Principles of Probabilistic Programming Lectures at EWSCS 2020 Winter School Joost-Pieter Katoen EWSCS 2020, Palmse, Estonia Joost-Pieter Katoen Principles of Probabilistic Programming 1/56
Expected runtime analysis Overview Expected runtime analysis 1 Analysing Bayesian networks 2 Epilogue 3 Joost-Pieter Katoen Principles of Probabilistic Programming 2/56
Expected runtime analysis Olivier Bournez Florent Garnier Nuances of termination . . . . . . certain termination . . . . . . termination with probability one almost-sure termination º . . . . . . in an expected finite number of steps “positive” almost-sure termination º . . . . . . in an expected infinite number of steps “null” almost-sure termination º Joost-Pieter Katoen Principles of Probabilistic Programming 3/56
Expected runtime analysis This lecture A weakest-precondition technique for proving “positive” almost-sure termination. In fact: A weakest-precondition technique for determining the expected runtime of a probabilistic program. Joost-Pieter Katoen Principles of Probabilistic Programming 4/56
Expected runtime analysis Expected run-time analysis Z What? AST+termination in finite expected time Z Generalise. How? Z Provide a weakest precondition calculus Z . . . . . . for expected run-times Z a compositional calculus to reason at program syntax level Z Why? Z Classical weakest-preconditions cannot be used Z Proving positive AST is a special instance Z Reason about the e ffi ciency of randomised algorithms Z Reason about simulative inference of Bayesian networks Joost-Pieter Katoen Principles of Probabilistic Programming 5/56
Expected runtime analysis The run time of a probabilistic program is random int i := 0; repeat {i++; (c := false [1/2] c := true )} until (c) The expected runtime is 1 + 3 � 1 / 2 + 5 � 1 / 4 + . . . + ( 2 n + 1 ) � 1 / 2 n = . . . < ô . Joost-Pieter Katoen Principles of Probabilistic Programming 6/56
Expected runtime analysis Hurdles in runtime analysis 1. Programs may diverge while having a finite expected runtime: while (x > 0) { x-- [1/2] skip } 2. Expected runtimes are extremely sensitive to variations in probabilities while (x > 0) { x-- [1/2+e] x++ } // 0 <= e <= 1/2 Z For e = 0, the expected runtime is infinite. Z For any e > 0, the expected runtime is finite. 3. Having a finite expected time is not compositional (cf. next slide) Joost-Pieter Katoen Principles of Probabilistic Programming 7/56
Expected runtime analysis PAST is not compositional Consider the two probabilistic programs: int x := 1; bool c := true ; while (c) { c := false [0.5] c := true ; x := 2*x } Finite expected termination time Joost-Pieter Katoen Principles of Probabilistic Programming 8/56
Expected runtime analysis = IT { k ) Pr iterations PAST is not compositional ! Te 2 ' , Consider the two probabilistic programs: , § Efx , ^ PAST k=o int x := 1; PAST bool c := true ; while (c) { while (x > 0) { ; c := false [0.5] c := true ; x-- x := 2*x } } Finite termination time Finite expected termination time - AST PAST n Joost-Pieter Katoen Principles of Probabilistic Programming 8/56
Expected runtime analysis PAST is not compositional Consider the two probabilistic programs: int x := 1; bool c := true ; while (c) { while (x > 0) { c := false [0.5] c := true ; x-- x := 2*x } } Finite termination time Finite expected termination time Running the right after the left program yields an infinite expected termination time Joost-Pieter Katoen Principles of Probabilistic Programming 8/56
Expected runtime analysis Using wp for expected runtimes? while ( true ) { x++ } Z Consider the post-expectation x Z Characteristic function Φ x ( X ) = X ( x ( x + 1 ) Z Candidate upper bound is I = 0 Z Induction: Φ x ( I ) = 0 ( x ⇥ = x + 1 ) = 0 = I & I We — wrongly — conclude that 0 is the runtime. Using weakest pre-expectations is unsound for expected run-time analysis. Joost-Pieter Katoen Principles of Probabilistic Programming 9/56
Expected runtime analysis Expected run-times Z Expected run-time of program P on input s : ô k � Pr ⌅ “ P terminates after k steps on input s ” ⌦ 9 k = 1 Z In general, ert () is a function t ⇥ Σ � R ' 0 < { ô } Z Let’s call this a run-time. Let T denote the set of run-times. Z Complete partial order on T : t 1 V t 2 i ff º s " Σ . t 1 ( s ) & t 2 ( s ) Joost-Pieter Katoen Principles of Probabilistic Programming 10/56
Expected runtime analysis Continuation passing - ④ ti - - r O O - = 9 Same principle as for weakest pre-conditions: reason backwards Joost-Pieter Katoen Principles of Probabilistic Programming 11/56
Expected runtime analysis Expected runtime transformer 0 Semantics ert ( P , t ) Syntax Z skip Z 1 + t Z diverge Z ô Z x := E Z 1 + t [ x ⇥ = E ] Z ert ( P 1 , ert ( P 2 , t )) Z P1 ; P2 Z 1 + [ G ] � ert ( P 1 , t ) + [ ¬ G ] � ert ( P 2 , t ) Z if (G) P1 else P2 Z 1 + p � ert ( P 1 , t ) + ( 1 � p ) � ert ( P 2 , t ) Z P1 [p] P2 Z lfp X . 1 + ([ G ] � ert ( P , X ) + [ ¬ G ] � t ) Z while (G)P Joost-Pieter Katoen Principles of Probabilistic Programming 12/56
Expected runtime analysis Expected runtime transformer Semantics ert ( P , t ) Syntax Z skip Z 1 + t Z diverge Z ô ⑤ Z x := E Z 1 + t [ x ⇥ = E ] Z ert ( P 1 , ert ( P 2 , t )) Z P1 ; P2 % Z 1 + [ G ] � ert ( P 1 , t ) + [ ¬ G ] � ert ( P 2 , t ) Z if (G) P1 else P2 Z 1 + p � ert ( P 1 , t ) + ( 1 � p ) � ert ( P 2 , t ) Z P1 [p] P2 Z lfp X . 1 + ([ G ] � ert ( P , X ) + [ ¬ G ] � t ) Z while (G)P lfp is the least fixed point operator wrt. the ordering & on run-times This simple mild twist of weakest pre-expectations is sound. Joost-Pieter Katoen Principles of Probabilistic Programming 12/56
Expected runtime analysis Example: straight-line code Joost-Pieter Katoen Principles of Probabilistic Programming 13/56
0 ) ) g ( x I ( t tin at at t g Cx HD ) 3- a = g Exc id O ① is x at > . . 4 2 P 7 6 53 2 It I t 1+1=2 on 00
Expected runtime analysis A simple rule for upper bounds We have ert ( while ( G ) P , t ) = lfp X . Φ t ( X ) with Φ t ( X ) = 1 + ([ G ] � ert ( P , X ) + [ ¬ G ] � t ) By Park’s lemma: if Φ t ( I ) V I then ert ( while ( G ) P , t ) V I . Joost-Pieter Katoen Principles of Probabilistic Programming 14/56
Expected runtime analysis Induction on loops while (c) { (x++ [1/2] c := false ) } Z Post runtime equals 0 Z Characteristic function: Φ 0 ( X ) = 1 + [ c = 1 ] � ⇤ 2 + 1 / 2 � ( X ( x ( x + 1 ) + X ( c ( 0 )) Z Candidate for upper bound: I = 1 + [ c = 1 ] � 6 Z Induction: Φ 0 ( I ) = 1 + [ c = 1 ] � ⇤ 2 + 1 / 2 � ( 1 + [ c = 1 ] � 6 + 1 + [ 0 = 1 ] � 6 ) = 1 + [ c = 1 ] � 6 & I By Park’s lemma: ert ( while . . . ) & 1 + [ c = 1 ] � 6 Joost-Pieter Katoen Principles of Probabilistic Programming 15/56
Expected runtime analysis Coupon collector’s problem 00 O O Joost-Pieter Katoen Principles of Probabilistic Programming 16/56
Expected runtime analysis Coupon collector’s problem cp := [0,...,0]; // no coupons yet i := 1; // coupon to be collected next x := 0: // number of coupons collected while (x < N) { while (cp[i] != 0) { i := uniform (1..N) // next coupon } cp[i] := 1; // coupon i obtained x++; // one coupon less to go } Using our ert-calculus one can prove that expected runtime is Θ ( N � log N ) . By systematic formal verification à la Floyd-Hoare. Machine checkable. Joost-Pieter Katoen Principles of Probabilistic Programming 17/56
Expected runtime analysis Elementary properties of the ert-calculus Z Continuity: ert ( P , t ) is continuous, that is for every chain T = t 0 & t 1 & t 2 & . . . ⇥ ert ( P , sup T ) = sup ert ( P , T ) t & t ¨ implies ert ( P , t ) & ert ( P , t ¨ ) Z Monotonicity: Z Constant propagation: ert ( P , k + t ) = k + ert ( P , t ) Z Preservation of ô : ert ( P , ô ) = ô Z Connection to wp: ert ( P , t ) = ert ( P , 0 ) + wp ( P , t ) Z A ffi inity: ert ( P , r � t + u ) = ert ( P , 0 ) + r � wp ( P , t ) + wp ( P , u ) Joost-Pieter Katoen Principles of Probabilistic Programming 18/56
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