mathematical logic for computer science

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Mathematical Logic for Computer Science(Third Revised Edition)

Springer, 2012

Mordechai (Moti) Ben-Arihttp://www.weizmann.ac.il/sci-tea/benari/

c© 2012 by Springer.

http://www.weizmann.ac.il/sci-tea/benari/

Example 1.2: A half-adder

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Bit2

Bit1

Sum

Carry

��

AAAAAAAA

$%$%��

$%

AA��e

Figure 2.1: Two formulas

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p q

JJJ

→

p

¬

q

¬

��\\\

→

""""

ZZZZ

↔

p

q

p

q

¬

��JJJ

→

¬

""""JJJ

↔

!!!!!!JJJ

→

Notation

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Operator Alternates Java language

¬ ∼ !

∧ & &, &&

∨ |, ||

→ ⊃,⇒

↔ ≡,⇔

⊕ 6≡ ^

↑ |

Figure 2.2: Derivation tree for p→q↔¬ p→¬q

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p

fml →

q

fml

��

QQQQ

fml ↔

¬

p

fml

��JJJ

fml →

¬

q

fml

��JJJ

fml

!!!!!!

aaaaaa

fml

""""

PPPPPPPP

fml

Figure 2.3: Truth values of formulas

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A v(A1) v(A2) v(A)

¬A1 T F¬A1 F T

A1∨A2 F F F

A1∨A2 otherwise T

A1∧A2 T T T

A1∧A2 otherwise F

A1→A2 T F F

A1→A2 otherwise T

A v(A1) v(A2) v(A)

A1 ↑ A2 T T F

A1 ↑ A2 otherwise T

A1 ↓ A2 F F T

A1 ↓ A2 otherwise F

A1↔A2 v(A1) = v(A2) T

A1↔A2 v(A1) 6= v(A2) F

A1⊕A2 v(A1) 6= v(A2) T

A1⊕A2 v(A1) = v(A2) F

Example 2.21

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p q p→q

T T T

T F F

F T T

F F T

Example 2.22

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p q p→q ¬ p ¬q ¬q→¬ p (p→q)↔ (¬q→¬ p)

T T T F F T T

T F F F T F T

F T T T F T T

F F T T T T T

Example 2.23

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(p → q) ↔ (¬ q → ¬ p)

T F F T

T F T F T

T F T F F T

T F T F F F T

T F F T F F F T

T F F T T F F F T

p q (p → q) ↔ (¬ q → ¬ p)

T T T T T T F T T F T

T F T F F T T F F F T

F T F T T T F T T T F

F F F T F T T F T T F

Example 2.27

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p q v(p∨q) v(q∨ p)

T T T T

T F T T

F T T T

F F F F

Figure 2.4: Subformulas

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p q

JJJ

→

p

¬

q

¬

��\\\

→

""""

ZZZZ

↔

p q

JJJ

→

p

¬

q

¬

��\\\

→

p

¬

p

¬

p q

Logically equivalent formulas (1)

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A∨ true ≡ true A∧ true ≡ A

A∨ false ≡ A A∧ false ≡ false

A→ true ≡ true true→A ≡ A

A→ false ≡ ¬A false→A ≡ true

A↔ true ≡ A A⊕ true ≡ ¬A

A↔ false ≡ ¬A A⊕ false ≡ A

A ≡ ¬¬A

A ≡ A∧A A ≡ A∨A

A∨¬A ≡ true A∧¬A ≡ false

A→A ≡ true

A↔A ≡ true A⊕A ≡ false¬A ≡ A ↑ A ¬A ≡ A ↓ A


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A∨B ≡ B∨A A∧B ≡ B∧A

A↔B ≡ B↔A A⊕B ≡ B⊕A

A ↑ B ≡ B ↑ A A ↓ B ≡ B ↓ A

A∨ (B∨C) ≡ (A∨B)∨C A∧ (B∧C) ≡ (A∧B)∧C

A↔ (B↔C) ≡ (A↔B)↔C A⊕ (B⊕C) ≡ (A⊕B)⊕C

A∨ (B∧C) ≡ (A∨B)∧ (A∨C)

A∧ (B∨C) ≡ (A∧B)∨ (A∧C)


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A↔B ≡ (A→B)∧ (B→A)

A⊕B ≡ ¬(A→B)∨¬(B→A)

A→B ≡ ¬A∨B

A→B ≡ ¬(A∧¬B)

A∨B ≡ ¬(¬A∧¬B)

A∧B ≡ ¬(¬A∨¬B)

A∨B ≡ ¬A→B

A∧B ≡ ¬(A→¬B)

One-place operators

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x ◦1 ◦2 ◦3 ◦4

T T T F F

F T F T F

Two-place operators

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x1 x2 ◦1 ◦2 ◦3 ◦4 ◦5 ◦6 ◦7 ◦8

T T T T T T T T T T

T F T T T T F F F F

F T T T F F T T F F

F F T F T F T F T F

x1 x2 ◦9 ◦10 ◦11 ◦12 ◦13 ◦14 ◦15 ◦16

T T F F F F F F F F

T F T T T T F F F F

F T T T F F T T F F

F F T F T F T F T F

Operator names

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op name symbol op name symbol

◦2 disjunction ∨ ◦15 nor ↓

◦8 conjunction ∧ ◦9 nand ↑

◦5 implication → ◦12

◦3 reverse implication ← ◦14

◦7 equivalence ↔ ◦10 exclusive or ⊕

Valid, satisfiable, falsifiable and unsatisfiable formulas

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'

&

$

%

'&

$%

True inall interpretations.

True in some interpretations;false in others.

False inall interpretations.

Falsifiable

6 6

Unsatisfiable

6

Satisfiable

6

6

Valid

6

Example 2.41

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p q p∨q ¬ p ¬q (p∨q)∧¬ p∧¬q

T T T F F F

T F T F T F

F T T T F F

F F F T T F

Semantic tableau forp∧ (¬q∨¬ p)

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p∧ (¬q∨¬ p)↓

p,¬q∨¬ pւ ց

p,¬q p,¬ p⊙ ×

Semantic tableaux(p∨q)∧ (¬ p∧¬q)

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(p∨q)∧ (¬ p∧¬q)↓

p∨q,¬ p∧¬q↓

p∨q,¬ p,¬qւ ց

p,¬ p,¬q q,¬ p,¬q× ×

(p∨q)∧ (¬ p∧¬q)↓

p∨q,¬ p∧¬qւ ցp,¬ p∧¬qq,¬ p∧¬q

↓ ↓p,¬ p,¬q q,¬ p,¬q

× ×

α-formulas

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α α1 α2

¬¬A1 A1

A1∧A2 A1 A2

¬(A1∨A2) ¬A1 ¬A2

¬(A1→A2) A1 ¬A2

¬(A1 ↑ A2) A1 A2

A1 ↓ A2 ¬A1 ¬A2

A1↔A2 A1→A2 A2→A1

¬(A1⊕A2) A1→A2 A2→A1

β -formulas

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β β1 β2

¬(B1∧B2) ¬B1 ¬B2

B1∨B2 B1 B2

B1→B2 ¬B1 B2

B1 ↑ B2 ¬B1 ¬B2

¬(B1 ↓ B2) B1 B2

¬(B1↔B2) ¬(B1→B2) ¬(B2→B1)

B1⊕B2 ¬(B1→B2) ¬(B2→B1)

Section 2.7.1: Inductive step in proof of soundness

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n′ : {A1,A2}∪U0

n : {A1∧A2}∪U0

n′ : {B1}∪U0 n′′ : {B2}∪U0

n : {B1∨B2}∪U0

��

@@@

@@

Example 2.73: Semantic tableau forp∧ (¬q∨¬ p)

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p∧ (¬q∨¬ p)↓

p,(¬q∨¬ p)ւ ց

p,¬q p,¬ p⊙ ×

Example 2.74: Semantic tableau forp∨ (q∧¬q)

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p∨ (q∧¬q)ւ ց

p q∧¬q⊙ ↓

q,¬q×

Figure 3.1: Classification ofα-formulas

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α α1 α2

¬¬A A

¬(A1∧A2) ¬A1 ¬A2

A1∨A2 A1 A2

A1→A2 ¬A1 A2

A1 ↑ A2 ¬A1 ¬A2

¬(A1 ↓ A2) A1 A2

¬(A1↔A2) ¬(A1→A2) ¬(A2→A1)

A1⊕A2 ¬(A1→A2) ¬(A2→A1)

Figure 3.1: Classification ofβ -formulas

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β β1 β2

B1∧B2 B1 B2

¬(B1∨B2) ¬B1 ¬B2

¬(B1→B2) B1 ¬B2

¬(B1 ↑ B2) B1 B2

B1 ↓ B2 ¬B1 ¬B2

B1↔B2 B1→B2 B2→B1

¬(B1⊕B2) B1→B2 B2→B1

Section 3.2.1:G and semantic tableaux

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¬ p,q, p ¬q,q, pց ւ¬(p∨q),q, p

↓¬(p∨q),(q∨ p)

↓(p∨q)→ (q∨ p)

¬[(p∨q)→ (q∨ p)]↓

p∨q,¬(q∨ p)↓

p∨q,¬q,¬ pւ ց

p,¬q,¬ p q,¬q,¬ p× ×

Hilbert system H

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Axiom 1 ⊢ (A→ (B→A)),

Axiom 2 ⊢ (A→ (B→C))→ ((A→B)→ (A→C)),

Axiom 3 ⊢ (¬B→¬A)→ (A→B).

Modus ponens⊢ A ⊢ A→B

⊢ B.

Derived rules of inference

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DeductionU∪{A}⊢BU⊢A→B

Contrapositive U⊢¬B→¬AU⊢A→B

Transitivity U⊢A→B U⊢B→CU⊢A→C

Exchange of antecedentU⊢A→(B→C)U⊢B→(A→C)

Double negation U⊢¬¬AU⊢A

Reductio ad absurdumU⊢¬A→false

U⊢A

Theorem 3.37: Soundness of axioms 1 and 3

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¬[A→ (B→A)]↓

A,¬(B→A)↓

A,B,¬A×

¬[(¬B→¬A)→ (A→B)]↓

¬B→¬A,¬(A→B)↓

¬B→¬A,A,¬Bւ ց

¬¬B,A,¬B ¬A,A,¬B↓ ×

B,A,¬B×

Figure 3.32: Rules of inference for sequents

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op Introduction into consequent Introduction into antecedent

∧U ⇒V ∪{A} U ⇒V ∪{B}

U ⇒V ∪{A∧B}U ∪{A,B}⇒V

U ∪{A∧B}⇒V

∨U ⇒V ∪{A,B}

U ⇒V ∪{A∨B}U ∪{A}⇒V U ∪{B}⇒V

U ∪{A∨B}⇒V

→U ∪{A}⇒V ∪{B}U ⇒V ∪{A→B}

U ⇒V ∪{A} U ∪{B}⇒VU ∪{A→B}⇒V

¬U ∪{A}⇒V

U ⇒V ∪{¬A}U ⇒V ∪{A}

U ∪{¬A}⇒V

Figure 3.3: A natural deduction proof

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1. A→¬A Assumption

2. ¬¬A Assumption3. A Double neg. 24. ¬A MP 1, 35. A→ (¬A→ false) Theorem 3.216. ¬A→ false MP 3, 57. false MP 4, 6

8. ¬¬A→ false Deduction 2, 79. ¬A Reductio ad absurdum 8

10. (A→¬A)→¬A Deduction 1, 9

Figure 4.1: Resolution refutation as a tree

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2

p̄ p

p̄q̄ p̄q

p̄q̄r r̄

��

HHHHHH

��

@@@@

��

AAAA

Figure 4.2: Incomplete resolution tree

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pqr̄ r pq̄r r̄

pq pq̄

p

��

@@@@

��

@@@@

��

HHHH

HH

Figure 4.3: Semantic tree

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��

XXXXXXXXXXp p̄q

qq i2

��

HHHHHq q̄

��

@@@ �

��

@@@r r̄ r r̄i3 i4 q q

qqi1

��

HHHH

Hq q̄

��

@@@@

��

@@@@

r r̄ r r̄q qqq

Figure 4.4: Inference and failure node

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n

n1 n2

l lc

��

@@@@

Section 4.5: Graph for Tseitin clauses

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��

0 t 0

q r s

0 p 1

Section 5.1: Efficient truth table for p∨ (q∧ r)

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p q r p∨ (q∧ r)

T T T T

T T F T

T F T T

T F F T

F T T T

F T F F

F F T F

F F F F

p q r p∨ (q∧ r)

T T ∗ T

T F ∗ T

p q r p∨ (q∧ r)

T ∗ ∗ T

p q r p∨ (q∧ r)

T ∗ ∗ T

F T T T

F T F F

F F ∗ F

Section 5.1: Efficient truth table for p⊕q⊕ r

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p q r p⊕q⊕ r

T T T T

T T F F

T F T F

T F F T

F T T F

F T F T

F F T T

F F F F

p q r p⊕q⊕ r

T T T T

T T F F

T F T F

T F F T

F T ∗ (See rows 3 and 4.)

F F ∗ (See rows 1 and 2.)

Figure 5.1: BDD for p∨ (q∧ r)

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kpkq kq

kr kr kr krT T T TF F F T

HHHH

HHHH

@@@

@@@

AAA

AAA

AAA

AAA

Example 5.6: First reduction of BDD for p∨ (q∧ r)

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kpkq kq

kr kr kr krF T

HHHH

HHHH

@@@

@@@

��

QQQQ

````````

`````

QQQQ

Example 5.6: Second reduction of BDD forp∨ (q∧ r)

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kpkq kqkr

F TQQ

QQ

@@@

Example 5.6: Third reduction of BDD for p∨ (q∧ r)

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kpkqkr

F TQQ

QQ

@@@

Example 5.7: BDD for p⊕q⊕ r after first reduction

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kpkq kq

kr kr kr kr

F T

HHHH

H

@@@

@@@

��

PPPP

PPPP

PPPP

PPP

��

��

Example 5.6: BDD for p⊕q⊕ r after second reduction

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kpkq kqkr kr

F T

@@@

""""""

bbbbbb

""""""

bbbbbb

BDD for (p⊕q)⊕ (p⊕ r) from BDDs for p⊕q and p⊕ r

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kpF T

@@@

BDD for p⊕q

kpkq kqF T

@@@

!!!!!!!

aaaa

aaa

⊕

BDD for p⊕ r

kpkr krF T

@@@

!!!!!!!

aaaa

aaa


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BDD for p⊕q

kpkq kq

BDD for F⊕q BDD for T ⊕q

@@@

��

@@@

��

@@@

⊕

BDD for p⊕ r

kpkr kr

BDD for F⊕ r BDD for T ⊕ r

@@@

��

@@@

��

@@@


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BDD for ¬q

kqF T

BDD for ¬T BDD for ¬F

��

⊕

BDD for ¬rkrF T


��

BDD for ¬F

T ⊕

BDD for ¬rirF T


��


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T F T⊕ ≡

BDD for ¬F BDD for ¬T BDD for ¬F⊕¬T

T T F⊕ ≡

BDD for ¬F BDD for ¬F BDD for ¬F⊕¬F

T ⊕ ≡

BDD for ¬F BDD for ¬r BDD for ¬F⊕¬rirF T��

irT F��


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ipiq iqir ir ir ir

F FT T F FT T

HHHH

HH

@@

��

AA

AA

��

��

BDD for q⊕ r

iqir irF T

@@

!!!!!!

aaaa

aa

Example 5.21: Restriction

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(a)

ipiqir

F TAAA

@@

(b)

ipiq

F T\\\\\\

(c)

ipiq

F T

(b)

ip

F T

Section 6.4: 4-queens problem

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1 2 3 4

4

3

2

1

Q

Q

Q

Q

Section 6.4.1: Encoding the 4-queens problem (1)

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11∨12∨13∨14,21∨22∨23∨24,31∨32∨33∨34,41∨42∨43∨44.

11∨12, 11∨13, 11∨14, 12∨13, 12∨14, 13∨14,21∨22, 21∨23, 21∨24, 22∨23, 22∨24, 23∨24,31∨32, 31∨33, 31∨34, 32∨33, 32∨34, 33∨34,41∨42, 41∨43, 41∨44, 42∨43, 42∨44, 43∨44,

Section 6.4.1: Encoding the 4-queens problem (2)

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11∨21, 11∨31, 11∨41, 21∨31, 21∨41, 31∨41,12∨22, 12∨32, 12∨42, 22∨32, 22∨42, 32∨42,13∨23, 13∨33, 13∨43, 23∨33, 23∨43, 33∨43,14∨24, 14∨34, 14∨44, 24∨34, 24∨44, 34∨44.

11∨22, 11∨33, 11∨44, 12∨21, 12∨23, 12∨34,13∨22, 13∨31, 13∨24, 14∨23, 14∨32, 14∨41,21∨32, 21∨43, 22∨31, 22∨33, 22∨44,23∨32, 23∨41, 23∨34, 24∨33, 22∨42,31∨42, 32∨41, 32∨43,33∨42, 33∨44, 34∨43.

Figure 7.1: Tree for ∀x(¬∃yp(x,y)∨¬∃yp(y,x))

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p(x,y)

∃y

¬

p(y,x)

∃y

¬

��

QQQQ

∨

∀x

Figure 7.2: Global and local variables

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class MyClass {int x;

void p() {int x;

x = 1;

// Print the value of x

}void q() {

// Print the value of x

}... void main(...) {x = 5;

p;

q;

}


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∀xA(x)↔¬∃x¬A(x) ∃xA(x)↔¬∀x¬A(x)

∀x∀yA(x,y)↔∀y∀xA(x,y) ∃x∃yA(x,y)↔∃y∃xA(x,y)

∃x∀yA(x,y)→∀y∃xA(x,y)

(∃xA(x)∨B)↔∃x(A(x)∨B) (∀xA(x)∨B)↔∀x(A(x)∨B)

(B∨∃xA(x))↔∃x(B∨A(x)) (B∨∀xA(x))↔∀x(B∨A(x))

(∃xA(x)∧B)↔∃x(A(x)∧B) (∀xA(x)∧B)↔∀x(A(x)∧B)

(B∧∃xA(x))↔∃x(B∧A(x)) (B∧∀xA(x))↔∀x(B∧A(x))


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∀x(A→B(x))↔ (A→∀xB(x))

∀x(A(x)→B)↔ (∃xA(x)→B)

(∃x(A(x)∨B(x))↔ (∃xA(x)∨∃xB(x))

∀x(A(x)∧B(x))↔ (∀xA(x)∧∀xB(x))

∀xA(x)∨∀xB(x))→∀x(A(x)∨B(x))

∃x(A(x)∧B(x))→ (∃xA(x)∧∃xB(x))

∀x(A(x)↔B(x))→ (∀xA(x)↔∀xB(x))

∀x(A(x)↔B(x))→ (∃xA(x)↔∃xB(x))

∃x(A(x)→B(x))↔ (∀xA(x)→∃xB(x))

(∃xA(x)→∀xB(x))→∀x(A(x)→B(x))


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∀x(A(x)∨B(x))→ (∀xA(x)∨∃xB(x))

∀x(A(x)→B(x))→ (∀xA(x)→∀xB(x))

∀x(A(x)→B(x))→ (∃xA(x)→∃xB(x))

∀x(A(x)→B(x))→ (∀xA(x)→∃xB(x))

Figure 7.3: Semantic tableau for a satisfiable formula

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¬(∀x(p(x)∨q(x))→ (∀xp(x)∨∀xq(x)))↓

∀x(p(x)∨q(x)), ¬(∀xp(x)∨∀xq(x))↓

∀x(p(x)∨q(x)), ¬∀xp(x), ¬∀xq(x)↓

∀x(p(x)∨q(x)), ¬∀xp(x), ¬q(a)↓

∀x(p(x)∨q(x)), ¬ p(a), ¬q(a)↓

p(a)∨q(a), ¬ p(a), ¬q(a)ւ ց

p(a), ¬ p(a), ¬q(a) q(a), ¬ p(a), ¬q(a)× ×

Figure 7.4: A tableau that should close, but doesn’t

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∀x∃yp(x,y) ∧ ∀x(q(x)∧¬q(x))↓

∀x∃yp(x,y), ∀x(q(x)∧¬q(x))↓

∀x∃yp(x,y), ∃yp(a1,y), ∀x(q(x)∧¬q(x))↓

∀x∃yp(x,y), p(a1,a2), ∀x(q(x)∧¬q(x))↓

∀x∃yp(x,y), p(a1,a2), ∀x(q(x)∧¬q(x))↓

∀x∃yp(x,y), ∃yp(a2,y), p(a1,a2), ∀x(q(x)∧¬q(x))↓

∀x∃yp(x,y), p(a2,a3), p(a1,a2), ∀x(q(x)∧¬q(x))

Rules for γ- and δ -formulas

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γ γ(a)∀xA(x) A(a)

¬∃xA(x) ¬A(a)

δ δ (a)∃xA(x) A(a)

¬∀xA(x) ¬A(a)

Figure 8.1: Semantic tableau in first-order logic

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¬((∀xp(x)∨∀xq(x))→∀x(p(x)∨q(x)))↓

∀xp(x)∨∀xq(x), ¬∀x(p(x)∨q(x))ւ ց

∀xp(x), ¬∀x(p(x)∨q(x)) ∀xq(x), ¬∀x(p(x)∨q(x))↓ ↓

∀xp(x), ¬(p(a)∨q(a)) ∀xq(x), ¬(p(a)∨q(a))↓ ↓

∀xp(x), ¬ p(a), ¬q(a) ∀xq(x), ¬ p(a), ¬q(a)↓ ↓

∀xp(x), p(a), ¬ p(a), ¬q(a) ∀xq(x), q(a), ¬ p(a), ¬q(a)× ×

Figure 8.2: Gentzen proof tree in first-order logic

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¬∀xp(x), ¬ p(a), p(a), q(a) ¬∀xq(x), ¬q(a), p(a), q(a)↓ ↓

¬∀xp(x), p(a), q(a) ¬∀xq(x), p(a), q(a)↓ ↓

¬∀xp(x), p(a)∨q(a) ¬∀xq(x), p(a)∨q(a)↓ ↓

¬∀xp(x), ∀x(p(x)∨q(x)) ¬∀xq(x), ∀x(p(x)∨q(x))ց ւ

¬(∀xp(x)∨∀xq(x)), ∀x(p(x)∨q(x))↓

(∀xp(x)∨∀xq(x))→∀x(p(x)∨q(x))

Rules for γ- and δ -formulas

67 / 136

U ∪{γ ,γ(a)}U ∪{γ}

U ∪{δ (a)}U ∪{δ}

,

The rule forδ -formulas can be applied only if the constanta does notoccur in any formula ofU .

γ γ(a)∃xA(x) A(a)

¬∀xA(x) ¬A(a)

δ δ (a)∀xA(x) A(a)

¬∃xA(x) ¬A(a)

Figure 8.3: δ -formulas follow γ-formulas

68 / 136

¬∀yp(a,y), ¬ p(a,b), ¬ p(a,a), ∃xp(x,b), p(a,b)↓

¬∀yp(a,y), ¬ p(a,a), ∃xp(x,b), p(a,b)↓

¬∀yp(a,y), ∃xp(x,b), p(a,b)↓

¬∀yp(a,y), ∃xp(x,b)↓

¬∀yp(a,y), ∀y∃xp(x,y)↓

¬∃x∀yp(x,y), ∀y∃xp(x,y)↓

∃x∀yp(x,y)→∀y∃xp(x,y)

Hilbert system H

69 / 136

Axiom 1 ⊢ (A→ (B→A)),

Axiom 2 ⊢ (A→ (B→C))→ ((A→B)→ (A→C)),

Axiom 3 ⊢ (¬B→¬A)→ (A→B),

Axiom 4 ⊢ ∀xA(x)→A(a),

Axiom 5 ⊢ ∀x(A→B(x))→ (A→∀xB(x)).

Modus ponens⊢ A→B ⊢ A

⊢ B

Generalization⊢ A(a)⊢ ∀xA(x)

.

In Axiom 5, B(x) is a formula with a free variablex, while x is not a freevariable of the formulaA.

Derived rules of inference

70 / 136

GeneralizationU⊢A(a)

U⊢∀xA(x), a does not appear inU .

DeductionU∪{A}⊢BU⊢A→B

Axiom 4U⊢∀xA(x)U⊢A(a)

Generalization⊢A(a)→B(a)⊢∀xA(x)→∀xB(x)

C-ruleU⊢∃xA(x)U⊢A(a) , a does not appear inU or ∃xA(x).

Figure 9.1: Finite sequences of terms (1)

71 / 136

n = 1 a

n = 2 b

n = 3 f (a), f (b), f ( f (a)), f ( f (b))

n = 4 f ( f ( f (a))), f ( f ( f (b),

g(a,a), g(a,b), g(a, f (a)), g(a, f (b)), g(a, f ( f (a))), g(a, f ( f (b))),

six similar terms withb as the first argument ofg,

g( f (a),a), g( f (a),b), g( f (a), f (a)), g( f (a), f (b)),

g( f (a), f ( f (a))), g( f (a), f ( f (b))),

six similar terms withf (b) as the first argument ofg,

Figure 9.1: Finite sequences of terms (2)

72 / 136

g( f ( f (a)),a), g( f ( f (a)),b), g( f ( f (a)), f (a)), g( f ( f (a)), f (b)),

g( f ( f (a)), f ( f (a))), g( f ( f (a)), f ( f (b))),

six similar terms withf ( f (b)) as the first argument ofg,

f (g(a,a)), f (g(a,b)), f (g(a, f (a)), f (g(a, f (b)),

twelve similar terms withb, f (a), f (b) as the first argument ofg,

f ( f (g(a,a))), f ( f (g(a,b))), f ( f (g(b,a))), f ( f (g(b,b))).

Example 9.20: Hebrand universes

73 / 136

S1 = {pa¬ pbqz, ¬qz¬ pbqz}

HS1 = {a, b}

S2 = {¬ px f (y), pwg(w)}

HS2 = {a, f (a), g(a),

f ( f (a)), g( f (a)), f (g(a)), g(g(a)), . . .}

S3 = {¬ pa f (x,y), pb f (x,y)}

HS3 = {a, b, f (a,a), f (a,b), f (b,a), f (b,b),

f (a, f (a,a)), f ( f (a,a),a), . . .}

Example 10.2: Ground resolution

74 / 136

{q( f (b)), r(a, f (b))} {p(a), ¬q( f (b)), r( f (a),b)}

��

HHHHHHHH

{p(a), r(a, f (b)), r( f (a),b)}

Example 10.7: Unification

75 / 136

〈 g(y) = x, f (x,h(x),y) = f (g(z),w,z) 〉

〈 f (g(y),h(g(y)),y) = f (g(z),w,z), x = g(y) 〉

〈 g(y) = g(z), h(g(y)) = w, y = z, x = g(y) 〉

〈 y = z, h(g(y)) = w, y = z, x = g(y) 〉

〈 h(g(z)) = w, z = z, x = g(z), y = z 〉

〈 z = z, x = g(z), y = z, w = h(g(z)) 〉

〈 x = g(z), y = z, w = h(g(z)) 〉

Example 10.29: Resolution refutation (1)

76 / 136

1. ¬ p(x)∨q(x)∨ r(x, f (x))

2. ¬ p(x)∨q(x)∨ s( f (x))

3. t(a)

4. p(a)

5. ¬r(a,y)∨ t(y)

6. ¬t(x)∨¬q(x)

7. ¬t(x)∨¬s(x)

Example 10.29: Resolution refutation (2)

77 / 136

8. ¬q(a) x← a 3,69. q(a)∨ s( f (a)) x← a 2,410. s( f (a)) 8,911. q(a)∨ r(a, f (a)) x← a 1,412. r(a, f (a)) 8,1113. t( f (a)) y← f (a) 5,1214. ¬s( f (a)) x← f (a) 7,1315. 2 10,14

Example 10.30: Resolution refutation

78 / 136

1. ¬ p(x,y)∨ p(y,x)

2. ¬ p(x,y)∨¬ p(y,z)∨ p(x,z)

3. p(x, f (x))

4. ¬ p(x,x)

3′. p(x′, f (x′)) Rename 35. p( f (x),x) σ1 = {y← f (x),x′← x} 1,3′

3′′. p(x′′, f (x′′)) Rename 36. ¬ p( f (x),z)∨ p(x,z) σ2 = {y← f (x),x′′← x} 2,3′′

5′′′. p( f (x′′′),x′′′) Rename 57. p(x,x) σ3 = {z← x,x′′′← x} 6,5′′′

4′′′′. ¬ p(x′′′′,x′′′′) Rename 48. 2 σ4 = {x′′′′← x} 7,4′′′′

Theorem 10.33: Lifting Lemma

79 / 136

C1,C2

C′1,C′2

-

-Resolution

Ground resolution

C

C′? ?

Instance Instance

Figure 10.1: Example for the lifting lemma (1)

80 / 136

C1 = {p(x), p( f (y)), p( f (z)), q(x)}

C2 = {¬ p( f (u)), ¬ p(w), r(u)}

θ1 = {x← f (a), y← a, z← a}

θ2 = {u← a, w← f (a)}

C′1 = C1θ1 = {p( f (a)), q( f (a))}

C′2 = C2θ2 = {¬ p( f (a)), r(a)}

C′ = Res(C1,C2) = {q( f (a)), r(a)}


81 / 136

L1 = {p(x), p( f (y)), p( f (z))}

λ1 = {x← f (y), z← y}

L1λ1 = {p( f (y))}

L2 = {¬ p( f (u)), ¬ p(w)}

λ2 = {w← f (u)}

L2λ2 = {¬ p( f (u))}

λ = λ1∪λ2 = {x← f (y),z← y,w← f (u)}

L1λ = {p( f (y))}

C1λ = {p( f (y)), q( f (y))}

L2λ = {¬ p( f (u))}

C2λ = {¬ p( f (u)), r(u)}


82 / 136

σ = {u← y}

C = Res(C1λ ,C2λ ) = {q( f (y)), r(y)}, usingσλσ = {x← f (y), z← y, w← f (y), u← y}

C1λσ = {p( f (y)), q( f (y))}

C2λσ = {¬ p( f (y)), r(y)}

C = Res(C1,C2) = {q( f (y)), r(y)}, usingλσ

θ ′1 = {y← a}

C′1 = C1θ1 = {p( f (a)), q( f (a))}=C1λσθ1

θ ′2 = {y← a}

C′2 = C2θ2 = {¬ p( f (a)), r(a)}=C2λσθ2

θ ′ = {y← a}

C′ = Res(C′1,C′2) = {q( f (a)), r(a)}

Figure 11.1: SLD-tree for selection of leftmost literal

83 / 136

(1) (2)

(5) (6) (7) (3) (10)

(1) (2) (1) (2)

q(y,b)��

PPPPPPPPp(y,b)

2 2 2

��

@@@

p(y,z),q(z,b)

. . .q(a,b) q(h,b)

p(a,b) p(a,z′),q(z′,b) p(h, j) p(h,z′′),q(z′′, j)× ... × ...

��

HHHHH

��

@@@

��

@@@

Theorem 12.3: Church’s Theorem

84 / 136

SM =

(n∧

i=0

Si∧ p0(a,a)

)

→∃z1∃z2pn(z1,z2).

Li Si

x = x + 1; ∀x∀y(pi(x,y)→ pi+1(s(x),y))

y = y + 1; ∀x∀y(pi(x,y)→ pi+1(x,s(y)))

if (x == 0) goto Lj; ∀x(pi(a,x)→ p j(a,x)) ∧

else x = x - 1; ∀x∀y(pi(s(x),y)→ pi+1(x,y))

if (y == 0) then goto Lj; ∀x(pi(x,a)→ p j(x,a)) ∧

else y = y - 1; ∀x∀y(pi(x,s(y))→ pi+1(x,y))

Figure 13.1: State transition diagram

85 / 136

��

p¬q

s0

-��

pq

s1

�--

� ��¬ pq

s2

��?

��¬ p¬q

s3

��?�

��

Figure 13.2: Alternate representation

86 / 136

��

p

s0

-��

pq

s1

�--

� ��

q

s2

��?

��s3

��?�

��

Definition 13.27: Linear temporal logic

87 / 136

��¬ p

s0

-��¬ p

s1

-��

p

s2

- . . .

Example 13.33

88 / 136

� ��ps0

-� ��qs1

-� ��ps2

-� ��qs3

-� ��ps4

- . . .

Commutativity

89 / 136

� ��¬ps0

-� ��¬ps1

-� ��ps2

-� ��¬ps3

-� ��¬ps4

-� ��ps5

- . . .

� ��¬ps0

-� ��¬ps1

-� ��¬ps2

-� ��¬ps3

-� ��ps4

-� ��ps5

- . . .

Section 13.51: Tableaux rules for LTL

90 / 136

α α1 α2

2A A #2A

¬3A ¬A ¬#3A

β β1 β2

3A A #3A

¬2A ¬A ¬#2A

X X1

#A A

¬#A ¬A

The rule for next formulas (1)

91 / 136

(p∨q)∧#(¬ p∧¬q)↓

p∨q, #(¬ p∧¬q)ւ ց

p, #(¬ p∧¬q) q, #(¬ p∧¬q)


92 / 136

(p∨q)∧#(¬ p∧¬q)↓

p∨q, #(¬ p∧¬q)ւ ց

p, #(¬ p∧¬q) q, #(¬ p∧¬q)

↓ ↓

¬ p∧¬q ¬ p∧¬q↓ ↓

¬ p, ¬q ¬ p, ¬q


93 / 136

��

p

s0

-��¬ p¬q

s1

��

q

s0

-��¬ p¬q

s1

��

p

s0

-��¬ p¬q

s1

��

q

s0

-��¬ p¬q

s1

��

Semantic tableau for¬(2(p∧q)→2p)

94 / 136

¬(2(p∧q)→2p)↓

2(p∧q), 3¬ p↓

p∧q, #2(p∧q), 3¬ p↓

p, q, #2(p∧q), 3¬ pւ ց

p, q, #2(p∧q), ¬ p p, q, #2(p∧q), #3¬ p×

Interpretations for ¬(2(p∧q)→2p)

95 / 136

��

pq

s0

-��

pq

s1

-��

pq

s2

-

��

pq

s0

��

Example 13.38

96 / 136

l0 : 23p↓

l1 : 3p, #23pւ ց

l2 : p, #23p l3 : #3p, #23p

↓ ↓l4 : 23p l5 : 3p, 23p

↓ ↓l6 : 3p, #23p l7 : 3p, #23pւ ց ւ ց

To l2 To l3 To l2 To l3

Example 13.43

97 / 136

l0 : 23p↓

l1 : 3p, #23pւ ց

l2 : p, #23p l3 : #3p, #23p

↓ ↓

To l0 To l1

��

p

s0

�--��s1

��

?

Figure 13.3: 2(3(p∧q)∧3(¬ p∧q)∧3(p∧¬q))

98 / 136

��

p¬q

s2

�-

��

pq

s0

�- ��¬ pq

s1

��

��s3

��-

�

-�

6

?

6

?��*��

HHHHHHHHHHHHHjHHH

HHHH

HHHH

HHY

Section 13.5.4: A linear, fulfilling Hintikka structure

99 / 136

��

pq

s0

-��¬ pq

s1

-��

p¬q

s2

��?

Figure 13.4: Strongly connected components

100 / 136

��s0

@@@@R

��

��s2 s3

s1

�� @@R ��

��s6 s7

s4 s5

�

-

6 ?

�

Figure 13.5: Component graph

101 / 136

��

@@R

�

G0

'&$%��

��s0

@@@@R

��G1

'

&

$

%��

s2 s3

s1

�� @@R G2

'

&

$

%��

s6 s7

s4 s5

�

-

6 ?

�

Figure 13.6: An SCC is contained in an MSCC

102 / 136

'

&

$

%

G′

G

'

&

$

%��

s2 s3

s1

�� @@R

-��

s6 s7

s4 s5

�

-

6 ?

�

Example 13.68

103 / 136

��

p

s0

-��

p

s1

-��

q

s2

- · · ·

��

p

s0

-��

p

s1

-��

p

s2

- · · ·

��

p

s0

-��¬ p

s1

-��

q

s2

- · · ·

Deductive systemL

104 / 136

Axiom 0 Prop Any substitution instance ofa valid propositional formula.

Axiom 1 Distribution of 2 ⊢2(A→B)→ (2A→2B).Axiom 2 Distribution of # ⊢#(A→B)→ (#A→#B).Axiom 3 Expansion of2 ⊢2A→ (A ∧ #A ∧ #2A).Axiom 4 Induction ⊢2(A→#A)→ (A→2A).Axiom 5 Linearity ⊢#A↔¬#¬A.

Modus ponens⊢ A ⊢ A→B

⊢ B.

Generalization⊢ A⊢2A

.

Example 14.9

105 / 136

��

p

s0

-��¬ p

s1

-��

p

s2

-��¬ p

s3

- · · ·

Semantic tableau for the negation of2p→#2p

106 / 136

¬(2p→#2p)↓

2p, #3¬ p↓

ls p, #2p, #3¬ p

↓2p, 3¬ p↓

p, #2p, 3¬ pւ ց

p, #2p, ¬ p (To nodels)×

Section 14.4: Axioms for binary temporal operators

107 / 136

Axiom 6 Expansion ofU ⊢ AU B↔ (B∨ (A∧#(AU B))).Axiom 7 Eventuality ⊢ AU B→3B.

Definition 15.8: H L (1)

108 / 136

Domain axiomsEvery true formula over the domain(s) of the program variables.

Assignment axiom

⊢ {p(x){x← t}}x := t{p(x)}.

Composition rule

⊢ {p}S1{q} ⊢ {q}S2{r}⊢ {p}S1;S2{r}

.

Definition 15.8: H L (2)

109 / 136

Alternative rule

⊢ {p∧B}S1{q} ⊢ {p∧¬B}S2{q}⊢ {p}if B then S1 else S2{q}

.

Loop rule⊢ {p∧B}S{p}

⊢ {p}while B do S{p∧¬B}.

Consequence rule

⊢ p1→ p ⊢ {p}S{q} ⊢ q→q1

⊢ {p1}S{q1}.

Section 15.3: Program to be verified

110 / 136

{true}x := 0;

{x = 0}y := b;

{x = 0∧ y = b}while y <> 0 do

{x = (b− y) ·a}begin x := x + a; y := y - 1 end;

{x = a ·b}

Section 15.4.1: Solution 1 (outline)

111 / 136

{0≤ a}x = E1(x,a);

while (B(x,a))

{0≤ x2≤ a}x = E2(x,a);

{0≤ x2≤ a < (x+1)2}.

Section 15.4.1: Solution 1 (final)

112 / 136

{0≤ a}x = 0;

while ((x+1)*(x+1) <= a)

{0≤ x2≤ a}x = x + 1;

{0≤ x2≤ a < (x+1)2}.

Section 15.4.1: Solution 2 (outline)

113 / 136

{0≤ a}x = 0;

y = a+1;

while (y != x+1)

{(0≤ x2≤ a < y2) ∧ (x < y≤ a+1)}E(x,y,a);

{0≤ x2≤ a < (x+1)2}.

Section 15.4.1: Solution 2 (final)

114 / 136

{0≤ a}x = 0;

y = a+1;

while (y != x+1)

{0≤ x2≤ a < y2 ∧ x < y≤ a+1}{z = (x+y) / 2;

if (z*z <= a)

x = z;

else

y = z;

}{0≤ x2≤ a < (x+1)2}.

Example 15.15: Weaker conditions

115 / 136

'

&

$

%��

' $

& %' $& %

y = 0 y = 1 y = 2 y = 3

y = 1∨ y = 3

y≤ 3

Definition 15.21: Sequential composition

116 / 136

'

&

$

%

'&

$%wp(S1, wp(S2, q)) -

'

&

$

%'&$%wp(S2, q) ��

��

��:

'

&

$

%

'&$%

q

Corollary 9.26: Excluded miracle

117 / 136

︸︷︷︸

wp(S, p)

︷︸︸︷

wp(S, ¬ p)'&

$%

p ∞ ¬ p

Theorem 9.28: Monotonicity

118 / 136

'

&

$

%

'

&

$

%wp(S, q)��wp(S, p) -

-

'

&

$

%

'&$%

q�� p

Lemma 15.37

119 / 136

'

&

$

%

'

&

$

%

'

&

$

%

'

&

$

%

'&

$%

��

��

0∨

k=0wp(Wk, p)

1∨

k=0wp(Wk, p)

2∨

k=0wp(Wk, p) · · ·

∞∨

k=0wp(Wk, p)

Example 16.2

120 / 136

int n = 0

Process p Process q

1: n = n + 1 1: n = n + 1

2: n = n + 1 2: n = n + 1

end: end:

Atomic operations

121 / 136

int n = 0

Process p Process q

1: n = n + 1 1: n = n + 1

end: end:

int n = 0

Process p Process q

int temp = 0 int temp = 0

1: temp = n 1: temp = n

2: temp = temp + 1 2: temp = temp + 1

3: n = temp 3: n = temp

end: end:

Peterson’s algorithm

122 / 136

boolean wantp = false, wantq = false

int turn = 1

Process p Process q

while (true) { while (true) {

non-critical-section non-critical-section

wantp = true wantq = true

turn = 1 turn = 2

wait until wait until

(!wantq or turn == 2) (!wantp or turn == 1)

critical-section critical-section

wantp = false wantq = false

} }

Abbreviated algorithm

123 / 136


int turn = 1

Process p Process q


tryp: wantp=true; turn=1 tryq: wantq=true; turn=2

waitp: wait until waitq: wait until

(!wantq or turn == 2) (!wantp or turn == 1)

csp: wantp = false csq: wantq = false

} }

Progress axioms

124 / 136

Statement Progress axioms

li: v := expression ⊢ li→3li+1

li+1:

li: if B then ⊢ li→3(lt ∨ l f )

lt: S1 ⊢ (li∧2B)→3ltelse

lf: S2 ⊢ (li∧2¬B)→3l f

li: while B do ⊢ li→3(lt ∨ l f )

lt: S1; ⊢ (li∧2B)→3ltlf: ⊢ (li∧2¬B)→3l f

Figure 16.3: Finite automata for Peterson’s algorithm

125 / 136

-��tryp -

wantp=trueturn=1 ��waitp -

!wantq orturn == 2��

��-

wantq andturn == 1

��csp

6

wantp=false

-��tryq -

wantq=trueturn=2 ��waitq -

!wantp orturn == 1��

��-

wantp andturn == 2

��csq

6

wantq=false

Product automaton for Peterson’s algorithm

126 / 136

?

"!# tryptryq

-

wantp=trueturn=1�

wantq=trueturn=2

"!# trypwaitq "!

# waitptryq

· · ·· · ·

Figure 16.2: State space for Peterson’s algorithm

127 / 136

1. (tryp,tryq,1)��

HHHHHHj2. (waitp,tryq,1)

��

@@@R

3. (tryp,waitq,2)

��

@@@R

4. (csp,tryq,1)

��i1 @@@R

5. (waitp,waitq,2)

��

@@R i56. (waitp,waitq,1)

��i6?

7. (tryp,csq,2)�� ?

8. (csp,waitq,2)

��i3 @@R i89. (waitp,csq,1)

��i9 @@R i210. (tryp,tryq,2)

��i2 @@R i3

Section 16.5.2: Dijkstra’s second attempt

128 / 136


Process p Process q


waitp: wait until !wantq waitq: wait until !wantp

tryp: wantp = true tryq: wantq = true

csp: wantp = false csq: wantq = false

} }

Figure 16.3: State space for the second attempt

129 / 136

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Section 16.6

130 / 136

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Section 16.6.1

131 / 136

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Figure 16.4: Model checking liveness

132 / 136


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4. (csp,tryq,1)

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Figure 16.5: The state space as a tree

133 / 136



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8. (t,t,1) 9. (c,w,2) 10. (w,w,2) 11. (w,w,1) 12. (w,c,1) 13. (t,t,2)

Figure 16.6: CTL model checking

134 / 136



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Section 16.8: Symbolic model checking

135 / 136

tryp p0∧ p1 tryq q0∧q1

waitp ¬ p0∧ p1 waitq ¬q0∧q1

csp p0∧¬ p1 csq q0∧¬q1

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Venn diagrams

136 / 136

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