Operational Semantics

Operational Semantics

Dening an Operational Semantics Granurality

Order of evaluation

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Big-step Semantics

Each rulecompletely 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

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

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+a⁰₂, 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 byn2. Interleaving : hc1, σi;hc⁰₁, σ⁰i

hc1||c2, σi;hc⁰₁||c2, σ⁰i

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From Small-step to Multi-step Semantics

The multi-step semantics is given by the relationt;^∗t⁰ which is the reexive and transitive closure oft;t⁰.

(P1) t;^∗t for everyt (P2) t;t⁰ impliest;^∗t⁰

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

Normal Forms

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

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) | letx=M inN (let)

Some constant function symbols : fst,snd, fix, ifthenelse,+,∗ . . .

Some constants :true, false,0,1,2, 3, . . .

Notations

M1M2. . . Mn ≡ (. . .((M1M2)M3). . . Mn−1)Mn

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

M +N ≡ +hM, Ni

if EthenM elseN ≡ ifthenelsehE,hM, Nii

Free variables

F V(x) = {x}

F V(ct) = ∅

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

F V(letx=M inN) = F V(M)∪F V(N)\ {x}

A term M is closedi it has no free variable, i.e.F V(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(letx=M inN) = 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) =α x(λz.z y)and letx=x⁰inx y=αletz=x⁰inz y.

Théorème : For every term tthere is a termt⁰ such that 1. t=α t⁰

2. Barendregt's Convention : F V(t⁰)∩BV(t⁰) =∅.

All the bound variables oft⁰ are distinct.

Substitution

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

σxi = ti Ifi∈ {1, . . . , n}

σy = y Ify /∈ {x1, . . . , xn}

σct = ct

σhM, Ni = hσM, σNi

σ(M N) = σM σN

σ(λx.M) = λx.σM If no capture of variables σ(letx=M inN) = letx=σM inσN If no capture of variables

Règles de réduction

(λx.M)N → M{x/N}

letx=N inM → M{x/N}

fix M → M (fix M)

fsthM, Ni → M

sndhM, Ni → N

if truethenM elseN → M if falsethenM elseN → N if 0thenM elseN → M

if nthenM elseN → N, n6= 0

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

(Values)V ::=ct| hV, Vi |λx.M |fix M Meaningless expressions such as (h1,1i3) or(true3) arenot 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

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N ⇓v V L{x/V} ⇓v W letx=N inL⇓v W

M ⇓v fix L N ⇓v W (L(fix 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 thenN elseL⇓v V

M ⇓v false L⇓v V if M thenN elseL⇓v V

M ⇓v 0 N ⇓v V if M thenN elseL⇓v V

M ⇓v n n6= 0 L⇓v V if M thenN elseL⇓v V

Particular case : closed pure terms

(Values)V ::=λx.M |fix M

V ⇓v V

M ⇓v λx.L N ⇓v W L{x/W} ⇓v V M N ⇓v V

N ⇓v V L{x/V} ⇓v W letx=N inL⇓v W

M ⇓v fix L (L(fix L))N ⇓v V M N ⇓v V

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An example

Let M =λf.λx.hx, f xiand 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} ⇓vλx.hx, N xi M N ⇓v λx.hx, N xi

1⇓v 1 N ⇓vN 1⇓v 1 y{y/1} ⇓v1 N 1⇓v1

h1, N 1i ⇓v h1,1i

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Call-by-value lambda calculus (small-step semantics) M ;v M⁰

M N ;v M⁰N

N ;v N⁰ V N ;v V N⁰

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

letx=N inL;v letx=N⁰inL letx=V inL;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

M ;v M⁰

if M thenN elseL;v if M⁰ thenN elseL

if truethenN elseL;v N if falsethenN elseL;v L

if 0thenN elseL;v N

n6= 0

if nthenN elseL;v L

The same example

LetM =λf.λx.hx, f xiand 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 |fix M

M ⇓nλx.L L{x/N} ⇓n P M N ⇓nP

P is a lazy form P ⇓n P L{x/N} ⇓n P

letx=N inL⇓n P

M ⇓n fix L (L(fix L))N ⇓n P M N ⇓nP

M ⇓n hM1, M2i M1 ⇓nP1

fst M ⇓n P1

M ⇓nhM1, M2i M2⇓nP2

snd M ⇓n P2

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M ⇓n true N ⇓n P if M thenN elseL⇓nP

M ⇓nfalse L⇓n P if M thenN elseL⇓nP M ⇓n0 N ⇓n P

if M thenN elseL⇓n P

M ⇓n n n6= 0 L⇓nP if M thenN elseL⇓nP

Particular case : closed pure terms

(Lazy Forms)P ::=λx.M |fix M

M ⇓n λx.L L{x/N} ⇓n P

M N ⇓n P P ⇓n P L{x/N} ⇓n P

letx=N inL⇓n P

M ⇓n fix L (L(fix L))N ⇓n P M N ⇓nP

An example

LetM =λf.λx.hx,(f x)i

fix M ⇓nfix M M (fix M) 1⇓nh1, fix M 1i fix M 1⇓nh1, fix M 1i

LetMf =fix M.

M ⇓n M (λx.hx, f xi){f/M_f} ⇓nλx.hx, M_f xi

M Mf⇓nλx.hx, Mf xi hx, Mfxi{x/1} ⇓nh1, Mf 1i M(Mf) 1⇓nh1, Mf 1i

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Exercice

Try to compute fix M 1⇓v?

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Call-by-name lambda calculus (small-step semantics) M ;nM⁰

M N ;nM⁰N

(λx.M)N ;nM{x/N} (fix M)N ;n(M (fix M))N

letx=M inL;n L{x/M}

M ;n M⁰

fst M ;n fst M⁰ fsthM, Ni;nM

M ;nM⁰

snd M ;nsnd M⁰ snd hM, Ni;nN M ;n M⁰

if M thenN elseL;n if M⁰thenN elseL

if truethenN elseL;n N if falsethenN elseL;n L

if 0thenN elseL;n N

n6= 0

if nthenN elseL;nL

The same example LetM =λf.λx.hx,(f x)i

fix M 1 ;n

M (fix M) 1 ;n

(λx.hx,(fix M x)i) 1 ;n

h1,(fix M 1)i

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Coherence of results

If M ⇓v N, then N is a value.

If M ⇓n N, then N is a lazy form.

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Deterministic properties

IfM ⇓v V and M ⇓v V⁰, then V =V⁰. IfM ⇓nP and M ⇓n P⁰, then P =P⁰. IfM ;v N and M ;v N⁰, then N =N⁰. IfM ;n N and M ;nN⁰, then N =N⁰.

Relating big and small-steps semantics If M ⇓v V, then M ;^∗_v V.

If M ⇓n P, 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 ⇓n N.