The Powerdomain of Continuous Random Variables Jean - PowerPoint PPT Presentation

The Powerdomain of Continuous Random Variables Jean Goubault-Larrecq, Daniele Varacca LSV - ENS Cachan, PPS - Paris Diderot LICS, June 21, 2011 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Semantics of Higher-Order Probabilistic Languages M , N , P ::= x 휏 ∣ 휆 x 휎 ⋅ M 휎, 휏 ::= 훾 ∣ MN ∣ 휎 → 휏 functions ∣ . . . ∣ V 휏 probability ∣ fair coin ⋇ ∣ . . . distributions ∣ val M ∣ let x = M in N sequence Open Problem: Does there exist a Cartesian closed category (=interpret 휎 → 휏 ) of continuous domains, closed under the probabilistic powerdomain (=interpret V 휏 )? We still do not know, but present an interesting alternative. Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Road Map Continuous Random Variables 1 The Classical Probabilistic Powerdomain The Definition of Continuous Random Variables In the CCC of BC-Domains Equational Theories Semantics 2 A Probabilistic Higher Order Language Semi-Decidability of Testing Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Road Map Continuous Random Variables 1 The Classical Probabilistic Powerdomain The Definition of Continuous Random Variables In the CCC of BC-Domains Equational Theories Semantics 2 A Probabilistic Higher Order Language Semi-Decidability of Testing Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Outline Continuous Random Variables 1 The Classical Probabilistic Powerdomain The Definition of Continuous Random Variables In the CCC of BC-Domains Equational Theories Semantics 2 A Probabilistic Higher Order Language Semi-Decidability of Testing Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Continuous Valuations Classical view [JonesPlotkin89]: interpret V 휏 as space of continuous valuations (=measures on a topology). Definition (Continuous Valuation) A function 휈 : Opens ( X ) → [ 0 , 1 ] with: 휈 ( ∅ ) = 0 (strictness) U ⊆ V ⇒ 휈 ( U ) ≤ 휈 ( V ) 휈 ( U ∪ V ) + 휈 ( U ∩ V ) = 휈 ( U ) + 휈 ( V ) ∪ ↑ sup ↑ 휈 ( i ∈ I U i ) = i ∈ I 휈 ( U i ) We shall also require 휈 ( X ) = 1 (probability). Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Dirac Valuations A Prominent Example. For any x ∈ X , the Dirac valuation 훿 x is defined as { 1 if x ∈ U 훿 x ( U ) = 0 otherwise Simple valuations are finite linear combinations of Dirac valuations n ∑ a i 훿 x i i = 1 with a 1 , . . . , a n ≥ 0, ∑ n i = 1 a i = 1. Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : the Cantor tree. 000 001 010 011 100 101 110 111 00 01 10 11 0 1 휖 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : the Cantor tree. 3 4 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 1 1 1 4 00 00 01 01 3 10 10 11 11 4 1 6 0 0 1 1 휖 휖 Evaluating 1 4 훿 00 + 1 6 훿 0 + 1 3 훿 01 + 1 4 훿 11 on ↑ 0 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : the Cantor tree. 1 3 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 1 1 1 4 00 00 01 01 3 10 10 11 11 4 1 6 0 0 1 1 휖 휖 Evaluating 1 4 훿 00 + 1 6 훿 0 + 1 3 훿 01 + 1 4 훿 11 on ↑ 01 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : the Cantor tree. 0 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 1 1 1 00 00 01 01 10 10 11 11 4 3 4 1 6 0 0 1 1 휖 휖 Evaluating 1 4 훿 00 + 1 6 훿 0 + 1 3 훿 01 + 1 4 훿 11 on ↑ 010 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : Any biased coin ( p , q ) with the Cantor tree. p + q = 1 induces a continuous valuation 휈 ( x ) = p a ( 1 − p ) b 0 . 2 where a is the number of 0’s in x , while b is the number of 1’s 000 001 010 011 100 101 110 111 00 01 10 11 0 1 휖 E.g., p = 0 . 2 , q = 0 . 8 . Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : Any biased coin ( p , q ) with the Cantor tree. p + q = 1 induces a continuous valuation 휈 ( x ) = p a ( 1 − p ) b 0 . 16 where a is the number of 0’s in x , while b is the number of 1’s 000 001 010 011 100 101 110 111 00 01 10 11 0 1 휖 E.g., p = 0 . 2 , q = 0 . 8 . Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : Any biased coin ( p , q ) with the Cantor tree. p + q = 1 induces a continuous valuation 휈 ( x ) = p a ( 1 − p ) b 0 . 032 where a is the number of 0’s in x , while b is the number of 1’s 000 001 010 011 100 101 110 111 00 01 10 11 0 1 휖 E.g., p = 0 . 2 , q = 0 . 8 . Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x Any biased coin ( p , q ) with { 0 , 1 } ≤ 휔 : p + q = 1 induces a continuous the Cantor tree. valuation 휈 ( x ) = p a ( 1 − p ) b where a is the number of 0’s in 1 2 x , while b is the number of 1’s If p = q = 1 / 2 the induced 000 001 010 011 100 101 110 111 valuation is the uniform 00 01 10 11 valuation Λ (on the top elts) 0 1 휖 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x Any biased coin ( p , q ) with { 0 , 1 } ≤ 휔 : p + q = 1 induces a continuous the Cantor tree. valuation 휈 ( x ) = p a ( 1 − p ) b where a is the number of 0’s in 1 4 x , while b is the number of 1’s If p = q = 1 / 2 the induced 000 001 010 011 100 101 110 111 valuation is the uniform 00 01 10 11 valuation Λ (on the top elts) 0 1 휖 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x Any biased coin ( p , q ) with { 0 , 1 } ≤ 휔 : p + q = 1 induces a continuous the Cantor tree. valuation 휈 ( x ) = p a ( 1 − p ) b 1 where a is the number of 0’s in 8 x , while b is the number of 1’s If p = q = 1 / 2 the induced 000 001 010 011 100 101 110 111 valuation is the uniform 00 01 10 11 valuation Λ (on the top elts) 0 1 휖 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : Any biased coin ( p , q ) with the Cantor tree. p + q = 1 induces a continuous valuation 휈 ( x ) = p a ( 1 − p ) b where a is the number of 0’s in x , while b is the number of 1’s 000 001 010 011 100 101 110 111 If p = q = 1 / 2 the induced 00 01 10 11 valuation is the uniform 0 1 valuation Λ (on the top elts) 휖 The support of Λ is The support supp 휈 , is the the whole Cantor tree complement of the largest U such that 휈 ( U ) = 0 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables

Continuous Random Variables Semantics Valuations Random Variables CCC Theories Examples Basic open sets: ↑ x for finite sequence x { 0 , 1 } ≤ 휔 : Any biased coin ( p , q ) with the Cantor tree. p + q = 1 induces a continuous valuation 휈 ( x ) = p a ( 1 − p ) b where a is the number of 0’s in x , while b is the number of 1’s 000 001 010 011 100 101 110 111 1 1 1 If p = q = 1 / 2 the induced 4 00 01 3 10 11 4 1 valuation is the uniform 6 0 1 valuation Λ (on the top elts) 휖 The support of The support supp 휈 , is the 4 훿 00 + 1 1 6 훿 0 + 1 3 훿 01 + 1 4 훿 11 complement of the largest U such that 휈 ( U ) = 0 Goubault-Larrecq, Varacca The Powerdomain of Continuous Random Variables