, 1}); a = E [X 0 ] = E [X T ] x ← x + z; end

Automatization Doob decomp. Extract Simplify OST Poly. Inputs Verify Hints.

Gambler’s ruin X i = X i-1 + Z i M 0 = X 0 M i = X i x[0] ← a; while ( 0 < x < b ) do z ← $ Bern(1/2, {-1, 1}); x=0 or x=b a = E [X 0 ] = E [X T ] x ← x + z; end

Automatization Doob decomp. Extract Simplify OST Poly. Inputs Verify Simplify Hints.

Gambler’s ruin X i = X i-1 + Z i M 0 = X 0 M i = X i x[0] ← a; while ( 0 < x < b ) do z ← $ Bern(1/2, {-1, 1}); x=0 or x=b a = E [X 0 ] = E [X T ] x ← x + z; end a = b P [x=b]