Convex Language Semantics for Nondeterministic Probabilistic - PowerPoint PPT Presentation

Convex Language Semantics for Nondeterministic Probabilistic Automata Gerco van Heerdt 1 Justin Hsu 1,2,3 el Ouaknine 4 Alexandra Silva 1 Jo¨ 1 University College London 2 Cornell University 3 University of Wisconsin–Madison 4 MPI-SWS and Oxford University October 19, 2018 1 / 17

Deterministic probabilistic automaton 1 will get home 2 1 2 sleep taste taste 2 / 17

Deterministic probabilistic automaton 1 will get home 2 1 2 sleep 1 0 taste taste � n � 1 Expected value after n wine tastings is 2 2 / 17

� Deterministic probabilistic automaton Given a finite alphabet A , a probabilistic automaton is a function Q with an initial state distribution [0 , 1] × ( D Q ) A Language of type A ∗ → [0 , 1] defined by determinization 3 / 17

� � � � Determinization NFAs: Q P Q ⇒ 2 × ( P Q ) A 2 × ( P Q ) A Probabilistic automata: D Q Q ⇒ [0 , 1] × ( D Q ) A [0 , 1] × ( D Q ) A Crucial: P and D are monads for which 2 and [0 , 1] are algebras 4 / 17

D -semantics: transitions State space D Q Transition D Q → DD Q ⇒ flatten with monad multiplication q 0 1 taste q 0 2 1 2 2 q 0 + 1 1 q 1 2 q 1 taste taste 1 4 q 0 + 3 taste 4 q 1 taste . . . 1 2 · ( 1 2 q 0 + 1 2 q 1 ) + 1 2 q 1 = 1 4 q 0 + 3 4 q 1 5 / 17

D -semantics: outputs State space D Q Output D Q → D [0 , 1] → [0 , 1] D -algebra : expected value 1 q 0 1 taste q 0 2 1 1 2 2 q 0 + 1 1 q 1 2 q 1 1 0 2 taste taste 1 4 q 0 + 3 1 taste 4 q 1 4 taste . . . 1 4 · 1 + 3 4 · 0 = 1 4 6 / 17

Nondeterministic probabilistic automaton 1 will get home 2 1 2 sleep 1 0 taste taste taste Expected value after n tastings is one of � 2 � 3 � n − 1 � n 1 , 1 � 1 � 1 � 1 � 1 2 , , , . . . , , 2 2 2 2 Which ones “make sense”? 7 / 17

Main results Nondeterministic probabilistic automata have ◮ two natural (categorical) semantics ◮ more expressive power than deterministic ones ◮ undecidable equivalence problem 8 / 17

� Nondeterministic probabilistic automaton A nondeterministic probabilistic automaton is a function Q with a set of initial state distributions [0 , 1] × ( PD Q ) A Language of type A ∗ → [0 , 1] But: PD is not a monad 9 / 17

Convex powerset For a convex set X , P c X = convex subsets of X (finitely generated, nonempty) 10 / 17

Convex powerset For a convex set X , P c X = convex subsets of X (finitely generated, nonempty) Convex algebra structure on these sets: pointwise pU + (1 − p ) V = { pu + (1 − p ) v | u ∈ U , v ∈ V } For example, in P c [0 , 1]: 0 . 3 · [0 . 1 , 0 . 5] + 0 . 7 · [0 . 2 , 1] = [0 . 17 , 0 . 85] 10 / 17

Adjusted monad Gives a monad P c D Multiplication: P c DP c D X → P c P c D X → P c D X convex algebra structure on P c 11 / 17

� � Two natural semantics P c D [0 , 1] Need: P c D -algebra on [0 , 1] P c E � [0 , 1] α Comes down to a P c -algebra α P c [0 , 1] 12 / 17

� � Two natural semantics P c D [0 , 1] Need: P c D -algebra on [0 , 1] P c E � [0 , 1] α Comes down to a P c -algebra α P c [0 , 1] Claim: α ∈ { min , max } Proof outline: � P c [0 , 1] generated by { 0 } , { 1 } , [0 , 1] α determined by α ([0 , 1]) α ( { 0 } ) = 0, α ( { 1 } ) = 1 α ([0 , 1]) 2 = α ([0 , 1]) ⇓ α ([0 , 1]) = 0 or α ([0 , 1]) = 1 12 / 17

Expressivity (min semantics) a , b q 1 0 a , b 1 2 1 a 2 1 1 2 a q 0 1 2 a , b 1 1 b q 2 q 3 Language: u �→ 2 − n , n = length of longest sequence of a ’s in u Not accepted by any WFA 13 / 17

Undecidability of equivalence (min semantics) Threshold: given probabilistic automaton X and κ ∈ [0 , 1], is there u ∈ A ∗ such that L X ( u ) < κ ? A A κ κ X κ Y = Z = A A 14 / 17

Approximate equivalence Given c ∈ [0 , 1) and l 1 , l 2 : A ∗ → [0 , 1], � c � | u | � d c ( l 1 , l 2 ) = | l 1 ( u ) − l 2 ( u ) | · | A | u ∈ A ∗ Given λ > 0 we can compute x such that | d c ( l 1 , l 2 ) − x | ≤ λ 15 / 17

Note on singleton alphabets Expressivity and undecidability proofs do not extend to | A | = 1 Separate proofs using linear recurrence sequences Expressivity result still holds ◮ Proof uses Skolem–Mahler–Lech and Cayley–Hamilton Deciding equivalence is at least hard ◮ Reduction from Positivity to Threshold 16 / 17

Conclusions Contributions ◮ Two natural semantics ◮ More expressivity ◮ Undecidability of equivalence Interplay of convex algebra, number theory, category theory Approximation techniques interesting for verification applications ◮ different metrics? 17 / 17