Random Variable Models of Computation Michael W. Mislove Tulane - PowerPoint PPT Presentation

Random Variable Models of Computation Michael W. Mislove Tulane University New Orleans, LA Samson@60 Fest Oxford, May 28 –30, 2013 Joint work with Tyler Barker Work sponsored by AFOSR & NSF

The Real Samson - MFPS XXIV 2007 – 08 UEFA Champions League Manchester United Chelsea 1 1 Manchester United won 6–5 on penalties Date 21 May 2008 Venue Luzhniki Stadium, Moscow UEFA Man of the Match: Edwin van der Sar (Manchester United) Fans’ Man of the Match: Cristiano Ronaldo (Manchester United)

The Real Samson - MFPS XXIV

Continuous Random Variables ◮ f : ( X , µ ) → ( Y , Ω) random variable ◮ ( X , µ ) probability space, ◮ ( Y , Ω) measure space ◮ f is measurable: f − 1 ( A ) measurable ( ∀ A ∈ Ω) ◮ Continuous if X and Y topological spaces, f continuous and X , Y endowed with Borel σ -algebras.

Continuous Random Variables ◮ f : ( X , µ ) → ( Y , Ω) random variable ◮ Assume X , Y domains endowed with Scott topology. CRV ( X , Y ) = { ( µ, f ) | µ ∈ Prob ( X ) , f : supp µ → Y } supp µ = � { X | µ ( X ) = 1 & X closed } .

Continuous Random Variables ◮ f : ( X , µ ) → ( Y , Ω) random variable ◮ Assume X , Y domains endowed with Scott topology. CRV ( X , Y ) = { ( µ, f ) | µ ∈ Prob ( X ) , f : supp µ → Y } ◮ Goubault-Larrecq & Varacca, LICS 2011: BCD closed under Θ RV ( C , P ) = { ( µ, f ) ∈ CRV ( C , P ) | µ thin } ( µ, f ) ≤ ( ν, g ) iff π supp µ ( ν ) = µ & f ◦ π supp µ ≤ g C - Cantor tree

Continuous Random Variables ◮ f : ( X , µ ) → ( Y , Ω) random variable ◮ Assume X , Y domains endowed with Scott topology. CRV ( X , Y ) = { ( µ, f ) | µ ∈ Prob ( X ) , f : supp µ → Y } ◮ Goubault-Larrecq & Varacca, LICS 2011: BCD closed under Θ RV ( C , P ) = { ( µ, f ) ∈ CRV ( C , P ) | µ thin } ( µ, f ) ≤ ( ν, g ) iff π supp µ ( ν ) = µ & f ◦ π supp µ ≤ g C - Cantor tree ◮ Goal: Understand Θ RV ( C , P ) construction for P ∈ BCD

Motivating the Order - Automata ◮ A (generative) probabilistic automaton A has a finite set S of states, a start state s 0 ∈ S , a finite set of actions, Act , and a transition relation − → ⊆ S × Prob( Act × S ). ◮ Here’s a simple example with one action, flip : Unfolding the automaton: 1 0 2 ǫ 1 2 1 1 2 2 0 1 start ǫ 1 1 1 1 1 1 2 2 2 2 2 2 1 2 0 1 0 1 . . . . . . . . . . . . 1 1 2

Motivating the Order - Trace Distributions ◮ Typically, such automata are modeled by their trace distributions: µ 0 = δ ǫ µ 1 = 1 2 δ 0 + 1 2 δ 1 µ 2 = 1 4 δ 00 + 1 4 δ 01 + 1 4 δ 10 + 1 4 δ 11 . . . ◮ Stripping away the probabilities, we have the following sets on which these measures are concentrated: X 0 = { ǫ } X 1 = { 0 , 1 } X 2 = { 00 , 01 , 10 , 11 } . . .

Motivating the Order - Trace Distributions ◮ Stripping away the probabilities, we have the following sets on which these measures are concentrated: X 0 = { ǫ } X 1 = { 0 , 1 } X 2 = { 00 , 01 , 10 , 11 } . . . ◮ Notice that the X n s are antichains , and X 0 ⊑ C X 1 ⊑ C X 2 ⊑ C · · · , where X ⊑ C Y ⇔ X ⊆ ↓ Y & Y ⊆ ↑ X ⇔ π X ( Y ) = X

The Underlying Structure - Domains and Trees ◮ A ∞ = A ∗ ∪ A ω is a domain under the prefix order. KA ∞ = A ∗ – the finite words If A is finite, then A ∞ is coherent Compact in the Lawson topology k ∈ A ∗ , F ⊆ A ∗ finite Open sets: U = ↑ k \ ↑ F ,

The Underlying Structure - Domains and Trees ◮ A ∞ = A ∗ ∪ A ω is a domain under the prefix order. ◮ AC ( A ∞ ) = ( { X | Lawson-compact antichain } , ⊑ C ) X ⊑ C Y ⇔ X ⊆ ↓ Y & Y ⊆ ↑ X ⇔ π X ( Y ) = X Subdomain of P C ( A ∞ ).

The Underlying Structure - Domains and Trees ◮ A ∞ = A ∗ ∪ A ω is a domain under the prefix order. ◮ AC ( A ∞ ) = ( { X | Lawson-compact antichain } , ⊑ C ) ◮ Theorem: AC ( A ∞ ) is a bounded complete domain: all nonempty subsets have infima. ( ∅ � = F ⊆ AC ( A ∞ ) ⇒ inf F = Max( � X ∈F ↓ X )

The Underlying Structure - Domains and Trees ◮ A ∞ = A ∗ ∪ A ω is a domain under the prefix order. ◮ AC ( A ∞ ) = ( { X | Lawson-compact antichain } , ⊑ C ) ◮ Theorem: AC ( A ∞ ) is a bounded complete domain: all nonempty subsets have infima. Moreover, given { X n } n ∈ N ⊆ AC ( A ∞ ) directed and X ∈ AC ( A ∞ ), TAE: ( i ) X = sup n X n ( ii ) X = lim n X n in the Vietoris topology on Γ( A ∞ ). ◮ In particular, any X ∈ AC ( A ∞ ) satisfies X = sup n π n ( X ) = lim n π n ( X ), where π n : A ∞ → A ≤ n is the canonical retraction.

Thin Probability Measures ◮ µ ∈ Prob ( A ∞ ) is thin if supp Λ µ ∈ AC ( A ∞ ). Note: supp Λ µ is in the Lawson topology. ◮ Define µ ≤ ν iff π supp Λ µ ( ν ) = µ Agrees with usual domain order ( qua valuations) / functional analysis order via cones. Θ Prob ( A ∞ ) = ( { µ ∈ Prob ( A ∞ ) | µ thin } , ≤ ).

Thin Probability Measures ◮ µ ∈ Prob ( A ∞ ) is thin if supp Λ µ ∈ AC ( A ∞ ). ◮ Proposition: (Θ Prob ( A ∞ ) , ≤ ) is a bounded complete domain: all nonempty subsets have infima. ( ∅ � = M ⊆ Θ Prob ( A ∞ ) ⇒ ∀ ν ∈ M , inf M = π M ( ν ) , M = inf µ ∈M supp Λ µ )

Thin Probability Measures ◮ µ ∈ Prob ( A ∞ ) is thin if supp Λ µ ∈ AC ( A ∞ ). ◮ Proposition: (Θ Prob ( A ∞ ) , ≤ ) is a bounded complete domain: all nonempty subsets have infima. Moreover, given { µ n } n ∈ N ⊆ Θ Prob ( A ∞ ) directed and µ ∈ Θ Prob ( A ∞ ), TAE: ( i ) µ = sup n µ n ( ii ) µ = lim n µ n in the weak ∗ -topology on Θ Prob ( A ∞ ). ◮ In particular, any µ ∈ Θ Prob ( A ∞ ) satisfies µ = sup n π n ( µ ) = lim n π n ( µ ), where π n : A ∞ → A ≤ n is the canonical retraction.

Adding Function Spaces ◮ X ⊑ C Y ∈ AC ( A ∞ ) , P ∈ BCD = ⇒ f �→ f ◦ π X : [ X − → P ] ֒ → [ Y − → P ] & g ( x ) = inf g ( π − 1 g �→ � g : [ Y − → P ] − → → [ X − → P ] by � X ( x )).

Adding Function Spaces ◮ X ⊑ C Y ∈ AC ( A ∞ ) , P ∈ BCD = ⇒ f �→ f ◦ π X : [ X − → P ] ֒ → [ Y − → P ] & g ( x ) = inf g ( π − 1 g �→ � g : [ Y − → P ] − → → [ X − → P ] by � X ( x )). ◮ X ∈ AC ( A ∞ ) , P ∈ BCD = ⇒ [ X − → P ] ∈ BCD: → P ] ≃ lim n P π n ( X ) . [ X − → P ] ≃ lim n [ π n ( X ) −

Adding Function Spaces ◮ X ∈ AC ( A ∞ ) , P ∈ BCD = ⇒ [ X − → P ] ∈ BCD: → P ] ≃ lim n P π n ( X ) . [ X − → P ] ≃ lim n [ π n ( X ) − ◮ ( � X ∈ AC ( A ∞ ) [ X − → P ] , ≤ R ) ∈ BCD: f ≤ R g iff dom f ⊑ C dom g & f ◦ π dom f ≤ g .

Adding Function Spaces ◮ X ∈ AC ( A ∞ ) , P ∈ BCD = ⇒ [ X − → P ] ∈ BCD: → P ] ≃ lim n P π n ( X ) . [ X − → P ] ≃ lim n [ π n ( X ) − ◮ ( � X ∈ AC ( A ∞ ) [ X − → P ] , ≤ R ) ∈ BCD: f ≤ R g iff dom f ⊑ C dom g & f ◦ π dom f ≤ g . Defining the Model ◮ Θ Prob ( A ∞ ) × � X ∈ AC ( A ∞ ) [ X − → P ] ∈ BCD if P ∈ BCD.

Adding Function Spaces ◮ X ∈ AC ( A ∞ ) , P ∈ BCD = ⇒ [ X − → P ] ∈ BCD: → P ] ≃ lim n P π n ( X ) . [ X − → P ] ≃ lim n [ π n ( X ) − ◮ ( � X ∈ AC ( A ∞ ) [ X − → P ] , ≤ R ) ∈ BCD: f ≤ R g iff dom f ⊑ C dom g & f ◦ π dom f ≤ g . Defining the Model ◮ Θ Prob ( A ∞ ) × � X ∈ AC ( A ∞ ) [ X − → P ] ∈ BCD if P ∈ BCD. ◮ For P ∈ BCD Θ RV ( A ∞ , P ) = { ( µ, f ) | µ ∈ Θ Prob ( A ∞ ) , f : supp Λ µ − → P } is a retract of Θ Prob ( A ∞ ) × � X ∈ AC ( A ∞ ) [ X − → P ] : ( µ, f ) �→ ( π Y ( µ ) , f ◦ π Y ) is the projection Y = supp Λ µ ∧ dom f .

The Monad Given P ∈ BCD, define ◮ η P : P → Θ RV ( A ∞ , P ) by η P ( x ) = ( δ ǫ , const x ). → Θ RV ( A ∞ , Q ) and ◮ Given h : P − ( � x ∈ F r x δ x , f ) ∈ Θ RV ( A ∞ , P ) define h † : Θ RV ( A ∞ , P ) − → Θ RV ( A ∞ , Q ) by h † ( � x ∈ F r x δ x , f ) = ( � x ∈ F r x δ x ∗ ( π 1 ◦ h ◦ f )( x ) , g ), where g : ( � x ∈ F x · supp Λ ( π 1 ◦ h ◦ f )( x )) → Q by g ( y ) = ( π 2 ◦ h ◦ f )( y )( y ).

The Monad Given P ∈ BCD, define ◮ η P : P → Θ RV ( A ∞ , P ) by η P ( x ) = ( δ ǫ , const x ). → Θ RV ( A ∞ , Q ) and ◮ Given h : P − ( � x ∈ F r x δ x , f ) ∈ Θ RV ( A ∞ , P ) define h † : Θ RV ( A ∞ , P ) − → Θ RV ( A ∞ , Q ) by h † ( � x ∈ F r x δ x , f ) = ( � x ∈ F r x δ x ∗ ( π 1 ◦ h ◦ f )( x ) , g ), where g : ( � x ∈ F x · supp Λ ( π 1 ◦ h ◦ f )( x )) → Q by g ( y ) = ( π 2 ◦ h ◦ f )( y )( y ). ◮ These constructs satisfy the monad laws

The Monad Given P ∈ BCD, define ◮ η P : P → Θ RV ( A ∞ , P ) by η P ( x ) = ( δ ǫ , const x ). → Θ RV ( A ∞ , Q ) and ◮ Given h : P − ( � x ∈ F r x δ x , f ) ∈ Θ RV ( A ∞ , P ) define h † : Θ RV ( A ∞ , P ) − → Θ RV ( A ∞ , Q ) by h † ( � x ∈ F r x δ x , f ) = ( � x ∈ F r x δ x ∗ ( π 1 ◦ h ◦ f )( x ) , g ), where g : ( � x ∈ F x · supp Λ ( π 1 ◦ h ◦ f )( x )) → Q by g ( y ) = ( π 2 ◦ h ◦ f )( y )( y ). ◮ These constructs satisfy the monad laws ◮ BUT h † is not monotone: h ( x ) = ( δ b , const y ) ( δ ǫ , f ) ≤ ( δ a , f ), but ( δ b , g 1 ) �≤ ( δ ab , g 2 ), if a � = b ∈ A .