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a a

a

K3

s t

φs4=a1∨a2∨a3s

4a

1 3

s a

φs4

D= (S, Φ) S

Φ = {ϕs|s∈S}

D= (S, Φ)

v D v :S7→ {t,f}v:S7→ {t,f,u}

VD

2VD

3D

u

t f

D= (S, Φ) ≤it f u

<iu<it u <if

D

v1≤iv2∀s∈S:v1(s)∈ {t,f}=⇒v1(s) = v2(s)

uituit=t f uif=f u

∈ VD

3(s) = us∈S v ∈ VD

3,≤iv

≤iVD

3

v∈ VD

3[v]D

2={w∈ VD

2|v≤iw}

[v]D

2v

D= (S, Φ) ΓD:VD

37→ VD

3

ΓD(v) : S7→ {t,f,u}s7→ ui{w(ϕs)|w∈[v]D

2}.

t f

u

adm cmp prf grd

D= (S, Φ) v∈ VD

3

v∈adm(D)v≤iΓD(v)

v∈cmp(D)v= ΓD(v)

v∈prf (D)v≤icmp(D)

v∈grd(D)v≤icmp (D)

D

prf (D)⊆cmp(D)⊆adm (D)grd(D)⊆cmp(D)⊆adm(D)

D= ({a, b, c},{φa=¬b, φb=¬a, φc=a})

a

¬b

b

¬a

c

a

D:

{ } =grd(D) = ΓD( )

≤i≤iVD

3

I1I2I1(a) = I1(b) = I1(c) = tI2(a) =

I2(b) = I2(c) = fI1(φa)uiI2(φa) = uI1(φa) = I1(¬b) = f

I2(φa) = I2(¬b) = tI1(φb)uiI2(φb) = u

I1(φc)uiI2(φc) = u= ΓD( )

adm(D) = {v1, v2, v3, v4,}cmp(D) = {v1, v3,}prf (D) = {v1, v3}

v1={a:t, b :f, c :t}v2={a:t, b :f, c :u}v3={a:f, b :t, c :f}

v4={a:f, b :t, c :u}

T

≤T

T

n

s t st

φsts t

D= (S, T, Φ) S T

Φ = {ϕst|s∈S, t ∈T}

s∈S t ∈T

n m

n·m

D= (S, T, Φ)

a, c ∈S[i, j]⊆T

ϕct=a[i,j]

≥1:= W

i≤k≤j

ak

c t a

[i, j]a c

i j

ϕct=a[i,j]

≥n:= W

{k1,...,kn}⊆[i,j]

|{k1,...,kn}|=n

ak1∧. . . ∧akn

c t a

n[i, j]a c n

[i, j]

ϕct=a[i,j]

≤n:= ¬(a[i,j]

≥n+1) = V

{k1,...,kn+1}⊆[i,j ]

|{k1,...,kn+1}|=n+1

¬ak1∨. . . ∨ ¬akn+1

c t a

n[i, j]n a

[i, j]c

ϕct=a[i,j]

≤1:= ¬(a[i,j]

≥2) = V

{k1,k2}⊆[i,j]

|{k1,k2}|=2

¬ak1∨ ¬ak2

a[i,j]

≤1c

t a [i, j]

ϕct=a[i,j]

=n:= ϕct=a[i,j]

≤n∧a[i,j]

≥n

c t a

n[i, j]

v

s

p l

t

D= (S, T, Φ)

S={v, s, t, p, l}T={1,2,3,4,5}n∈T

nth

ϕl4

ϕl4

ϕt4

ϕp4

ϕsi=sii∈ {1,2,3,4,5}

v1

s1

t1

p1

l1

v2

s2

t2

p2

l2

v3

s3

t3

p3

l3

v4

s4

t4

p4

l4

v5

s5

t5

p5

l5

D

atϕat

v1, v2, v3, v5⊥

v4s[1,3]

≥1∧(p4∨t4)

t1, t2, t3, t5>

t4>∧¬p4≡ ¬p4

p1, p2⊥

p3l3

p4l[3,4]

≥1∧ ¬t4

p5l[3,5]

≥1

l1, l2⊥

l3¬l[4,5]

≥1

l4¬l3∧ ¬l5

l5¬l[3,4]

≥1

sisi

D

D

D

v1

prf (D)l3l4l5p3p4p5s1s2s3t4v4

v1f f t f f t f f f t f

v2f f t f f t f f t t t

v3f f t f f t f t f t t

v4f f t f f t f t t t t

v5f f t f f t t f f t t

v6f f t f f t t f t t t

v7f f t f f t t t f t t

v8f f t f f t t t t t t

D

v v0

s

ϕsv

s

v0L3ϕs

L3

L3

ϕ v

vL3(ϕ) = ui{w(ϕ)|w∈[v]2}.

A={a, b, c, ...}

σ(ϕ)ϕ

ϕ=a∨ ¬a σ(ϕ) = {a}

F

A⊆F

ϕ∈ F ¬ϕ∈ F

ϕ, ψ ∈ F σ(ϕ)∩σ(ψ) = ∅ϕ∨ψ, ϕ ∧ψ∈ F

a b a ∨b a ∧b¬a

t t t t f

t f t f f

t u t u f

f t t f t

f f f f t

f u u f t

u t t u u

u f u f u

u u u u u

K3

ϕ∈ F

F K3

ϕ∈ F v

vK3(ϕ) = ui{w(ϕ)|w∈[v]2}.