Convex Language Semantics for Nondeterministic Probabilistic - - PowerPoint PPT Presentation

Convex Language Semantics for Nondeterministic Probabilistic Automata

Gerco van Heerdt1 Justin Hsu1,2,3 Jo¨ el Ouaknine4 Alexandra Silva1

1University College London 2Cornell University 3University of Wisconsin–Madison 4MPI-SWS and Oxford University

October 19, 2018

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Deterministic probabilistic automaton

will get home

taste

1 2 1 2 sleep

taste

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Deterministic probabilistic automaton

1

will get home

taste

1 2 1 2 sleep

taste Expected value after n wine tastings is

• 1

2

n

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Deterministic probabilistic automaton

Given a finite alphabet A, a probabilistic automaton is a function Q

• [0, 1] × (DQ)A

with an initial state distribution Language of type A∗ → [0, 1] defined by determinization

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Determinization

NFAs: Q

• 2 × (PQ)A

⇒ PQ

• 2 × (PQ)A

Probabilistic automata: Q

• [0, 1] × (DQ)A

⇒ DQ

• [0, 1] × (DQ)A

Crucial: P and D are monads for which 2 and [0, 1] are algebras

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D-semantics: transitions

State space DQ Transition DQ → DDQ ⇒ flatten with monad multiplication

q0

taste

1 2 1 2 q1

taste . . .

q0

taste

1 2 q0 + 1 2 q1

taste

1 4 q0 + 3 4 q1

taste

1 2 · (1 2q0 + 1 2q1) + 1 2q1 = 1 4q0 + 3 4q1

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D-semantics: outputs

State space DQ Output DQ → D[0, 1] → [0, 1] D-algebra: expected value 1

q0

taste

1 2 1 2 q1

taste 1

1 2 1 4

. . .

q0

taste

1 2 q0 + 1 2 q1

taste

1 4 q0 + 3 4 q1

taste

1 4 · 1 + 3 4 · 0 = 1 4

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Nondeterministic probabilistic automaton

1

will get home

taste taste

1 2 1 2 sleep

taste Expected value after n tastings is one of 1, 1 2,

1

2

2

,

1

2

3

, . . . ,

1

2

n−1

,

1

2

n

Which ones “make sense”?

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Main results

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

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Nondeterministic probabilistic automaton

A nondeterministic probabilistic automaton is a function Q

• [0, 1] × (PDQ)A

with a set of initial state distributions Language of type A∗ → [0, 1] But: PD is not a monad

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Convex powerset

For a convex set X, PcX = convex subsets of X (finitely generated, nonempty)

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Convex powerset

For a convex set X, PcX = 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 Pc[0, 1]: 0.3 · [0.1, 0.5] + 0.7 · [0.2, 1] = [0.17, 0.85]

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Adjusted monad

Gives a monad PcD Multiplication: PcDPcDX → PcPcDX → PcDX convex algebra structure on Pc

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Two natural semantics

Need: PcD-algebra on [0, 1] Comes down to a Pc-algebra α PcD[0, 1]

• PcE
• Pc[0, 1]

α

[0, 1]

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Two natural semantics

Need: PcD-algebra on [0, 1] Comes down to a Pc-algebra α PcD[0, 1]

• PcE
• Pc[0, 1]

α

[0, 1]

Claim: α ∈ {min, max} Proof outline: Pc[0, 1] generated by {0}, {1}, [0, 1] α({0}) = 0, α({1}) = 1

• α determined by α([0, 1])

α([0, 1])2 = α([0, 1]) ⇓ α([0, 1]) = 0 or α([0, 1]) = 1

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Expressivity (min semantics)

1 1 1 a, b a

q0 1 2 1 2

a, b

q1

a b

q2 1 2 1 2

a, b

q3

Language: u → 2−n, n = length of longest sequence of a’s in u Not accepted by any WFA

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Undecidability of equivalence (min semantics)

Threshold: given probabilistic automaton X and κ ∈ [0, 1], is there u ∈ A∗ such that LX(u) < κ? Y = κ κ X A A A Z = κ A

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Approximate equivalence

Given c ∈ [0, 1) and l1, l2 : A∗ → [0, 1], dc(l1, l2) =

• u∈A∗

|l1(u) − l2(u)| ·

c

|A|

|u|

Given λ > 0 we can compute x such that |dc(l1, l2) − x| ≤ λ

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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

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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?

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