Introduction to Probabilistic and Quantum Programming ...dallago/BISS2014/slides-III.pdf ·...

Introduction to Probabilisticand Quantum Programming

Part III

Ugo Dal Lago

BISS 2014, Bertinoro

Section 1

The λ-Calculus as a Functional Language

Minimal Syntax and Dynamics

I Terms:M ::= x | MN | λx.M.

I Substitution:

xx/M =M yx/M = y

NLx/M = (Nx/M)(Lx/M) λy.Nx/M = λy.(Nx/M)

I CBN Operational Semantics:

(λx.M)N →Mx/NM → N

ML→ NL

I Values:V ::= λx.M

I CBV Operational Semantics:

(λx.M)V →Mx/V M → N

ML→ NLM → N

VM → V N

I Substitution:

xx/M =M yx/M = y

ML→ NL

ML→ NLM → N

VM → V N

I Substitution:

xx/M =M yx/M = y

ML→ NL

ML→ NLM → N

VM → V N

I Substitution:

xx/M =M yx/M = y

ML→ NL

ML→ NLM → N

VM → V N

I Substitution:

xx/M =M yx/M = y

ML→ NL

ML→ NLM → N

VM → V N

ObservationsI Not all redexes are reduced.I For every M there is at most one N such that M → N .I It is easy to find a term M such that

M = N0 → N1 → N2 → . . .

I There is a problem with substitutions:

(λx.λy.x)(yz)→ λy.yz.(λx.λy.x)(yz)→ λw.yz.

I The standard solution is to consider terms modulo so-calledα-equivalence: renaming bound variables turns a terminto one which is indistinguishable.

λx.λy.xxy ≡ λz.λy.zzy.

I α-equivalence is not necessary if we assume to work withclosed terms.

M = N0 → N1 → N2 → . . .

Computing Simple Function on N, in CBV

I Natural Numbers

p0q = λx.λy.x;

pn+ 1q = λx.λy.ypnq.

I Finite Set A = a1, . . . , an

paiq = λx1 · · ·λxn.xi.

I Pairs〈M,N〉 = λx.xMN

I Fixed-Point Combinator

Θ = λx.λy.y(λz.(xx)yz);

H = ΘΘ.

Computing Simple Functions on N, in CBV

I SuccessorSUCC = λz.λx.λy.yz

Indeed:SUCCpnq→ λx.λy.ypnq

I Projections

PROJki = λx.x(λy1 · λyk.yi)

Indeed:

PROJki 〈pn1q, . . . , pnkq〉 → 〈pn1q, . . . , pnkq〉(λy1 · · ·λyk.yi)→ (λy1 · · ·λyk.yi)pn1q · · · pnkq→∗ pniq.

I Projections

Indeed:

I Projections

Indeed:

I Projections

Indeed:

A Simple Form of Recursion

I Suppose you have a term Mf which computes the functionf : N→ N, and you want to iterate over it, i.e. you want tocompute g : N× N→ N such that

g(0, n) = n;

g(m+ 1, n) = f(g(m,n)).

I We need a form of recursion in the language. Observe that:

HV = (ΘΘ)V → (λy.y(λz.Hyz))V → V (λz.HV z)

I The function g can be computed by way of

Mg = H(λx.λy.y(λz.λw.zw(λq.Mf (x〈q, w〉)))

g(0, n) = n;

g(m+ 1, n) = f(g(m,n)).

g(0, n) = n;

g(m+ 1, n) = f(g(m,n)).

Mg〈pm+ 1q, pnq〉 →∗ 〈pm+ 1q, pnq〉(λz.λw.zw(λq.Mf ((λx.Mgx)〈q, w〉x))→ (λz.λw.zw(λq.Mf ((λx.Mgx)〈q, pnq〉))pm+ 1qpnq

→∗ pm+ 1qpnq(λq.Mf ((λx.Mgx)〈q, pnq〉))→∗ (λq.Mf ((λx.Mgx)〈q, pnq〉))pmq

→∗ Mf ((λx.Mgx)〈pmq, pnq〉)→∗ Mf (Mg〈pmq, pnq〉)

Universality

I One could go on, and show that the following schemes canall be encoded:

I Composition;I Primitive Recursion;I Minimization.

I This implies that all partial recursive functions can becomputed in the λ-calculus.

TheoremThe λ-calculus is Turing-complete, both when CBV and CBNare considered.

Why is This a Faithful Model of FunctionalComputation?

I Data, conditionals, basic functions, and recursion can all beencoded into the λ-calculus. We can then give programs“informally”, in our favourite functional language.

I Take, as an example, the usual recursive programcomputing the factorial of a natural numberlet rec fact x = if (x==0) then 1 else x*(fact x-1)

I This can indeed be turned into a λ-term (where M∗ andM−1 are terms for multiplication and the predecessor,respectively):

let rec fact x = if (x==0) then 1 else x*(fact x-1)

H(λfact.λx.if (x==0) then 1 else x*(fact x-1))

H(λfact.λx.x p1q (x*(fact x-1)))

H(λfact.λx.x p1q (M∗ x (fact x-1)))

H(λfact.λx.x p1q (M∗ x (fact (M−1x))))

More About SemanticsI The semantics we have adopted so far is said to besmall-step, since it derives assertions in the form M → N ,each corresponding to one atomic computation step.

I There is an equivalent way of formulating the sameconcept, big-step semantics, in which we “jump directlyto the result”:

V ⇓ VM ⇓ λx.N Nx/L ⇓ V

ML ⇓ V

V ⇓ VM ⇓ λx.N L ⇓ V Nx/V ⇓W

ML ⇓W

TheoremIn both CBN and CBV, M →∗ V if and only if M ⇓ V .

I If M ⇓ V for some V , then we write M ⇓, otherwise M ⇑

ML ⇓ V

ML ⇓W

ML ⇓ V

ML ⇓W

Program Equivalence

I A fundamental question in programming language theory isthe following: when can two programs be consideredequivalent?

I A satisfactory answer to this question would enable:I To validate program optimizations; you could be sure

that the transformations you are performing turn programsinto equivalent ones.

I To perform program verification; just compare theprogram at hand with another (abstract) programcorresponding to the underlying specification.

I But what is the right notion of program equivalence?I M ≡ N iff M = N ; trivial!I M ≡ N iff either M ⇓ V and N ⇓ V for some V or M ⇑ andN ⇑; does not equate values which behave the same.

I M ≡ N iff M and N “have the same semantics”; nice, butwe do not have time to talk about program’s semantics.

Program Equivalence

Context Equivalence

I What we need is a notion of equivalence which equatesterms with the same behaviour.

I More specifically, we would like to equate those termswhich have the same observable behavior in every context.

I Observable Behaviour. Two terms M and N have thesame observable behaviour whenver M ⇓ iff N ⇓ .

I Contexts. They are defined as terms, but they contain aunique placeholder [·]:

C ::= [·] | CM | MC | λx.C.

The term C[M ] is the one obtained from C by substituting[·] with M .

I We can then stipulate that M ≡ N iff for every C, C[M ] ⇓iff C[N ] ⇓.

Context Equivalence

C ::= [·] | CM | MC | λx.C.

Context Equivalence

C ::= [·] | CM | MC | λx.C.

Context Equivalence

I Examples:λx.x ≡ λx.(λy.y)x

(λx.xx)(λx.xx) = Ω 6≡ λx.Ω

I Exercise: is it true that for every closed M , M ≡ λx.Mx?I Proving that two terms are not context equivalent seems to

be easier than proving that they are.I We need some new techniques to overcome this problem.

Context Equivalence

Applicative Bisimulation

Terms Values

M Veval

Terms Values

M Veval

Terms Values

M Veval

λx.NNx/W W

I Simulation.I If M R N and M ⇓ λx.L, then N ⇓ λx.P and for every

argument Q, (LQ/x) R (PQ/x).I Bisimulation.

I If M R N then:I If M ⇓ λx.L, then N ⇓ λx.P and for every argument Q,

(LQ/x) R (PQ/x).I If N ⇓ λx.L, then M ⇓ λx.P and for every argument Q,

(LQ/x) R (PQ/x).I Similarity: union of all simulations, denoted -;I Bisimilarity: union of all bisimulation, denoted ∼.

Theorem (Full Abstraction)M ≡ N iff M ∼ N .

I Simulation.I If M R N and M ⇓ λx.L, then N ⇓ λx.P and for every

argument Q, (LQ/x) R (PQ/x).I Bisimulation.

I If M R N then:I If M ⇓ λx.L, then N ⇓ λx.P and for every argument Q,

(LQ/x) R (PQ/x).I If N ⇓ λx.L, then M ⇓ λx.P and for every argument Q,

(LQ/x) R (PQ/x).I Similarity: union of all simulations, denoted -;I Bisimilarity: union of all bisimulation, denoted ∼.

Theorem (Full Abstraction)M ≡ N iff M ∼ N .

I Suppose we want to prove that, indeed,

M = λx.x ≡ λx.(λy.y)x = N.

I In principle, we should consider all possible contexts.I But we can also exploit full-abstraction, and just exhibitone applicative bisimulation containing (M,N).

I Here it is:

R = (M,N)∪(L, (λx.x)L) | L is a closed term∪((λx.x)L,L) | L is a closed term∪(L,L) | L is a closed term

I Exercise: prove that R is indeed an applicativebisimulation.

I Here it is:

Type Systems

I We haven’t considered any form of static type for ourprograms, so far.

I Some concrete programming languages are designed as todo type-checking only at runtime, i.e., they do not haveany static type system.

I Scheme, Python, etc.I Actually, endowing the λ-calculus with a type system is

relatively simple.I Types: A,B ::= bool | nat | A→ B | A×B | · · ·I Judgments: x1 : A1, . . . , xn : An `M : B.I Rules (Examples):

Γ `M : A→ B Γ ` N : AΓ `MN : B

Γ, x : A `M : B

Γ ` λx.M : A→ B

Γ, x : A ` x : A Γ ` H : (A→ A)→ A

Type Systems

Γ `M : A→ B Γ ` N : AΓ `MN : B

Γ, x : A `M : B

Γ ` λx.M : A→ B

Γ, x : A ` x : A Γ ` H : (A→ A)→ A

Type Systems

Γ `M : A→ B Γ ` N : AΓ `MN : B

Γ, x : A `M : B

Γ ` λx.M : A→ B

Γ, x : A ` x : A Γ ` H : (A→ A)→ A

Type Systems

Γ `M : A→ B Γ ` N : AΓ `MN : B

Γ, x : A `M : B

Γ ` λx.M : A→ B

Γ, x : A ` x : A Γ ` H : (A→ A)→ A

Type Systems

Γ `M : A→ B Γ ` N : AΓ `MN : B

Γ, x : A `M : B

Γ ` λx.M : A→ B

Γ, x : A ` x : A Γ ` H : (A→ A)→ A

Section 2

Probabilistic λ-Calculi

Probabilistic λ-Calculus: Syntax and OperationalSemantics

I Terms: M,N ::= x | λx.M | MN | M ⊕N ;I Values: V ::= λx.M ;I How should we define the semantics of a probabilistic termM?

I Informally, M ⊕N rewrites to M or to N with probability12 : this is just like flippling a coin and choosing one of thetwo terms according to the result.

I A value distribution is a function D from the set V ofvalues to R such ∑

V ∈VD(V ) ≤ 1.

I The fact the sum can be strictly smaller than 1 is a wayto reflect (the probability of) divergence.

I The empty value distribution is denoted as ∅.I Useful notation for finite distributions: V p11 , . . . , V pnn .I Value distributions can be ordere, pointwise.I S(D) is the support of D , i.e. the set of values to which D

assigns nonnull probability.

V ∈VD(V ) ≤ 1.

I Approximation (Big-Step) Semantics, CBN:

M ⇓ ∅ V ⇓ V 1M ⇓ D N ⇓ E

M ⊕N ⇓ 12D + 1

M ⇓ D Px/N ⇓ Eλx.P λx.P∈S(D)

MN ⇓∑

V ∈S(F)

D(V )Eλx.P

I Example: consider M = I ⊕ ((II)⊕ Ω), where I = λx.xand Ω = (λx.xx)(λx.xx).

M ⇓ ∅ M ⇓ I12 M ⇓ I

I Semantics: [[M ]] = supM⇓D D ;I Variations: Small-Step Semantics, CBV.

M ⇓ ∅ V ⇓ V 1M ⇓ D N ⇓ E

M ⊕N ⇓ 12D + 1

MN ⇓∑

V ∈S(F)

D(V )Eλx.P

M ⇓ ∅ M ⇓ I12 M ⇓ I

M ⇓ ∅ V ⇓ V 1M ⇓ D N ⇓ E

M ⊕N ⇓ 12D + 1

MN ⇓∑

V ∈S(F)

D(V )Eλx.P

M ⇓ ∅ M ⇓ I12 M ⇓ I

M ⇓ ∅ V ⇓ V 1M ⇓ D N ⇓ E

M ⊕N ⇓ 12D + 1

MN ⇓∑

V ∈S(F)

D(V )Eλx.P

M ⇓ ∅ M ⇓ I12 M ⇓ I

M ⇓ ∅ V ⇓ V 1M ⇓ D N ⇓ E

M ⊕N ⇓ 12D + 1

MN ⇓∑

V ∈S(F)

D(V )Eλx.P

M ⇓ ∅ M ⇓ I12 M ⇓ I

M ⇓ ∅ V ⇓ V 1M ⇓ D N ⇓ E

M ⊕N ⇓ 12D + 1

MN ⇓∑

V ∈S(F)

D(V )Eλx.P

M ⇓ ∅ M ⇓ I12 M ⇓ I

I As an example, consider the following recursive program,and call it M :let rec fancy x = x (+) (fancy x+1)

I One can easily check that:

Mp0q ⇓ ∅ Mp0q ⇓ p0q12 Mp0q ⇓ p0q

12 , p1q

14 · · ·

I As a consequence:

[[Mp0q]] = supMp0q⇓D

D = p0q12 , p1q

14 , p2q

18 , . . .

I As an exercise, find a term N such that[[N ]] = p1q

12 , p4q

14 , p9q

18 , . . .

12 , p1q

14 · · ·

I As a consequence:

D = p0q12 , p1q

14 , p2q

18 , . . .

12 , p4q

14 , p9q

18 , . . .

12 , p1q

14 · · ·

I As a consequence:

D = p0q12 , p1q

14 , p2q

18 , . . .

12 , p4q

14 , p9q

18 , . . .

12 , p1q

14 · · ·

I As a consequence:

D = p0q12 , p1q

14 , p2q

18 , . . .

12 , p4q

14 , p9q

18 , . . .

12 , p1q

14 · · ·

I As a consequence:

D = p0q12 , p1q

14 , p2q

18 , . . .

12 , p4q

14 , p9q

18 , . . .

12 , p1q

14 · · ·

I As a consequence:

D = p0q12 , p1q

14 , p2q

18 , . . .

12 , p4q

14 , p9q

18 , . . .

Contextual Equivalence in a Probabilistic Setting

I In analogy to the deterministic setting, we observeconvergence, which now becomes quantitative.

I The observable behaviour of M is given by∑[[M ]] =

∑V ∈V [[M ]](V ).

I But∑

[[M ]] is nothing more than the probability ofconvergence for the term M .

I Contexts are extended to reflect the presence ofprobabilistic choice

C ::= [·] | CM | MC | λx.C | C ⊕M | M ⊕ C.I Analogously to the deterministic case, M ≡ N iff for everyC, the following equality holds:∑

[[C[M ]]] =∑

[[C[N ]]].

Motivating Example: Perfect Security

I We want to formalize the notion of perfect security for anencryption scheme.

I TakeM and C as the random variables modelingmessages and cryptograms, respectively.

I Of course, the two variables are not independent, at least inprinciple.

I If we want the underlying scheme to be secure, we need tobe sure that they are as independent as possible.

Pr(M = m | C = c) = Pr(M = m).

Pr(C = c | M = m) = Pr(C = c).