On Higher-Order Probabilistic Computation: Relational Reasoning, - - PowerPoint PPT Presentation

On Higher-Order Probabilistic Computation:

Relational Reasoning, Termination, and Bayesian Programming

Ugo Dal Lago (Based on joint work with Michele Alberti, Raphaëlle Crubillé, Charles Grellois, Davide Sangiorgi,. . . ) IFIP WG 2.2 Annual Meeting, Brno, September 17th

Probabilistic Models

◮ The environment is supposed not to behave

deterministically, but probabilistically.

Probabilistic Models

◮ The environment is supposed not to behave

deterministically, but probabilistically.

◮ Crucial when modeling uncertainty.

Probabilistic Models

◮ The environment is supposed not to behave

deterministically, but probabilistically.

◮ Crucial when modeling uncertainty. ◮ Useful to handle complex domains.

Probabilistic Models

◮ The environment is supposed not to behave

deterministically, but probabilistically.

◮ Crucial when modeling uncertainty. ◮ Useful to handle complex domains. ◮ Example:

q0 q1 q2 q3

1 4 3 4 1 1 2 1 2 1 3 2 3

Probabilistic Models

◮ The environment is supposed not to behave

deterministically, but probabilistically.

◮ Crucial when modeling uncertainty. ◮ Useful to handle complex domains. ◮ Example:

q0 q1 q2 q3

1 4 3 4 1 1 2 1 2 1 3 2 3 ◮ Abstractions:

◮ (Labelled) Markov Chains.

Probabilistic Models

ROBOTICS

Probabilistic Models

ARTIFICIAL INTELLIGENCE

Probabilistic Models

NATURAL LANGUAGE PROCESSING

Randomized Computation

◮ Algorithms and automata are assumed to have the ability

to sample from a distribution [dLMSS1956,R1963].

Randomized Computation

◮ Algorithms and automata are assumed to have the ability

to sample from a distribution [dLMSS1956,R1963].

◮ This is a powerful tool when solving computational

problems.

Randomized Computation

◮ Algorithms and automata are assumed to have the ability

to sample from a distribution [dLMSS1956,R1963].

◮ This is a powerful tool when solving computational

problems.

◮ Example:

Randomized Computation

◮ Algorithms and automata are assumed to have the ability

to sample from a distribution [dLMSS1956,R1963].

◮ This is a powerful tool when solving computational

problems.

◮ Example:

Randomized Computation

◮ Algorithms and automata are assumed to have the ability

to sample from a distribution [dLMSS1956,R1963].

◮ This is a powerful tool when solving computational

problems.

◮ Example: ◮ Abstractions:

◮ Randomized algorithms; ◮ Probabilistic Turing machines. ◮ Labelled Markov chains.

Randomized Computation

ALGORITHMICS

Randomized Computation

CRYPTOGRAPHY

Randomized Computation

PROGRAM VERIFICATION

Higher-Order Computation

◮ Mainly useful in programming.

Higher-Order Computation

◮ Mainly useful in programming. ◮ Functions are first-class citizens:

◮ They can be passed as arguments; ◮ They can be obtained as results.

Higher-Order Computation

◮ Mainly useful in programming. ◮ Functions are first-class citizens:

◮ They can be passed as arguments; ◮ They can be obtained as results.

◮ Motivations:

◮ Modularity; ◮ Code reuse; ◮ Conciseness.

Higher-Order Computation

◮ Mainly useful in programming. ◮ Functions are first-class citizens:

◮ They can be passed as arguments; ◮ They can be obtained as results.

◮ Motivations:

◮ Modularity; ◮ Code reuse; ◮ Conciseness.

◮ Example:

Higher-Order Computation

◮ Mainly useful in programming. ◮ Functions are first-class citizens:

◮ They can be passed as arguments; ◮ They can be obtained as results.

◮ Motivations:

◮ Modularity; ◮ Code reuse; ◮ Conciseness.

◮ Example:

Higher-Order Computation

◮ Mainly useful in programming. ◮ Functions are first-class citizens:

◮ They can be passed as arguments; ◮ They can be obtained as results.

◮ Motivations:

◮ Modularity; ◮ Code reuse; ◮ Conciseness.

◮ Example: ◮ Models:

◮ λ-calculus

Higher-Order Computation

FUNCTIONAL PROGRAMMING

Higher-Order Computation

FUNCTIONAL DATA STRUCTURES

Higher-Order Computation

λ-CALCULUS

Higher-Order Probabilistic Computation

Does it Make Sense?

Higher-Order Probabilistic Computation

Does it Make Sense? What Kind of Metatheory Does it Have?

Higher-Order Probabilistic Computation

Does it Make Sense? What Kind of Metatheory Does it Have? Applications?

1980 1990 2000 2010 [Saheb-Djaromi] [JonesPlotkin] [JungTix] [DanosHarmer]

1980 1990 2000 2010 [Saheb-Djaromi] [JonesPlotkin] [JungTix] [DanosHarmer] . . . too many

Outline Part I Relational Reasoning Part II Bayesian Functional Programming Part III Termination

Part I Relational Reasoning

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M;

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M;

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M; ◮ Value Distributions:

V

D

− → D(V ) ∈ R[0,1]

• D =
• V

D(V ) ≤ 1.

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M; ◮ Value Distributions:

V

D

− → D(V ) ∈ R[0,1]

• D =
• V

D(V ) ≤ 1.

◮ Semantics: M = supM⇓D D;

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M; ◮ Value Distributions:

V

D

− → D(V ) ∈ R[0,1]

• D =
• V

D(V ) ≤ 1.

◮ Semantics: M = supM⇓D D;

M ⇓ ∅ V ⇓ {V 1} M ⇓ D N ⇓ E M ⊕ N ⇓ 1

2D + 1 2E

M ⇓ K {P[N/x] ⇓ E P }λx.P ∈SK MN ⇓

• λx.P ∈SK

K (λx.P) · EP

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M; ◮ Value Distributions:

V

D

− → D(V ) ∈ R[0,1]

• D =
• V

D(V ) ≤ 1.

◮ Semantics: M = supM⇓D D; ◮ Context Equivalence: M ≡ N iff for every context C it

holds that C[M] = C[N].

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M; ◮ Value Distributions:

V

D

− → D(V ) ∈ R[0,1]

• D =
• V

D(V ) ≤ 1.

◮ Semantics: M = supM⇓D D; ◮ Context Equivalence: M ≡ N iff for every context C it

holds that C[M] = C[N]. C ::= [·] | λx.C | CM | MC | C ⊕M | M ⊕C

Syntax and Operational Semantics of Λ⊕

◮ Terms: M ::= x | λx.M | MM | M ⊕ M; ◮ Values: V ::= λx.M; ◮ Value Distributions:

V

D

− → D(V ) ∈ R[0,1]

• D =
• V

D(V ) ≤ 1.

◮ Semantics: M = supM⇓D D; ◮ Context Equivalence: M ≡ N iff for every context C it

holds that C[M] = C[N].

◮ Context Distance:

δC(M, N) = supC | C[M] − C[N]|.

Examples I ⊕ Ω vs. I

Examples I ⊕ Ω vs. I

λx.x

Examples I ⊕ Ω vs. I

∆∆ = (λx.xx)(λx.xx)

Examples I ⊕ Ω vs. I

Not Context Equivalent: C = [·]. Context Distance? Consider Cn = (λx. x . . . x n times )[·].

Examples I ⊕ Ω vs. I I ⊕ Ω vs. Ω

Examples I ⊕ Ω vs. I I ⊕ Ω vs. Ω

Not Context Equivalent: C = [·]. Context Distance? Cannot Easily Amplify.

Examples I ⊕ Ω vs. I I ⊕ Ω vs. Ω (λx.I) ⊕ (λx.Ω) vs. λx.I ⊕ Ω

Examples I ⊕ Ω vs. I I ⊕ Ω vs. Ω (λx.I) ⊕ (λx.Ω) vs. λx.I ⊕ Ω

Not Context Equivalent in CBV: C = (λx.x(xI))[·] Apparently Context Equivalent in CBN.

Examples I ⊕ Ω vs. I I ⊕ Ω vs. Ω (λx.I) ⊕ (λx.Ω) vs. λx.I ⊕ Ω Y1 vs. Y2

Examples I ⊕ Ω vs. I I ⊕ Ω vs. Ω (λx.I) ⊕ (λx.Ω) vs. λx.I ⊕ Ω Y1 vs. Y2

Y1M →∗ M(Y2M) ⊕ M(Y3M) Y2M →∗ M(Y1M) ⊕ M(Y3M) Y3M →∗ M(Y1M) ⊕ M(Y2M)

A Labelled Markov Chain for Λ⊕

Terms

A Labelled Markov Chain for Λ⊕

Terms Values

A Labelled Markov Chain for Λ⊕

Terms Values M

A Labelled Markov Chain for Λ⊕

Terms Values M V W Z . . . eval, M(V ) eval, M(W) eval, M(Z)

A Labelled Markov Chain for Λ⊕

Terms Values λx.N

A Labelled Markov Chain for Λ⊕

Terms Values λx.N N{W/x} W, 1

Probabilistic Applicative Bisimulation

λx.M R λx.N

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R M R N

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R M R N M eval

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R M R N M eval N eval

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R M R N M eval N eval M(E)

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R M R N M eval N eval M(E) N(E)

Probabilistic Applicative Bisimulation

λx.M R λx.N M{L/x} L N{L/x} L R M R N M eval N eval M(E) N(E) =

Applicative Bisimilarity vs. Context Equivalence

◮ Bisimilarity: the union ∼ of all bisimulation relations. ◮ Is it that ∼ is included in ≡? How to prove it? ◮ Natural strategy: is ∼ a congruence?

◮ If this is the case:

M ∼ N = ⇒ C[M] ∼ C[N] = ⇒

• C[M] =
• C[N]

= ⇒ M ≡ N.

◮ This is a necessary sanity check anyway.

◮ The naïve proof by induction fails, due to application:

from M ∼ N, one cannot directly conclude that LM ∼ LN.

Howe’s Technique

R RH

Howe’s Technique

R RH ⊆

Howe’s Technique

R RH ⊆ RH is a Congruence whenever R is an equivalence

Howe’s Technique

⊆ ∼H is a Congruence ∼ ∼H

Howe’s Technique

⊆ ∼H is a Congruence ∼ ∼H ⊇ Key Lemma

Our Neighborhood

◮ Λ, where we observe convergence

∼ ⊆ ≡ ≡ ⊆ ∼ CBN

• CBV
• [Abramsky1990, Howe1993]

◮ Λ⊕ with nondeterministic semantics, where we observe

convergence, in its may or must flavors.

∼ ⊆ ≡ ≡ ⊆ ∼ CBN

• ×

CBV

• ×

[Ong1993, Lassen1998]

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

∼ ⊆ ≡ ≡ ⊆ ∼ CBN

• ×

CBV

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

∼ ⊆ ≡ ≡ ⊆ ∼ CBN

• ×

CBV

• ◮ Counterexample for CBN: (λx.I) ⊕ (λx.Ω) ∼ λx.I ⊕ Ω

◮ Where these discrepancies come from?

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

∼ ⊆ ≡ ≡ ⊆ ∼ CBN

• ×

CBV

• ◮ Counterexample for CBN: (λx.I) ⊕ (λx.Ω) ∼ λx.I ⊕ Ω

◮ Where these discrepancies come from? ◮ From testing!

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

∼ ⊆ ≡ ≡ ⊆ ∼ CBN

• ×

CBV

• ◮ Counterexample for CBN: (λx.I) ⊕ (λx.Ω) ∼ λx.I ⊕ Ω

◮ Where these discrepancies come from? ◮ From testing! ◮ Bisimulation can be characterized by testing equivalence as

follows: Calculus Testing Λ T ::= ω | a · T PΛ⊕ T ::= ω | a · T | T, T NΛ⊕ T ::= ω | a · T | ∧i∈I Ti | . . .

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

⊆ ≤ ≤ ⊆ CBN

• ×

CBV

• ×

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

⊆ ≤ ≤ ⊆ CBN

• ×

CBV

• ×

◮ Probabilistic simulation can be characterized by testing as

follows: T ::= ω | a · T | T, T | T ∨ T

The Probabilistic Case

◮ Λ⊕ with probabilistic semantics.

⊆ ≤ ≤ ⊆ CBN

• ×

CBV

• ×

◮ Probabilistic simulation can be characterized by testing as

follows: T ::= ω | a · T | T, T | T ∨ T

◮ Full abstraction can be recovered if endowing Λ⊕ with

parallel disjunction [CDLSV2015].

⊆ ≤ ≤ ⊆ CBN

• ×

CBV

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first.

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω;

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω; ◮ Terms: any preterm M such that Γ ⊢ M.

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω; ◮ Terms: any preterm M such that Γ ⊢ M.

Γ, x ⊢ x x, Γ ⊢ M Γ ⊢ λx.M Γ ⊢ M ∆ ⊢ N Γ, ∆ ⊢ MN Γ ⊢ M Γ ⊢ N Γ ⊢ M ⊕ N

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω; ◮ Terms: any preterm M such that Γ ⊢ M. ◮ Behavioural Distance δb.

◮ The metric analogue to bisimilarity.

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω; ◮ Terms: any preterm M such that Γ ⊢ M. ◮ Behavioural Distance δb.

◮ The metric analogue to bisimilarity.

◮ Trace Distance δt.

◮ The maximum distance induced by traces, i.e., sequences of

actions: δt(M, N) = supT |Pr(M, T) − Pr(N, T)|.

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω; ◮ Terms: any preterm M such that Γ ⊢ M. ◮ Behavioural Distance δb.

◮ The metric analogue to bisimilarity.

◮ Trace Distance δt.

◮ The maximum distance induced by traces, i.e., sequences of

actions: δt(M, N) = supT |Pr(M, T) − Pr(N, T)|.

◮ Soundness and Completeness Results:

δb ≤ δc δc ≤ δb δt ≤ δc δc ≤ δt

• ×

Context Distance: the Affine Case [CDL2015]

◮ Let us consider a simple fragment of Λ⊕, first. ◮ Preterms: M, N ::= x | λx.M | MM | M ⊕ M | Ω; ◮ Terms: any preterm M such that Γ ⊢ M. ◮ Behavioural Distance δb.

◮ The metric analogue to bisimilarity.

◮ Trace Distance δt.

◮ The maximum distance induced by traces, i.e., sequences of

actions: δt(M, N) = supT |Pr(M, T) − Pr(N, T)|.

◮ Soundness and Completeness Results:

δb ≤ δc δc ≤ δb δt ≤ δc δc ≤ δt

• ×
• ◮ Example: δt(I, I ⊕ Ω) = δt(I ⊕ Ω, Ω) = 1

2.

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying.

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying. ◮ A Tuple LMC.

◮ Preterms:

M ::= x | λx.M | λ!x.M | MM | M ⊕ M | !M

◮ Terms: any preterm M such that Γ ⊢ M. ◮ States: sequences of terms, rather than terms. ◮ Actions not only model parameter passing, but also

copying of terms.

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying. ◮ A Tuple LMC.

◮ Preterms:

M ::= x | λx.M | λ!x.M | MM | M ⊕ M | !M

◮ Terms: any preterm M such that Γ ⊢ M.

!Γ, x ⊢ x !Γ, !x ⊢ x x, Γ ⊢ M Γ ⊢ λx.M !x, Γ ⊢ M Γ ⊢ λ!x.M !Γ ⊢ M !Γ ⊢!M Γ, !Θ ⊢ M ∆, !Θ ⊢ N Γ, ∆, Θ ⊢ MN Γ ⊢ M Γ ⊢ N Γ ⊢ M ⊕ N

◮ States: sequences of terms, rather than terms. ◮ Actions not only model parameter passing, but also

copying of terms.

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying. ◮ A Tuple LMC.

◮ Preterms:

M ::= x | λx.M | λ!x.M | MM | M ⊕ M | !M

◮ Terms: any preterm M such that Γ ⊢ M. ◮ States: sequences of terms, rather than terms. ◮ Actions not only model parameter passing, but also

copying of terms.

◮ Soundness and Completeness Results:

δt ≤ δc δc ≤ δt

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying. ◮ A Tuple LMC.

◮ Preterms:

M ::= x | λx.M | λ!x.M | MM | M ⊕ M | !M

◮ Terms: any preterm M such that Γ ⊢ M. ◮ States: sequences of terms, rather than terms. ◮ Actions not only model parameter passing, but also

copying of terms.

◮ Soundness and Completeness Results:

δt ≤ δc δc ≤ δt

• ◮ Examples: δt(!(I ⊕ Ω), !Ω) = 1

2

δt(!(I ⊕ Ω), !I) = 1.

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying. ◮ A Tuple LMC.

◮ Preterms:

M ::= x | λx.M | λ!x.M | MM | M ⊕ M | !M

◮ Terms: any preterm M such that Γ ⊢ M. ◮ States: sequences of terms, rather than terms. ◮ Actions not only model parameter passing, but also

copying of terms.

◮ Soundness and Completeness Results:

δt ≤ δc δc ≤ δt

• ◮ Examples: δt(!(I ⊕ Ω), !Ω) = 1

2

δt(!(I ⊕ Ω), !I) = 1.

◮ Trivialisation: the context distance collapses to an

equivalence in strongly normalising fragments or in presence

• f parellel disjuction.

Context Distance: the General Case [CDL2016]

◮ The LMC we have have worked so far with induces

unsound metrics for Λ⊕. . .

◮ . . . because it does not adequately model copying. ◮ A Tuple LMC.

◮ Preterms:

M ::= x | λx.M | λ!x.M | MM | M ⊕ M | !M

◮ Terms: any preterm M such that Γ ⊢ M. ◮ States: sequences of terms, rather than terms. ◮ Actions not only model parameter passing, but also

copying of terms.

◮ Soundness and Completeness Results:

δt ≤ δc δc ≤ δt

• ◮ Examples: δt(!(I ⊕ Ω), !Ω) = 1

2

δt(!(I ⊕ Ω), !I) = 1.

◮ Trivialisation: the context distance collapses to an

equivalence in strongly normalising fragments or in presence

• f parellel disjuction.

What would a sensible notion of distance look like?

Part II Bayesian Functional Programming

1. normalize( 2. let x = sample(bern 5 7

• ) in

3. let r = if x then 10 else 3 in 4.

• bserve 4 from poisson(r);

5. return(x)) x = true x = false

5 7 2 7

r = 10 r = 3 x = true x = false

1. normalize( 2. let x = sample(bern 5 7

• ) in

3. let r = if x then 10 else 3 in 4.

• bserve 4 from poisson(r);

5. return(x)) x = true x = false

5 7 2 7

r = 10 r = 3 x = true x = false

1. normalize( 2. let x = sample(bern 5 7

• ) in

3. let r = if x then 10 else 3 in 4.

• bserve 4 from poisson(r);

5. return(x)) x = true x = false

5 7 2 7

r = 10 r = 3 x = true x = false

1. normalize( 2. let x = sample(bern 5 7

• ) in

3. let r = if x then 10 else 3 in 4.

• bserve 4 from poisson(r);

5. return(x)) x = true x = false

5 7 2 7

r = 10 r = 3 x = true x = false

1. normalize( 2. let x = sample(bern 5 7

• ) in

3. let r = if x then 10 else 3 in 4.

• bserve 4 from poisson(r);

5. return(x)) x = true x = false

5 7 2 7

r = 10 r = 3 x = true x = false poisson(10)(4) ∼ 0.016 poisson(3)(4) ∼ 0.168

1. normalize( 2. let x = sample(bern 5 7

• ) in

3. let r = if x then 10 else 3 in 4.

• bserve 4 from poisson(r);

5. return(x)) x = true x = false

5 7 2 7

r = 10 r = 3 x = true x = false 0.22 0.78

Bayesian Functional Programming

ANGLICAN

Bayesian Functional Programming

HAKARU

1. normalize( 2. let x = sample(gauss (0, 1)) in 4.

• bserve d from exp(1/f(x));

5. return(x))

1. normalize( 2. let x = sample(gauss (0, 1)) in 4.

• bserve d from exp(1/f(x));

5. return(x))

Bayesian Programming: Semantics

◮ Giving semantics to programming languages like Anglican

• r Hakaru is nontrivial:

◮ Real numbers; ◮ Sampling from continuous distributions; ◮ Conditioning.

Bayesian Programming: Semantics

◮ Giving semantics to programming languages like Anglican

• r Hakaru is nontrivial:

◮ Real numbers; ◮ Sampling from continuous distributions; ◮ Conditioning.

◮ Key ingredients:

◮ In M ⇓ D, we need D to be a measure, because the set of

term is not countable anymore.

Bayesian Programming: Semantics

◮ Giving semantics to programming languages like Anglican

• r Hakaru is nontrivial:

◮ Real numbers; ◮ Sampling from continuous distributions; ◮ Conditioning.

◮ Key ingredients:

◮ In M ⇓ D, we need D to be a measure, because the set of

term is not countable anymore.

◮ Terms must thus be equipped with the structure of a

measurable space.

Bayesian Programming: Semantics

◮ Giving semantics to programming languages like Anglican

• r Hakaru is nontrivial:

◮ Real numbers; ◮ Sampling from continuous distributions; ◮ Conditioning.

◮ Key ingredients:

◮ In M ⇓ D, we need D to be a measure, because the set of

term is not countable anymore.

◮ Terms must thus be equipped with the structure of a

measurable space.

◮ From

M ⇓ K {P[N/x] ⇓ E P }λx.P ∈SK MN ⇓

• λx.P ∈SK

K (λx.P) · EP we go to M ⇓ K {P[N/x] ⇓ E P }λx.P ∈SK MN ⇓

• EP · dK (λx.P)

Bayesian Programming: Semantics

◮ Giving semantics to programming languages like Anglican

• r Hakaru is nontrivial:

◮ Real numbers; ◮ Sampling from continuous distributions; ◮ Conditioning.

◮ Key ingredients:

◮ In M ⇓ D, we need D to be a measure, because the set of

term is not countable anymore.

◮ Terms must thus be equipped with the structure of a

measurable space.

◮ From

M ⇓ K {P[N/x] ⇓ E P }λx.P ∈SK MN ⇓

• λx.P ∈SK

K (λx.P) · EP we go to M ⇓ K {P[N/x] ⇓ E P }λx.P ∈SK MN ⇓

• EP · dK (λx.P)

◮ This Lebesgue integral

does not necessarily exist.

◮ We must ensure that ⇓

gives rise to a stochastic kernel.

◮ In presence of conditioning,

we need even more.

Part III Termination

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

◮ Sound for termination, in absence

• f recursion.

◮ Poor expressive power. ◮ Intuitionistic Logic.

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

◮ Second-order Logic. ◮ Very expressive, extensionally. ◮ Still poor, intensionally.

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

◮ Motivated by Semantics. ◮ Complete for termination. ◮ Type inference is undecidable.

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

The Landscape: Type Theory

Simple Types Polymorphic Types Intersection Types Sized Types

τ ::= ι | τ → τ τ ::= · · · | α | ∀α.τ τ ::= · · · | τ ∧ τ τ ::= · · · | ι[ξ]

◮ Reasonably expressive,

intensionally.

◮ Type inference remains decidable

The Landscape: Recursion Theory

Determinism Ms →∗ Ns

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds

Ds can be smaller than 1.

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF

Undecidable; Σ0

1-complete.

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF Ds = 1

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF Ds = 1

Almost-Sure Termination Π0

2-complete.

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF Ds = 1 Uniform Termination ∀s.∃Ns ∈ NF

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF Ds = 1 Uniform Termination ∀s.∃Ns ∈ NF

Π0

2-complete.

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF Ds = 1 Uniform Termination ∀s.∃Ns ∈ NF ∀s. Ds = 1

The Landscape: Recursion Theory

Determinism Probabilism Ms →∗ Ns Ms = Ds Termination ∃Ns ∈ NF Ds = 1 Uniform Termination ∀s.∃Ns ∈ NF ∀s. Ds = 1

Π0

2-complete.

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ For every type τ, define a set of

reducible terms Redτ.

◮ Prove that all reducible terms are

• normalizing. . .

◮ . . . and that all typable terms are

reducible.

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder?

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder?

(fix x.M)V → M{fix x.M/x}V

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder? ◮ All the termination properties are lost, for very good

reasons.

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder? ◮ All the termination properties are lost, for very good

reasons.

◮ Is everything lost?

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder? ◮ All the termination properties are lost, for very good

reasons.

◮ Is everything lost? ◮ NO!

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder? ◮ All the termination properties are lost, for very good

reasons.

◮ Is everything lost? ◮ NO!

fix f λx : ι

f(x − 1) f(x) f(x) f(x + 1)

M fix f λx : ι

f(x − 1) f(x − 2) f(x − 3)

M

BAD! GOOD!

Deterministic Sized Types

◮ Pure λ-calculus with simple types is terminating.

◮ This can be proved in many ways, including by

reducibility.

◮ But useless as a programming language.

◮ What if we endow it with full recursion as a fix binder? ◮ All the termination properties are lost, for very good

reasons.

◮ Is everything lost? ◮ NO!

fix f λx : ι

f(x − 1) f(x) f(x) f(x + 1)

M fix f λx : ι

f(x − 1) f(x − 2) f(x − 3)

M

BAD! GOOD!

fix f λx : ι

f(x − 1) f(x − 2) f(x − 3)

M

GOOD!

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ.

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ. Index Terms

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ.

◮ Typing Fixpoints.

Γ, x : ι[a] → τ ⊢ M : ι[a + 1] → τ Γ ⊢ fix x.M : ι[ξ] → τ

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ.

◮ Typing Fixpoints.

Γ, x : ι[a] → τ ⊢ M : ι[a + 1] → τ Γ ⊢ fix x.M : ι[ξ] → τ

◮ Quite Powerful.

◮ Can type many forms of structural recursion.

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ.

◮ Typing Fixpoints.

Γ, x : ι[a] → τ ⊢ M : ι[a + 1] → τ Γ ⊢ fix x.M : ι[ξ] → τ

◮ Quite Powerful.

◮ Can type many forms of structural recursion.

◮ Termination.

◮ Proved by Reducibility. ◮ . . . but of an indexed form.

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ.

◮ Typing Fixpoints.

Γ, x : ι[a] → τ ⊢ M : ι[a + 1] → τ Γ ⊢ fix x.M : ι[ξ] → τ

◮ Quite Powerful.

◮ Can type many forms of structural recursion.

◮ Termination.

◮ Proved by Reducibility. ◮ . . . but of an indexed form.

◮ Reducibility sets are of the form Redθ τ. ◮ θ is an environment for index variables. ◮ Proof of reducibility for fix x.M is

rather delicate.

Deterministic Sized Types, Technically

◮ Types.

ξ ::= a | ω | ξ + 1; τ ::= ι[ξ] | τ → τ.

◮ Typing Fixpoints.

Γ, x : ι[a] → τ ⊢ M : ι[a + 1] → τ Γ ⊢ fix x.M : ι[ξ] → τ

◮ Quite Powerful.

◮ Can type many forms of structural recursion.

◮ Termination.

◮ Proved by Reducibility. ◮ . . . but of an indexed form.

◮ Type Inference.

◮ It is indeed decidable. ◮ But nontrivial.

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

Unbiased Random Walk

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

Unbiased Random Walk Biased Randomn Walk

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

◮ Non-Examples:

fix f.λx.if FairCoin then f(x − 1) else (f(x + 1); f(x + 1)); fix f.λx.if BiasedCoin then f(x + 1) else f(x − 1);

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

◮ Non-Examples:

fix f.λx.if FairCoin then f(x − 1) else (f(x + 1); f(x + 1)); fix f.λx.if BiasedCoin then f(x + 1) else f(x − 1);

Unbiased Random Walk, with two upward calls.

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

◮ Non-Examples:

fix f.λx.if FairCoin then f(x − 1) else (f(x + 1); f(x + 1)); fix f.λx.if BiasedCoin then f(x + 1) else f(x − 1);

Unbiased Random Walk, with two upward calls. Biased Random Walk, the “wrong” way.

Probabilistic Termination

◮ Examples:

fix f.λx.if x > 0 then if FairCoin then f(x − 1) else f(x + 1); fix f.λx.if x > 0 then if BiasedCoin then f(x − 1) else f(x + 1); fix f.λx.if BiasedCoin then f(x + 1) else x.

◮ Non-Examples:

fix f.λx.if FairCoin then f(x − 1) else (f(x + 1); f(x + 1)); fix f.λx.if BiasedCoin then f(x + 1) else f(x − 1);

◮ Probabilistic termination is thus:

◮ Sensitive to the actual distribution from which we sample. ◮ Sensitive to how many recursive calls we perform.

One-Counter Blind Markov Chains

◮ They are automata of the form (Q, δ) where

◮ Q is a finite set of states. ◮ δ : Q → Dist(Q × {−1, 0, 1}).

◮ They are a very special form of One-Counter Markov

Decision Processeses [BBEK2011].

◮ The model is fully probabilistic, there is no nondeterminism. ◮ The counter value is ignored.

One-Counter Blind Markov Chains

◮ They are automata of the form (Q, δ) where

◮ Q is a finite set of states. ◮ δ : Q → Dist(Q × {−1, 0, 1}).

◮ They are a very special form of One-Counter Markov

Decision Processeses [BBEK2011].

◮ The model is fully probabilistic, there is no nondeterminism. ◮ The counter value is ignored.

◮ The probability of reaching a configuration where the

counter is 0 can be approximated arbitrarily well in polynomial time.

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ Every higher-order variable occurs at most once.

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ

◮ Typing Fixpoints.

Γ | x : σ ⊢ V : ι[a + 1] → τ OCBMC(σ) terminates. Γ | x : σ ⊢ V : ι[ξ] → τ

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ

◮ Typing Fixpoints.

Γ | x : σ ⊢ V : ι[a + 1] → τ OCBMC(σ) terminates. Γ | x : σ ⊢ V : ι[ξ] → τ

This is sufficient for typing:

◮ Unbiased random walks; ◮ Biased random walks.

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ

◮ Typing Fixpoints.

Γ | x : σ ⊢ V : ι[a + 1] → τ OCBMC(σ) terminates. Γ | x : σ ⊢ V : ι[ξ] → τ

◮ Typing Probabilistic Choice

Γ | ∆ ⊢ M : τ Γ | Ω ⊢ N : ρ Γ | 1

2∆ + 1 2Ω ⊢ M ⊕ N : 1 2τ + 1 2ρ

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ

◮ Typing Fixpoints.

Γ | x : σ ⊢ V : ι[a + 1] → τ OCBMC(σ) terminates. Γ | x : σ ⊢ V : ι[ξ] → τ

◮ Typing Probabilistic Choice

Γ | ∆ ⊢ M : τ Γ | Ω ⊢ N : ρ Γ | 1

2∆ + 1 2Ω ⊢ M ⊕ N : 1 2τ + 1 2ρ

◮ Termination.

◮ By a quantitative nontrivial refinement of reducibility.

Probabilistic Sized Types [DLGrellois2017]

◮ Basic Idea: craft a sized-type system in such a way as to

mimick the recursive structure by a OCBMC.

◮ Judgments.

Γ | ∆ ⊢ M : µ

◮ Typing Fixpoints.

Γ | x : σ ⊢ V : ι[a + 1] → τ OCBMC(σ) terminates. Γ | x : σ ⊢ V : ι[ξ] → τ

◮ Typing Probabilistic Choice

Γ | ∆ ⊢ M : τ Γ | Ω ⊢ N : ρ Γ | 1

2∆ + 1 2Ω ⊢ M ⊕ N : 1 2τ + 1 2ρ

◮ Termination.

◮ By a quantitative nontrivial refinement of reducibility.

◮ Reducibility sets are now on the form Redθ,p τ ◮ p stands for the probability of being reducible. ◮ Reducibility sets are continuous:

Redθ,p

τ

=

• q<p

Redθ,q

τ

Deterministic Intersection Types

◮ Question: what are simple types missing as a way to

precisely capture termination?

Deterministic Intersection Types

◮ Question: what are simple types missing as a way to

precisely capture termination?

◮ Very simple examples of normalizing terms which canoot be

typed: ∆ = λx.xx ∆(λx.x).

Deterministic Intersection Types

◮ Question: what are simple types missing as a way to

precisely capture termination?

◮ Very simple examples of normalizing terms which canoot be

typed: ∆ = λx.xx ∆(λx.x).

◮ Types

τ ::= ⋆ | A → B A ::= {τ1, . . . , τn}

Deterministic Intersection Types

◮ Question: what are simple types missing as a way to

precisely capture termination?

◮ Very simple examples of normalizing terms which canoot be

typed: ∆ = λx.xx ∆(λx.x).

◮ Types

τ ::= ⋆ | A → B A ::= {τ1, . . . , τn}

◮ Typing Rules: Examples

{Γ ⊢ M : τi}1≤i≤n Γ ⊢ M : {τ1, . . . , τn} Γ ⊢ M : {A → B} Γ ⊢ N : A Γ ⊢ MN : B

Deterministic Intersection Types

◮ Question: what are simple types missing as a way to

precisely capture termination?

◮ Very simple examples of normalizing terms which canoot be

typed: ∆ = λx.xx ∆(λx.x).

◮ Types

τ ::= ⋆ | A → B A ::= {τ1, . . . , τn}

◮ Typing Rules: Examples

{Γ ⊢ M : τi}1≤i≤n Γ ⊢ M : {τ1, . . . , τn} Γ ⊢ M : {A → B} Γ ⊢ N : A Γ ⊢ MN : B

◮ Termination

◮ Again by reducibility.

Deterministic Intersection Types

◮ Question: what are simple types missing as a way to

precisely capture termination?

◮ Very simple examples of normalizing terms which canoot be

typed: ∆ = λx.xx ∆(λx.x).

◮ Types

τ ::= ⋆ | A → B A ::= {τ1, . . . , τn}

◮ Typing Rules: Examples

{Γ ⊢ M : τi}1≤i≤n Γ ⊢ M : {τ1, . . . , τn} Γ ⊢ M : {A → B} Γ ⊢ N : A Γ ⊢ MN : B

◮ Termination

◮ Again by reducibility.

◮ Completeness

◮ By subject expansion, the dual of subject reduction.

Oracle Intersection Types [BreuvartDL2017]

◮ Probabilistic choice can be seen as a form of read operation:

M ⊕ N = if BitInput then M else N

Oracle Intersection Types [BreuvartDL2017]

◮ Probabilistic choice can be seen as a form of read operation:

M ⊕ N = if BitInput then M else N

◮ Types

τ ::= ⋆ | A → s · B A ::= {τ1, . . . , τn} s ∈ {0, 1}∗

Oracle Intersection Types [BreuvartDL2017]

◮ Probabilistic choice can be seen as a form of read operation:

M ⊕ N = if BitInput then M else N

◮ Types

τ ::= ⋆ | A → s · B A ::= {τ1, . . . , τn} s ∈ {0, 1}∗

◮ Typing Rules: Examples

Γ ⊢ M : s · A Γ ⊢ M ⊕ N : 0s · A Γ ⊢ M : r · {A → s · B} Γ ⊢ N : q · A Γ ⊢ MN : (rqs) · B

Oracle Intersection Types [BreuvartDL2017]

◮ Probabilistic choice can be seen as a form of read operation:

M ⊕ N = if BitInput then M else N

◮ Types

τ ::= ⋆ | A → s · B A ::= {τ1, . . . , τn} s ∈ {0, 1}∗

◮ Typing Rules: Examples

Γ ⊢ M : s · A Γ ⊢ M ⊕ N : 0s · A Γ ⊢ M : r · {A → s · B} Γ ⊢ N : q · A Γ ⊢ MN : (rqs) · B

◮ Termination and Completeness

◮ Formulated in a rather unusual way. ◮ Proved as usual, but relative to a single probabilistic branch

Oracle Intersection Types [BreuvartDL2017]

◮ Probabilistic choice can be seen as a form of read operation:

M ⊕ N = if BitInput then M else N

◮ Types

τ ::= ⋆ | A → s · B A ::= {τ1, . . . , τn} s ∈ {0, 1}∗

◮ Typing Rules: Examples

Γ ⊢ M : s · A Γ ⊢ M ⊕ N : 0s · A Γ ⊢ M : r · {A → s · B} Γ ⊢ N : q · A Γ ⊢ MN : (rqs) · B

◮ Termination and Completeness

◮ Formulated in a rather unusual way. ◮ Proved as usual, but relative to a single probabilistic branch

P(M ↓) =

• ⊢M:s·⋆

2|s|

Oracle Intersection Types [BreuvartDL2017]

◮ Probabilistic choice can be seen as a form of read operation:

M ⊕ N = if BitInput then M else N

◮ Types

τ ::= ⋆ | A → s · B A ::= {τ1, . . . , τn} s ∈ {0, 1}∗

◮ Typing Rules: Examples

Γ ⊢ M : s · A Γ ⊢ M ⊕ N : 0s · A Γ ⊢ M : r · {A → s · B} Γ ⊢ N : q · A Γ ⊢ MN : (rqs) · B

◮ Termination and Completeness

◮ Formulated in a rather unusual way. ◮ Proved as usual, but relative to a single probabilistic branch

P(M ↓) =

• ⊢M:s·⋆

2|s| This is unavoidable, due to recursion theory.

Intersection Types and Computations

M V

Intersection Types and Computations

M V

Intersection Types

Intersection Types and Computations

M V M W V . . . . . .

Intersection Types and Computations

M V M W V . . . . . .

Oracle Intersection Types

Intersection Types and Computations

M V M W V . . . . . . M W V . . . . . .

Intersection Types and Computations

M V M W V . . . . . . M W V . . . . . .

Monadic Intersection Types [BDL2017]

◮ They are a combination of oracle and

sized types.

◮ Intersections are needed for preciseness. ◮ Distributions of types allow to analyse

more than one probabilistic branch in the same type derivation.

These Slides, and More...

These Slides, and More...

Questions?