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

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 2/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1/2 1/2 1 a 0 1 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating if there are no infinite paths 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating if there are no infinite paths R is confluent if for all paths s 1 ← ∗ s → ∗ s 2 there is a t such that s 1 → ∗ t ← ∗ s 2 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating if there are no infinite paths R is confluent if for all paths s 1 ← ∗ s → ∗ s 2 there is a t such that s 1 → ∗ t ← ∗ s 2 R is convergent if R is confluent and terminating 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating if there are no infinite paths R is confluent if for all paths s 1 ← ∗ s → ∗ s 2 there is a t such that s 1 → ∗ t ← ∗ s 2 R is convergent if R is confluent and terminating 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating R is almost-sure terminating if there are no infinite paths if the probability of reaching a normal form is 1. R is confluent if for all paths s 1 ← ∗ s → ∗ s 2 there is a t such that s 1 → ∗ t ← ∗ s 2 R is convergent if R is confluent and terminating 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating R is almost-sure terminating if there are no infinite paths if the probability of reaching a normal form is 1. R is confluent ∞ � (1 / 2) i if for all paths s 1 ← ∗ s → ∗ s 2 i =0 there is a t such that s 1 → ∗ t ← ∗ s 2 R is convergent if R is confluent and terminating 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating R is almost-sure terminating if there are no infinite paths if the probability of reaching a normal form is 1. R is confluent ∞ (1 / 2) i = 1 � if for all paths s 1 ← ∗ s → ∗ s 2 i =0 there is a t such that s 1 → ∗ t ← ∗ s 2 R is convergent if R is confluent and terminating 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems Abstract Reduction System 1/2 is a pair R =( A , → ) where A is 1/2 1 a 0 1 countable and →⊆ A × A . R is terminating R is almost-sure terminating if there are no infinite paths if the probability of reaching a normal form is 1. R is confluent ∞ (1 / 2) i = 1 � if for all paths s 1 ← ∗ s → ∗ s 2 i =0 there is a t such that s 1 → ∗ t ← ∗ s 2 R is almost-surely convergent if for all s 1 ← ∗ s → ∗ s 2 there R is convergent is a normal form t such that s 1 → ∗ t ← ∗ s 2 and if R is confluent and P ( s 1 → ∗ t ) = P ( s 2 → ∗ t ) = 1 terminating 3/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1 / 3 1 / 3 1 / 2 0 1 1 / 2 a 1 / 6 1 / 6 4/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1 / 3 1 / 3 1 / 2 0 1 1 / 2 a 1 / 6 1 / 6 Probabilistic Abstract Reduction System R P = ( R , P ) where R = ( A , → ) is an ARS. For each s ∈ A \ R NF , � P ( s → t ) = 1 . s → t For all s and t , P ( s → t ) > 0 if and only if s → t . 4/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1 / 3 1 / 3 1 / 2 0 1 1 / 2 a 1 / 6 1 / 6 Probability of a finite path s 0 → s 1 → . . . → s n with n ≥ 0 P ( s 0 → s 1 → . . . → s n ) = � n i =1 P ( s i − 1 → s i ) . 4/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1 / 3 1 / 3 1 / 2 0 1 1 / 2 a 1 / 6 1 / 6 4/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1 / 3 1 / 3 1 / 2 0 1 1 / 2 a 1 / 6 1 / 6 Probability of s reaching t ∈ R NF ( s ) P ( s → ∗ t ) = � δ ∈ ∆( s , t ) P ( δ ) where ∆( s , t ) = { δ | δ = s → . . . → t } . Probability of diverging from s P ( s → ∗ t ) . � P ( s → ∞ ) = 1 − t ∈ R NF ( s ) 4/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems 1 / 3 1 / 3 ... 0 0 0 0 ... 1 / 3 1 / 3 1 / 2 ... 1 1 / 2 ... 1 / 2 a 1 / 6 0 1 ... 1 / 2 1 0 ... ... 1 a 1 / 6 1 / 6 ... a a 1 / 2 1 0 0 ... ... 1 / 6 1 / 3 ... 1 ... a 1 / 6 ... 1 0 ... ... 1 ... a a a Probability of diverging from s P ( s → ∗ t ) . � P ( s → ∞ ) = 1 − t ∈ R NF ( s ) 4/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Probabilistic Abstract Reduction Systems R is almost-surely convergent For all s 1 ← ∗ s → ∗ s 2 there is a normal form t such that s 1 → ∗ t ← ∗ s 2 and P ( s 1 → ∗ t ) = P ( s 2 → ∗ t ) = 1 . 1/2 1/2 1/2 1/2 a 0 1 b 1/2 1 a 0 1 1/2 1 − 1 / 4 2 1 − 1 / 4 3 1 − 1 / 4 4 1 / 3 1 / 3 1 − 1 / 4 . . . 0 1 2 3 1 / 2 1 / 4 3 1 / 4 2 0 1 / 2 1 1 / 4 1 / 4 4 . . . a a 1 / 6 1 / 6 5/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Almost-surely convergent R is almost-surely convergent For all s 1 ← ∗ s → ∗ s 2 there is a normal form t such that s 1 → ∗ t ← ∗ s 2 and P ( s 1 → ∗ t ) = P ( s 2 → ∗ t ) = 1 . Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. 6/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Almost-surely convergent R is almost-surely convergent For all s 1 ← ∗ s → ∗ s 2 there is a normal form t such that s 1 → ∗ t ← ∗ s 2 and P ( s 1 → ∗ t ) = P ( s 2 → ∗ t ) = 1 . Theorem A PARS is almost-surely terminating and confluent if and only if it is almost-surely convergent. Two tasks: 6/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems

Almost-surely convergent R is almost-surely convergent For all s 1 ← ∗ s → ∗ s 2 there is a normal form t such that s 1 → ∗ t ← ∗ s 2 and P ( s 1 → ∗ t ) = P ( s 2 → ∗ 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 6/10 Maja H. Kirkeby, Henning Christiansen Confluence in Probabilistically Terminating Reduction Systems