Operational Semantics

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Operational Semantics

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Dening an Operational Semantics

Granurality

Order of evaluation

Delia Kesner (Université Paris Diderot) Typage 2018 2 / 34

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Big-step Semantics

Each rule completely evaluates the expression to avalue.

hn, σi ⇓ n hX, σi ⇓σ(X) ha₁, σi ⇓ n₁ ha₂, σi ⇓n₂ n is n₁ plus n₂

ha₁+a₂, σi ⇓ n

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Properties

Abstract

Allows to avoid details

No specication of evaluation order (e.g.(1+3) + (5−3)) No specication of control of errors

No specication of interleaving

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Small-step Semantics

Evaluation is given by a sequence of state changes of an abstract machine which terminates when the state cannot be reduced further.

ha1, σi ha⁰₁, σ⁰i ha1+a2, σi ha⁰₁+a2, σ⁰i

ha2, σi ha⁰₂, σ⁰i hn1+a2, σi hn1+a₂⁰, σ⁰i

hX, σi hσ(X), σi

n is n₁ plus n₂ hn₁+n₂, σi hn, σi

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Properties

Less abstract

Specication of order of evaluation Control of errors : n₂ 6=0

n₁/n₂ n , where n is n₁ divided by n₂. Interleaving : hc1, σi hc₁⁰, σ⁰i

hc1||c2, σi hc₁⁰||c2, σ⁰i

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From Small-step to Multi-step Semantics

The multi-step semantics is given by the relation t ^∗ t⁰ which is the reexive and transitive closure of t t⁰.

(P1) t ^∗ t for every t (P2) t t⁰ implies t ^∗ t⁰

(P3) t ^∗ t⁰ and t⁰ ^∗t⁰⁰ implies t ^∗ t⁰⁰

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Normal Forms

Anormal formis a term that cannot be evaluated any further : is a state where the abstract machine is halted (result of the evaluation).

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Properties of the small and big step semantics

The relation is deterministic.

The relation⇓ is deterministic.

t ⇓v i t ^∗ v, where v is a value.

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Big-step versus small-step semantics

In small-step semantics evaluation stops at errors. In big-step semantics errors occur deeply inside derivation trees.

The order of evaluation is explicit in small-step semantics but implicit in big-step semantics.

Big-step semantics is more abstract, but less precise.

Small-step semantics allows to make dierence between non-termination and "getting stuck".

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A functional langage

M,N ::= x (variable) |

ct (constant) |

hM,Ni (pair) | M N (application) | λx.M (abstraction) | let x =M in N (let)

Some constant function symbols : fst, snd, x, ifthenelse,+,∗. . . Some constants : true, false, 0, 1, 2, 3, . . .

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Notations

M₁M₂. . .M_n ≡ (. . .((M₁M₂)M₃). . .M_n−1)M_n N M~ ≡ (. . .(((N M₁)M₂)M₃). . .M_n−1)M_n

M+N ≡ +hM,Ni

if E then M else N ≡ ifthenelsehE,hM,Nii

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Free variables

FV(x) = {x}

FV(ct) = ∅

FV(hM,Ni) = FV(M)∪FV(N) FV(M N) = FV(M)∪FV(N) FV(λx.M) = FV(M)\ {x}

FV(let x =M in N) = FV(M)∪FV(N)\ {x}

A term M isclosedi it has no free variable, i.e. FV(M) =∅. For example, λz.((λx.x z)(λy.y))is closed but (λx.x z)(λy.y) is not.

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Bound variables

BV(x) = ∅

BV(ct) = ∅

BV(hM,Ni) = BV(M)∪BV(N) BV(M N) = BV(M)∪BV(N) BV(λx.M) = BV(M)∪ {x}

BV(let x =M in N) = BV(M)∪BV(N)∪ {x} A variable may be free and bound : x (λx.x).

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Alpha-conversion

Alpha-conversion is the operation which consists in renaming some bound variables.

Thus for example x (λx.x y) =αx (λz.z y)and let x =x⁰ in x y =_α let z=x⁰ in z y.

Théorème : For every term t there is a term t⁰ such that

1 t =α t⁰

2 Barendregt's Convention :

I FV(t⁰)∩BV(t⁰) =∅.

I All the bound variables of t⁰ are distinct.

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Substitution

The application of a substitution σ={x₁/t₁, . . . ,x_n/t_n} to a term M is dened by induction as follows :

σx_i = t_i If i∈ {1, . . . ,n}

σy = y If y ∈ {/ x₁, . . . ,x_n}

σct = ct

σhM,Ni = hσM, σNi

σ(M N) = σM σN

σ(λx.M) = λx.σM If no capture of variables σ(let x =M in N) = let x =σM inσN If no capture of variables

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Reduction Rules

(λx.M)N → M{x/N} let x =N in M → M{x/N}

x M → M (x M)

fsthM,Ni → M

sndhM,Ni → N

if true then M else N → M if false then M else N → N if 0 then M else N → M

if n then M else N → N, n 6=0

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Call-by-value lambda-calculus (big-step semantics)

(Values) V ::=ct | hV,Vi |λx.M |x M

Meaningless expressions such as (h1,1i 3) or(true 3) are not considered as values.

V is a value V ⇓_v V

M1 ⇓_v V1 M2 ⇓_v V2

hM1,M2i ⇓_v hV1,V2i M ⇓_v λx.L N ⇓_v W L{x/W} ⇓_v V

M N ⇓_v V

N ⇓_v V L{x/V} ⇓_v W let x =N in L⇓_v W

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M ⇓_v x L N ⇓_v W (L(x L))W ⇓_v V M N ⇓_v V

M ⇓_v fst N ⇓_v hV1,V2i M N ⇓_v V1

M ⇓_v snd N ⇓_v hV1,V2i M N ⇓_v V2

M ⇓_v true N ⇓_v V if M then N else L⇓_v V

M ⇓_v false L⇓_v V if M then N else L⇓_v V M ⇓_v 0 N⇓_v V

if M then N else L⇓_v V

M ⇓_v n n 6=0 L⇓_v V if M then N else L⇓_v V

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Particular case : closed pure lambda-terms

(Values) V ::=λx.M

V ⇓_v V

M ⇓_v λx.L N ⇓_v W L{x/W} ⇓_v V M N ⇓_v V

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An example

M =λf.λx.hx,f xi and N =λy.y.

M N⇓_v λx.hx,N xi 1⇓_v 1 h1,N 1i⇓_v h1,1i M N 1⇓_vh1,1i

M ⇓_v M N ⇓_v N λx.hx,f xi{f/N} ⇓_v λx.hx,N xi M N⇓_v λx.hx,N xi

1⇓_v 1 N ⇓_v N 1⇓_v 1 y{y/1} ⇓_v 1 N 1⇓_v 1

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Call-by-value lambda calculus (small-step semantics)

M v M⁰ M N v M⁰ N

N v N⁰ V N v V N⁰

(λx.M)V _v M{x/V} (x M)V _v (M (x M))V N v N⁰

let x =N in L v let x =N⁰ in L let x =V in L v L{x/V} M _v M⁰

hM,Ni _v hM⁰,Ni

N _v N⁰ hV,Ni _v hV,N⁰i

fst hV1,V2i _v V1 snd hV1,V2i _v V2

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M _v M⁰

if M then N else L _v if M⁰then N else L

if true then N else L v N if false then N else L v L

if 0 then N else L _v N

n6=0

if n then N else L _v L

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The same example

M =λf.λx.hx,f xi and N =λy.y.

M N 1 _v

(λx.hx,N xi) 1 _v h1,N 1i _v h1,1i

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Call-by-name lambda-calculus (big-step semantics)

(Lazy Forms)P ::=ct | hM,Ni |λx.M |x M M ⇓_nλx.L L{x/N} ⇓_nP

M N ⇓_nP

P is a lazy form P ⇓_n P L{x/N} ⇓_nP

let x =N in L⇓_nP

M ⇓_nx L (L(x L))N ⇓_nP M N ⇓_nP

M⇓_n hM₁,M₂i M₁ ⇓_nP₁ fst M ⇓_nP₁

M ⇓_nhM₁,M₂i M₂⇓_nP₂ snd M ⇓_nP₂

M ⇓_ntrue N ⇓_nP M ⇓_nfalse L⇓_nP

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Particular case : closed pure lambda-terms

(Lazy Forms)P ::=λx.M

P ⇓_n P

M ⇓_nλx.L L{x/N} ⇓_nP M N ⇓_nP

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An example

Let M =λf.λx.hx,(f x)i

x M ⇓_nx M M (x M)1⇓_nh1,x M 1i x M 1⇓_nh1,x M 1i

Let M_f =x M.

M ⇓_n M (λx.hx,f xi){f/Mf} ⇓_nλx.hx,Mf xi

M Mf ⇓_nλx.hx,Mf xi hx,Mf xi{x/1} ⇓_nh1,Mf 1i M(M_f)1⇓_nh1,M_f 1i

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Exercice

Try to compute x M 1⇓_v?

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Call-by-name lambda calculus (small-step semantics)

M n M⁰ M N n M⁰ N

(λx.M)N _nM{x/N} (x M)N _n(M (x M))N

let x =M in L nL{x/M} M _nM⁰

fst M _nfst M⁰ fst hM,Ni _nM

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M _n M⁰

if M then N else L _n if M⁰ then N else L

if true then N else L nN if false then N else L nL

if 0 then N else L _nN

n 6=0

if n then N else L _nL

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The same example

M =λf.λx.hx,(f x)i.

x M 1 _n

M (x M)1 _n

(λx.hx,(x M x)i) 1 n

h1,(x M 1)i

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Coherence of results

If M ⇓_v N, then N is a value.

If M ⇓_nN, then N is a lazy form.

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Deterministic properties

If M ⇓_v V and M ⇓_v V⁰, then V =V⁰. If M ⇓_nP and M ⇓_nP⁰, then P=P⁰. If M v N and M v N⁰, then N =N⁰. If M nN and M nN⁰, then N=N⁰.

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Relating big and small-steps semantics

If M ⇓_v V , then M ^∗_v V . If M ⇓_nP, then M ^∗_nP.

If M ^∗_v N and N is a value, then M⇓_v N.

If M ^∗_nN and N is a lazy form, then M ⇓_nN.