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Confluence and Convergence in Probabilistically Terminating Reduction Systems

Maja H. Kirkeby Henning Christiansen

Computer Science, Roskilde University, Denmark

LOPSTR – 2017 – Namur, Belgium

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Probabilistic Abstract Reduction Systems

PARS cover probabilistic algorithms and programs scheduling strategies protocols . . . Background Almost-sure convergence and almost-sure termination introduced for a subset of probabilistic programs by Hart et al 1983 PARS formulated by Bournez and Kirchner 2002 Almost-surely confluence formulated by Fr¨ uhwirt et al 2002, Bournez and Kirchner 2002

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Probabilistic Abstract Reduction Systems

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Probabilistic Abstract Reduction Systems

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Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A.

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Probabilistic Abstract Reduction Systems

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Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths

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Probabilistic Abstract Reduction Systems

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Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2

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Probabilistic Abstract Reduction Systems

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1 1/2 1/2

Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2 R is convergent if R is confluent and terminating

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Probabilistic Abstract Reduction Systems

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1 1/2 1/2

Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2 R is convergent if R is confluent and terminating

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Probabilistic Abstract Reduction Systems

1 a

1 1/2 1/2

R is almost-sure terminating if the probability of reaching a normal form is 1. Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2 R is convergent if R is confluent and terminating

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Probabilistic Abstract Reduction Systems

1 a

1 1/2 1/2

R is almost-sure terminating if the probability of reaching a normal form is 1.

∞

• i=0

(1/2)i Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2 R is convergent if R is confluent and terminating

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Probabilistic Abstract Reduction Systems

1 a

1 1/2 1/2

R is almost-sure terminating if the probability of reaching a normal form is 1.

∞

• i=0

(1/2)i = 1 Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2 R is convergent if R is confluent and terminating

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Probabilistic Abstract Reduction Systems

1 a

1 1/2 1/2

R is almost-sure terminating if the probability of reaching a normal form is 1.

∞

• i=0

(1/2)i = 1 R is almost-surely convergent if for all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1 Abstract Reduction System is a pair R =(A, →) where A is countable and →⊆ A × A. R is terminating if there are no infinite paths R is confluent if for all paths s1 ←∗ s →∗ s2 there is a t such that s1 →∗ t ←∗ s2 R is convergent if R is confluent and terminating

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Probabilistic Abstract Reduction Systems

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Probabilistic Abstract Reduction Systems

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Probabilistic Abstract Reduction System RP = (R, P) where R = (A, →) is an ARS. For each s ∈ A \ RNF,

• s→t

P(s → t) = 1. For all s and t, P(s → t) > 0 if and only if s → t.

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Probabilistic Abstract Reduction Systems

a 1

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Probability of a finite path s0 → s1 → . . . → sn with n ≥ 0 P(s0 → s1 → . . . → sn) = n

i=1 P(si−1 → si).

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Probabilistic Abstract Reduction Systems

a 1

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Probabilistic Abstract Reduction Systems

a 1

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Probability of s reaching t ∈ RNF(s) P(s →∗ t) =

δ∈∆(s,t) P(δ)

where ∆(s, t) = {δ | δ = s → . . . → t}. Probability of diverging from s P(s →∞) = 1 −

• t∈RNF (s)

P(s →∗ t).

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Probabilistic Abstract Reduction Systems

a 1

1/2 1/2 1/3 1/3 1/6 1/6

1 a 1 a 1 a 1 a

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

1 a 1 a 1 a

1/3 1/2 1/6 1/3 1/2 1/6 1/2 1/3 1/6

Probability of diverging from s P(s →∞) = 1 −

• t∈RNF (s)

P(s →∗ t).

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Probabilistic Abstract Reduction Systems

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. 1 a

1 1/2 1/2

a 1 b

1/2 1/2 1/2 1/2

a 1

1/2 1/2 1/3 1/3 1/6 1/6

1 2 3 . . . . . . a

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Almost-surely convergent

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent.

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Almost-surely convergent

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. Two tasks:

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Almost-surely convergent

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. Two tasks: almost-sure termination

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Almost-surely convergent

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. Two tasks: almost-sure termination confluence

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Almost-surely convergent

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. Two tasks: almost-sure termination: e.g. Lyapunov ranking functions.1 confluence

1L.M.Ferrer Fioriti, H.Hermanns, Probabilistic Termination POPL ’15. Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

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Almost-surely convergent

R is almost-surely convergent For all s1 ←∗ s →∗ s2 there is a normal form t such that s1 →∗ t ←∗ s2 and P(s1 →∗ t) = P(s2 →∗ t) = 1. Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. Two tasks: almost-sure termination: e.g. Lyapunov ranking functions.1 confluence: e.g. transform to systems suited for automatic confluence analysis.2

1L.M.Ferrer Fioriti, H.Hermanns, Probabilistic Termination POPL ’15. 2P.-L.Curien, G.Ghelli, On Confluence for Weakly Normalizing Systems ’91. Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

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Almost-sure termination

Definition A measure V : A → R+ is a Lyapunov ranking function if ∀s ∈ A, ∃ǫ > 0: V(s) ≥

• s→s′

P(s → s′) · V(s′) + ǫ Lemma (Ferrer Fioriti, Hermanns POPL ’15 ) A PARS RP = ((A, →), P) is a-s. terminating if there is a Lyapunov ranking function over A.

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Almost-sure termination

Definition A measure V : A → R+ is a Lyapunov ranking function if ∀s ∈ A, ∃ǫ > 0: V(s) ≥

• s→s′

P(s → s′) · V(s′) + ǫ Lemma (Ferrer Fioriti, Hermanns POPL ’15 ) A PARS RP = ((A, →), P) is a-s. terminating if there is a Lyapunov ranking function over A. a 1

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Almost-sure termination

Definition A measure V : A → R+ is a Lyapunov ranking function if ∀s ∈ A, ∃ǫ > 0: V(s) ≥

• s→s′

P(s → s′) · V(s′) + ǫ Lemma (Ferrer Fioriti, Hermanns POPL ’15 ) A PARS RP = ((A, →), P) is a-s. terminating if there is a Lyapunov ranking function over A. a 1

1/2 1/2 1/3 1/3 1/6 1/6

V(0) > 1/3V(0) + 1/2V(1) + 1/6V(a)

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Almost-sure termination

Definition A measure V : A → R+ is a Lyapunov ranking function if ∀s ∈ A, ∃ǫ > 0: V(s) ≥

• s→s′

P(s → s′) · V(s′) + ǫ Lemma (Ferrer Fioriti, Hermanns POPL ’15 ) A PARS RP = ((A, →), P) is a-s. terminating if there is a Lyapunov ranking function over A. a 1

1/2 1/2 1/3 1/3 1/6 1/6

V(0) = 100 V(1) = 100 V(a) = 1 V(0) > 1/3V(0) + 1/2V(1) + 1/6V(a)

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Almost-sure termination

Definition A measure V : A → R+ is a Lyapunov ranking function if ∀s ∈ A, ∃ǫ > 0: V(s) ≥

• s→s′

P(s → s′) · V(s′) + ǫ Lemma (Ferrer Fioriti, Hermanns POPL ’15 ) A PARS RP = ((A, →), P) is a-s. terminating if there is a Lyapunov ranking function over A. a 1

1/2 1/2 1/3 1/3 1/6 1/6

V(0) = 100 V(1) = 100 V(a) = 1 100 = V(0) > 1/3V(0) + 1/2V(1) + 1/6V(a) = 83 + 1/2

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Confluence

Lemma (Curien, Ghelli RTA ’91) Given two ARS R = (A, →R) and R′ = (A, →R′) and a mapping G : A → A′, then R is confluent if the following holds. R′ is confluent, R is normalizing, if s →R t then G(s) ↔∗

R′G(t),

∀t ∈ RNF, G(t) ∈ R′

NF, and

∀t, u ∈ RNF, G(t) = G(u) ⇒ t = u.

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Confluence

Lemma (Curien, Ghelli RTA ’91) Given two ARS R = (A, →R) and R′ = (A, →R′) and a mapping G : A → A′, then R is confluent if the following holds. R′ is confluent, R is normalizing, if s →R t then G(s) ↔∗

R′G(t),

∀t ∈ RNF, G(t) ∈ R′

NF, and

∀t, u ∈ RNF, G(t) = G(u) ⇒ t = u. a 1 1 a

1/2 1/2 1/3 1/3 1/6 1/6

number′ a′

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Confluence

Lemma (Curien, Ghelli RTA ’91) Given two ARS R = (A, →R) and R′ = (A, →R′) and a mapping G : A → A′, then R is confluent if the following holds. R′ is confluent, R is normalizing, if s →R t then G(s) ↔∗

R′G(t),

∀t ∈ RNF, G(t) ∈ R′

NF, and

∀t, u ∈ RNF, G(t) = G(u) ⇒ t = u. a 1 1 a

1/2 1/2 1/3 1/3 1/6 1/6

number′ a′

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Confluence iff

Lemma Given two ARS R = (A, →R) and R′ = (A, →R′) and a mapping G : A → A′, satisfying (surjective) ∀s′ ∈ A′, ∃s ∈ A, G(s) = s′, R and R′ are normalizing, if s →R t then G(s) ↔∗

R′ G(t), and

if G(s) ↔∗

R′ G(t) then s ↔∗ R t,

∀t ∈ RNF, G(t) ∈ R′

NF, and ∀t′ ∈ R′ NF, G −1(t′) ⊆ RNF,

∀t, u ∈ RNF, G(t) = G(u) ⇒ t = u, then R is confluent iff R′ is confluent. a 1 b

1/2 1/2 1/2 1/2

a 1 b a′ number′ b′

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Contributions

Self-contained definitions of PARS; proving basic properties Almost-sure convergence generalized to PARS Theorem: almost-sure convergence ⇔ almost-sure termination ∧ confluence Unfolded methods for (dis-) proving almost-sure convergence

Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems