Flat Base-Domains

Aktie "Flat Base-Domains"

(1)

Flat Base-Domains

(ZZ ^⊥ , IB ^⊥ , C ^⊥ , F ^⊥ )

d ⊑ D d ^′ iff d = ⊥ D or d = d ^′

❅❅

❅❅❅

❏❏

❏❏❏

❇❇

❇❇❇

✂✂

✂✂✂

✡✡

✡

. . . -1 0 1 2 3 . . .

⊥

Product Domains

(ZZ ^⊥ × ZZ ^⊥ , etc.)

( d 1 , . . . , d ⁿ ) ⊑ D ( d ^′ ₁ , . . . d ^′ _n ) iff d ⁱ ⊑ D _i d ^′ _i for all 1 ≤ i ≤ n

✟✟

.

✟

❍❍❍❍

❍❍❍

✟✟

✟

❍❍❍❍

❍❍❍

✟✟

✟

❍❍❍❍

❍❍❍

❅❅

PPPPPP

PPPPP

✏✏

✏

(0, ⊥ ) ( ⊥ ,0) ( ⊥ ,1) (1, ⊥ )

(0,0) (0,1) (1,0) (1,1)

( ⊥ , ⊥ )

Function Domains

(subset of ZZ ^⊥ → ZZ ^⊥ , etc.)

f ⊑ D ₁ →D ₂ g iff f (d) ⊑ D ₂ g(d) for all d ∈ D ¹ .

29

(2)

Flat Base-Domains

Flat Base-Domains

(ZZ ^⊥ , IB ^⊥ , C ^⊥ , F ^⊥ )

d ⊑ D d ^′ iff d = ⊥ D or d = d ^′

. . . -1 0 1 2 3 . . .

⊥

Product Domains

(ZZ ^⊥ × ZZ ^⊥ , etc.)

( d 1 , . . . , d ⁿ ) ⊑ D ( d ^′ ₁ , . . . d ^′ _n ) iff d ⁱ ⊑ D _i d ^′ _i for all 1 ≤ i ≤ n

.

(0, ⊥ ) ( ⊥ ,0) ( ⊥ ,1) (1, ⊥ )

(0,0) (0,1) (1,0) (1,1)

( ⊥ , ⊥ )

Function Domains

(subset of ZZ ^⊥ → ZZ ^⊥ , etc.)

f ⊑ D ₁ →D ₂ g iff f (d) ⊑ D ₂ g(d) for all d ∈ D ¹ .

29

Strict and Non-Strict Functions

f : D 1 × . . . × D ⁿ → D is strict iff

f (d ¹ , . . . , d ⁿ ) = ⊥ D whenever d ⁱ = ⊥ D _i for some i.

one :: Int -> Int one x = 1

Choices for the semantics of one:

(a) f ¹ (x) = 1 for x ∈ ZZ f ¹ ( ⊥ ) = 1 (non-strict) (b) f 2 (x) = 1 for x ∈ ZZ f 2 ( ⊥ ) = ⊥ (strict)

(c) f ³ (x) = 1 for x ∈ ZZ f ³ ( ⊥ ) = 0 (not computable)

30

Flat Base-Domains

Flat Base-Domains

(ZZ ⊥ , IB ⊥ , C ⊥ , F ⊥ )

d ⊑ D d ′ iff d = ⊥ D or d = d ′

. . . -1 0 1 2 3 . . .

⊥

Product Domains

(ZZ ⊥ × ZZ ⊥ , etc.)

( d 1 , . . . , d n ) ⊑ D ( d ′ 1 , . . . d ′ n ) iff d i ⊑ D i d ′ i for all 1 ≤ i ≤ n

.

(0, ⊥ ) ( ⊥ ,0) ( ⊥ ,1) (1, ⊥ )

(0,0) (0,1) (1,0) (1,1)

( ⊥ , ⊥ )

Function Domains

(subset of ZZ ⊥ → ZZ ⊥ , etc.)

f ⊑ D 1 →D 2 g iff f (d) ⊑ D 2 g(d) for all d ∈ D 1 .

29

Strict and Non-Strict Functions

f : D 1 × . . . × D n → D is strict iff

f (d 1 , . . . , d n ) = ⊥ D whenever d i = ⊥ D i for some i.

one :: Int -> Int one x = 1

Choices for the semantics of one:

(a) f 1 (x) = 1 for x ∈ ZZ f 1 ( ⊥ ) = 1 (non-strict) (b) f 2 (x) = 1 for x ∈ ZZ f 2 ( ⊥ ) = ⊥ (strict)

(c) f 3 (x) = 1 for x ∈ ZZ f 3 ( ⊥ ) = 0 (not computable)

30

(ZZ ^⊥ , IB ^⊥ , C ^⊥ , F ^⊥ )

d ⊑ D d ^′ iff d = ⊥ D or d = d ^′

(ZZ ^⊥ × ZZ ^⊥ , etc.)

( d 1 , . . . , d ⁿ ) ⊑ D ( d ^′ ₁ , . . . d ^′ _n ) iff d ⁱ ⊑ D _i d ^′ _i for all 1 ≤ i ≤ n

(subset of ZZ ^⊥ → ZZ ^⊥ , etc.)

f ⊑ D ₁ →D ₂ g iff f (d) ⊑ D ₂ g(d) for all d ∈ D ¹ .

f : D 1 × . . . × D ⁿ → D is strict iff

f (d ¹ , . . . , d ⁿ ) = ⊥ D whenever d ⁱ = ⊥ D _i for some i.

(a) f ¹ (x) = 1 for x ∈ ZZ f ¹ ( ⊥ ) = 1 (non-strict) (b) f 2 (x) = 1 for x ∈ ZZ f 2 ( ⊥ ) = ⊥ (strict)

(c) f ³ (x) = 1 for x ∈ ZZ f ³ ( ⊥ ) = 0 (not computable)