On Higher-Order Probabilistic Subrecursion

(1)

On Higher-Order Probabilistic Subrecursion

Flavien BREUVART, Ugo Dal Lago

Focus, Inria team, Bologna

LC&A: 2016 September 6

(2)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

Finite Representation prob

³

M_→k^´₌ ^f_g(k)^(k) _∈Q Functional Representation prob

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

Parametrization ForM:_N_→_N

When is (infinite) randomization useful?

Monte Carlo

f(n)reached with prob_>¹₂₊_² Las Vegas

f(n)reached and ascertained with prob_>_²

Dynamic/Strict bounds Make²dynamic/null

Different classes

Complexity Primitive Rec. HO Prim. Rec. Recursive

BPP

ZPP

NP,PP Exponential

Blow Up

future work Depending on

Prob. Operators

(3)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

(4)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

Complexity Primitive Rec. HO Prim. Rec. Recursive BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

(5)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

(6)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

(7)

Gödel’s System _T :

A Language for HO Primitive Recursion

Gödel’s System

_T

(Call-by-Value)

M,N,L ::₌ x _| _λx.M _| M N _| rec _| 0 _| S

recM N 0 _→ M recM N (Sn)_→N n (recM N n)

Γ,x:A_`x:A

Γ,x:A_`M:B Γ`λx.M:A_→B

Γ`M:A_→B _Γ_`N:A Γ`M N:B

Γ`0:_N _Γ_`S:_N_→_N _Γ_`rec:A_→(_N_→A_→A)_→_N_→A

A Powerful System

•

Corresponds to provably recursive functions in Peano arithmetic

•

Encodes any monad (states, exceptions, continuations...)

(8)

Probabilistic Systems _T

Different Probabilistic Extensions for System

T M,N,L ::₌ x _| λx.M _| M N _| rec _| 0 _| S _| M_⊕N _| X _| R

(3_⊕4)_⊕2

3_⊕4 2

3 4

1

2 1

2

1

2 1

2

Finite

tree X S 0

0 S(X S 0) 1 SS(X S 0)

2 . ..

1 2

1

2 1

2

Finitely branching

Finite paths

R

0 1 2 3 4

1

2 1

4 1

8 1

24

T^L ⊂ T^R = T^X

(9)

Almost Sure Convergence

T^L^,^R^,^X

is Almost Surely Converging

∀M_∈_T^L^,^R^,^X, Prob(M_⇓)₌1

Idea of Reducibility

1)By induction onA, we define Red_A_⊆{M_{| `}M:A}, 2)By induction on_→:

M_→N_∈Red_A _⇒ M_∈Red_A, 2⁰)Red_A is inhabited (ind. onA)

3)By induction onA:

Red_A_⊆ASC, 4)By induction onM:

`M:A _⇒ M_∈Red_A,

Specificities for Probabilities

Item 3)becomes:

3)Red_A are exactly the ASC terms that are evaluated into

[[M]]_⊆Red_A.

We also need a technical lemma 0)The application is continuous:

[[M N]] ₌ ^h^h[[M]] [[N]]ⁱⁱ.

(10)

Finite Representation of _T

^L

’s Distributions

∀M_∈_T^L(_N),∃N_∈_T(_N_→_Q), Prob(M_→k) :₌NF(N k)

Using a Finite Random Tape Using a Single Memory Cell s :

_N

(N₁_⊕N₂)^†:₌if(even!s) (s:₌!s/2;N₁) (s:₌!s/2;N₂)

State Passing Transformation

M^† _λs.M^‡

Counting Occurrences

P(M_→k) :₌

#ⁿs_≤2^m_|M^‡_→k^o 2^m

formsufficiently large.

T-Computable (simple recursion on s)

Exists since the execution of M_∈_T^L is bounded

Parametrization: How do we calculate the bound 2

^m

?

Feasible but more technical: we use a more complex monad.

(11)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

(12)

Functional Representation of Probabilistic recursive functions

What is a functional representation?

LetM a probabilistic program that output an integer with probability 1.

We look for a recursive functionf:_N_→_R

N→N

such that:

prob(M_→k)₌f(k)

X n

f(k)(n) 2ⁿ

let f M k n :=

let tab : Rational[] in let rec simul M p:=

eval M with

"return k" -> tab[k] += 1/(2^p);

"if coin then L else N" -> { newThread{simul L (p+1)}; newThread{simul N (p+1)}; };

in newthread{simul M 0}; newThread{

while ( sum tab < 1-1/n ) do skip; return tab[k];

}

tab: dynamical approximation Updating tabon terminating

branches

Run probabilistic choicesin parallel Waitingfor a reasonable

approximation

(13)

Functional Representation of Probabilistic recursive functions

What is a functional representation?

We look for a recursive functionf:_N_→_N_→_N such that:

prob(M_→k)₌^X

n

f(k)(n) 2ⁿ

let f M k n :=

eval M with

"return k" -> tab[k] += 1/(2^p);

"if coin then L else N" -> { newThread{simul L (p+1)}; newThread{simul N (p+1)}; };

in newthread{simul M 0}; newThread{

while ( sum tab < 1-1/n ) do skip; return tab[k];

}

branches

approximation

(14)

Functional Representation of Probabilistic recursive functions

What is a functional representation?

We look for a recursive functionf:_N_→_N_→_Q such that:

prob(M_→k)₌lim

n f(k)(n) ^³1₋^X

k

f(k)(n)^´_≤1 n

let f M k n :=

eval M with

"return k" -> tab[k] += 1/(2^p);

"if coin then L else N" -> { newThread{simul L (p+1)};

newThread{simul N (p+1)};

};

in newthread{simul M 0};

newThread{

while ( sum tab < 1-1/n ) do skip;

return tab[k];

}

branches

approximation

(15)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

(16)

Functional representation of _T

^X

Remark: Finite representation does not hold

Prob[X(rec 0(_λxy.0_⊕SSx))1 _→ 0] is not rational

Objective: Functional representation of

T^X

in

T^L ForM_∈_T^X(_N), findM^∗_∈_T^L(_N_→_N),[[M^∗ n]]_'1

2n [[M]]

Naive idea: Counter

StopM afterf(n)steps

〈M, f(n)_〉^M_{−→ 〈}^→^M⁰ M⁰,f(n)₋1_〉

· · ·

→ 〈M⁰⁰,0_〉

→0

f(1)

...

_f₍₂₎

...

f(3)

...

f has to be_T-definable! May not be possible...

(17)

Functional representation of _T

^X

Remark: Finite representation does not hold

Prob[X(rec 0(_λxy.0_⊕SSx))1 _→ 0] is not rational

Objective: Functional representation of

T^X

in

T^L ForM_∈_T^X(_N), findM^∗_∈_T^L(_N_→_N),[[M^∗ n]]_'1

2n [[M]]

Naive idea: Counter

StopM afterf(n)steps

〈M, f(n)_〉^M_{−→ 〈}^→^M⁰ M⁰,f(n)₋1_〉

· · ·

→ 〈M⁰⁰,0_〉

→0

f(1)

...

_f₍₂₎

...

f(3)

...

f has to be_T-definable!

May not be possible...

(18)

Functional representation of _T

^R

Better Idea:

Truncating R

Limiting

themth

each apparition of the operatorRto[0, ...,n

∗m

]

R:₌











0 _7→ ¹₂

· · · n _7→ ₂n¹+1

− 7→ Err











Bounded Probability of error

Prob(M[R/R]_→Err) _≤ 1₋^Y

n≥0

(1₋ 1

2^m^∗ⁿ) _≤ 1 n

(19)

Functional representation of _T

^R

The Real Idea:

Increasing Truncature

Limitingthemthapparition of the operatorRto[0, ...,n_∗m]

R:₌(m:₌m₊1);











0 _7→ ¹₂

· · · n_∗m _7→ ₂n∗¹m+1

− 7→ Err











Bounded Probability of error

Prob(M[R/R]_→Err) _≤ 1₋^Y

n_≥0

(1₋ 1

2^m^∗ⁿ) _≤ 1 n

(20)

When are the Probabilities Computationally Necessary?

Are all distributions deterministically representable?

³

M_→k^´₌^P_n^f⁽ⁿ₂n^,^k) ∈R

When is (infinite) randomization useful?

Monte Carlo

Different classes

BPP

ZPP

NP,PP Exponential

Blow Up

Prob. Operators

On Higher-Order Probabilistic Subrecursion