Logic as a Tool 3mm Chapter 4: Deductive Reasoning in ......October 2016. Goranko Revision: CNF and...

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Logic as a Tool

Chapter 4: Deductive Reasoning in First-Order Logic4.4 Prenex normal form.

Skolemization. Clausal form.

Valentin GorankoStockholm University

October 2016

Goranko

Revision: CNF and DNF of propositional formulae

• A literal is a propositional variable or its negation.

• An elementary disjunction is a disjunction of literals.An elementary conjunction is a conjunction of literals.

• A disjunctive normal form (DNF) is a disjunction of elementaryconjunctions.

• A conjunctive normal form (CNF) is a conjunction of elementarydisjunctions.

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Conjunctive and disjunctive normal formsof first-order formulae

An open first-order formula is in disjunctive normal form (resp.,conjunctive normal form) if it is a first-order instance of a propositionalformula in DNF (resp. CNF), obtained by uniform substitution of atomicformulae for propositional variables.

Examples:(¬P(x) ∨ Q(x , y)) ∧ (P(x) ∨ ¬R(y))

is in CNF, as it is a first-order instance of (¬p ∨ q) ∧ (p ∨ ¬r);

(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

is in DNF, as it is a first-order instance of (¬p ∧ q ∧ r) ∨ ¬p.

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

are not in either CNF or DNF.

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(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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Examples:

(¬P(x) ∨ Q(x , y)) ∧ (P(x) ∨ ¬R(y))

(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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(P(x) ∧ Q(x , y) ∧ R(y)) ∨ ¬P(x)

∀xP(x) ∨ Q(x , y)

and¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y)

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Prenex normal forms

A first-order formula Q1x1...QnxnA, where Q1, ...,Qn are quantifiers andA is an open formula, is in a prenex form.

The quantifier string Q1x1...Qnxn is called the prefix,and the formula A is the matrix of the prenex form.

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

is in prenex form, while

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

are not in prenex form.

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Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

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Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

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Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

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Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

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Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

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Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

Goranko

Prenex normal forms

Examples:

∀x∃y(x > 0→ (y > 0 ∧ x = y2))

∃x(x = 0) ∧ ∃y(y < 0)

and∀x(x > 0 ∨ ∃y(y > 0 ∧ x = y2))

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Prenex conjunctive and disjunctive normal forms

If A is in DNF then Q1x1...QnxnA is in prenex disjunctive normal form(PDNF); if A is in CNF then Q1x1...QnxnA is in prenex conjunctivenormal form (PCNF).

Examples:∀x∃y(¬x > 0 ∨ y > 0)

is both in PDNF and in PCNF.

∀x∃y(¬x > 0 ∨ (y > 0 ∧ ¬x = y2))

is in PDNF, but not in PCNF.

∀x∀y(¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y))

is neither in PCNF nor in PDNF.

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∀x∃y(¬x > 0 ∨ (y > 0 ∧ ¬x = y2))

∀x∀y(¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y))

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Examples:

∀x∃y(¬x > 0 ∨ y > 0)

∀x∃y(¬x > 0 ∨ (y > 0 ∧ ¬x = y2))

∀x∀y(¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y))

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∀x∃y(¬x > 0 ∨ (y > 0 ∧ ¬x = y2))

∀x∀y(¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y))

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∀x∃y(¬x > 0 ∨ (y > 0 ∧ ¬x = y2))

∀x∀y(¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y))

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∀x∃y(¬x > 0 ∨ (y > 0 ∧ ¬x = y2))

∀x∀y(¬P(x) ∨ (Q(x , y) ∧ R(y)) ∧ ¬R(y))

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Transformation to prenex normal forms

THEOREM: Every first-order formula is equivalent to a formula in aprenex disjunctive normal form (PDNF) and to a formula in a prenexconjunctive normal form (PCNF).

Here is an algorithmic procedure:

1. Eliminate all occurrences of → and ↔.

2. Import all negations inside all other logical connectives.

3. Use the equivalences:

(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

to pull some quantifiers outwards and, after renaming one of the boundvariables if necessary.

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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(a) ∀xP ∧ ∀xQ ≡ ∀x(P ∧ Q),

(b) ∃xP ∨ ∃xQ ≡ ∃x(P ∨ Q),

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Transformation to prenex normal forms cont’d

4. To pull all quantifiers in front of the formula and thus transform it intoa prenex form, use the following equivalences, where x is not free in Q:

(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

If necessary, use renaming in order to apply these.

Example: ∀xP(x) ∧ ∃xQ(x) ≡ ∀x(P(x) ∧ ∃xQ(x)) ≡∀x(P(x) ∧ ∃yQ(y)) ≡ ∀x∃y(P(x) ∧ Q(y)).

Better: ∀xP(x) ∧ ∃xQ(x) ≡ ∃x(∀xP(x) ∧ Q(x)) ≡∃x(∀yP(y) ∧ Q(x)) ≡ ∃x∀y(P(y) ∧ Q(x)).

5. Finally, transform the matrix in a DNF or CNF, just like apropositional formula.

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Example: ∀xP(x) ∧ ∃xQ(x)

≡ ∀x(P(x) ∧ ∃xQ(x)) ≡∀x(P(x) ∧ ∃yQ(y)) ≡ ∀x∃y(P(x) ∧ Q(y)).

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Example: ∀xP(x) ∧ ∃xQ(x) ≡ ∀x(P(x) ∧ ∃xQ(x))

≡∀x(P(x) ∧ ∃yQ(y)) ≡ ∀x∃y(P(x) ∧ Q(y)).

Goranko

(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Example: ∀xP(x) ∧ ∃xQ(x) ≡ ∀x(P(x) ∧ ∃xQ(x)) ≡∀x(P(x) ∧ ∃yQ(y))

≡ ∀x∃y(P(x) ∧ Q(y)).

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Goranko

(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Better: ∀xP(x) ∧ ∃xQ(x)

≡ ∃x(∀xP(x) ∧ Q(x)) ≡∃x(∀yP(y) ∧ Q(x)) ≡ ∃x∀y(P(y) ∧ Q(x)).

Goranko

(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Better: ∀xP(x) ∧ ∃xQ(x) ≡ ∃x(∀xP(x) ∧ Q(x))

≡∃x(∀yP(y) ∧ Q(x)) ≡ ∃x∀y(P(y) ∧ Q(x)).

Goranko

(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

Better: ∀xP(x) ∧ ∃xQ(x) ≡ ∃x(∀xP(x) ∧ Q(x)) ≡∃x(∀yP(y) ∧ Q(x))

≡ ∃x∀y(P(y) ∧ Q(x)).

Goranko

(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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(c) ∀xP ∧ Q ≡ Q ∧ ∀xP ≡ ∀x(P ∧ Q),

(d) ∀xP ∨ Q ≡ Q ∨ ∀xP ≡ ∀x(P ∨ Q),

(e) ∃xP ∨ Q ≡ Q ∨ ∃xP ≡ ∃x(P ∨ Q),

(f) ∃xP ∧ Q ≡ Q ∧ ∃xP ≡ ∃x(P ∧ Q),

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Transformation to prenex normal forms: example

A = ∃z(∃xQ(x , z) ∨ ∃xP(x))→ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x)).

1. Eliminating →:A ≡ ¬∃z(∃xQ(x , z) ∨ ∃xP(x)) ∨ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x))2. Importing the negation:A ≡ ∀z(¬∃xQ(x , z) ∧ ¬∃xP(x)) ∨ (¬¬∃xP(x) ∨ ¬∀x∃zQ(z , x))≡ ∀z(∀x¬Q(x , z) ∧ ∀x¬P(x)) ∨ (∃xP(x) ∨ ∃x∀z¬Q(z , x)).3. Using the equivalences (a) and (b):A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃x(P(x) ∨ ∀z¬Q(z , x)).4. Renaming:A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃y(P(y) ∨ ∀w¬Q(w , y)).5. Using the equivalences (c)-(f) to pull the quantifiers in front:A ≡ ∀z∀x∃y∀w((¬Q(x , z) ∧ ¬P(x)) ∨ P(y) ∨ ¬Q(w , y)).

6. The resulting formula is in a prenex DNF.

For a prenex CNF we have to distribute the ∨ over ∧:A ≡∀z∀x∃y∀w((¬Q(x , z)∨P(y)∨¬Q(w , y))∧(¬P(x)∨P(y)∨¬Q(w , y))).

Goranko

1. Eliminating →:A ≡ ¬∃z(∃xQ(x , z) ∨ ∃xP(x)) ∨ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x))

2. Importing the negation:A ≡ ∀z(¬∃xQ(x , z) ∧ ¬∃xP(x)) ∨ (¬¬∃xP(x) ∨ ¬∀x∃zQ(z , x))≡ ∀z(∀x¬Q(x , z) ∧ ∀x¬P(x)) ∨ (∃xP(x) ∨ ∃x∀z¬Q(z , x)).3. Using the equivalences (a) and (b):A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃x(P(x) ∨ ∀z¬Q(z , x)).4. Renaming:A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃y(P(y) ∨ ∀w¬Q(w , y)).5. Using the equivalences (c)-(f) to pull the quantifiers in front:A ≡ ∀z∀x∃y∀w((¬Q(x , z) ∧ ¬P(x)) ∨ P(y) ∨ ¬Q(w , y)).

Goranko

1. Eliminating →:A ≡ ¬∃z(∃xQ(x , z) ∨ ∃xP(x)) ∨ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x))2. Importing the negation:A ≡ ∀z(¬∃xQ(x , z) ∧ ¬∃xP(x)) ∨ (¬¬∃xP(x) ∨ ¬∀x∃zQ(z , x))

≡ ∀z(∀x¬Q(x , z) ∧ ∀x¬P(x)) ∨ (∃xP(x) ∨ ∃x∀z¬Q(z , x)).3. Using the equivalences (a) and (b):A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃x(P(x) ∨ ∀z¬Q(z , x)).4. Renaming:A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃y(P(y) ∨ ∀w¬Q(w , y)).5. Using the equivalences (c)-(f) to pull the quantifiers in front:A ≡ ∀z∀x∃y∀w((¬Q(x , z) ∧ ¬P(x)) ∨ P(y) ∨ ¬Q(w , y)).

Goranko

1. Eliminating →:A ≡ ¬∃z(∃xQ(x , z) ∨ ∃xP(x)) ∨ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x))2. Importing the negation:A ≡ ∀z(¬∃xQ(x , z) ∧ ¬∃xP(x)) ∨ (¬¬∃xP(x) ∨ ¬∀x∃zQ(z , x))≡ ∀z(∀x¬Q(x , z) ∧ ∀x¬P(x)) ∨ (∃xP(x) ∨ ∃x∀z¬Q(z , x)).

3. Using the equivalences (a) and (b):A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃x(P(x) ∨ ∀z¬Q(z , x)).4. Renaming:A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃y(P(y) ∨ ∀w¬Q(w , y)).5. Using the equivalences (c)-(f) to pull the quantifiers in front:A ≡ ∀z∀x∃y∀w((¬Q(x , z) ∧ ¬P(x)) ∨ P(y) ∨ ¬Q(w , y)).

Goranko

1. Eliminating →:A ≡ ¬∃z(∃xQ(x , z) ∨ ∃xP(x)) ∨ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x))2. Importing the negation:A ≡ ∀z(¬∃xQ(x , z) ∧ ¬∃xP(x)) ∨ (¬¬∃xP(x) ∨ ¬∀x∃zQ(z , x))≡ ∀z(∀x¬Q(x , z) ∧ ∀x¬P(x)) ∨ (∃xP(x) ∨ ∃x∀z¬Q(z , x)).3. Using the equivalences (a) and (b):A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃x(P(x) ∨ ∀z¬Q(z , x)).

4. Renaming:A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃y(P(y) ∨ ∀w¬Q(w , y)).5. Using the equivalences (c)-(f) to pull the quantifiers in front:A ≡ ∀z∀x∃y∀w((¬Q(x , z) ∧ ¬P(x)) ∨ P(y) ∨ ¬Q(w , y)).

Goranko

1. Eliminating →:A ≡ ¬∃z(∃xQ(x , z) ∨ ∃xP(x)) ∨ ¬(¬∃xP(x) ∧ ∀x∃zQ(z , x))2. Importing the negation:A ≡ ∀z(¬∃xQ(x , z) ∧ ¬∃xP(x)) ∨ (¬¬∃xP(x) ∨ ¬∀x∃zQ(z , x))≡ ∀z(∀x¬Q(x , z) ∧ ∀x¬P(x)) ∨ (∃xP(x) ∨ ∃x∀z¬Q(z , x)).3. Using the equivalences (a) and (b):A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃x(P(x) ∨ ∀z¬Q(z , x)).4. Renaming:A ≡ ∀z∀x(¬Q(x , z) ∧ ¬P(x)) ∨ ∃y(P(y) ∨ ∀w¬Q(w , y)).

5. Using the equivalences (c)-(f) to pull the quantifiers in front:A ≡ ∀z∀x∃y∀w((¬Q(x , z) ∧ ¬P(x)) ∨ P(y) ∨ ¬Q(w , y)).

Goranko

For a prenex CNF we have to distribute the ∨ over ∧:

A ≡∀z∀x∃y∀w((¬Q(x , z)∨P(y)∨¬Q(w , y))∧(¬P(x)∨P(y)∨¬Q(w , y))).

Goranko

Skolemization I: Skolem constants

Skolemization: procedure for systematic elimination of the existentialquantifiers in a first-order formula in a prenex form, by introducing newconstant and functional symbols, called Skolem constants and Skolemfunctions, in the formula.

I Simple case: the result of Skolemization of the formula ∃x∀y∀zA isthe formula ∀y∀zA[c/x ], where c is a new (Skolem) constant.

.. For instance, the result of Skolemization of the formula∃x∀y∀z(P(x , y)→ Q(x , z)) is ∀y∀z(P(c , y)→ Q(c , z)).

I More generally, the result of Skolemization of the formula∃x1 · · · ∃xk∀y1 · · · ∀ynA is ∀y1 · · · ∀ynA[c1/x1, . . . , ck/xk ],where c1, . . . , ck are new (Skolem) constants.

Note that the resulting formula is not equivalent to the original one, butis equally satisfiable with it.

Goranko

.. For instance, the result of Skolemization of the formula∃x∀y∀z(P(x , y)→ Q(x , z))

is ∀y∀z(P(c , y)→ Q(c , z)).

Goranko

I More generally, the result of Skolemization of the formula∃x1 · · · ∃xk∀y1 · · · ∀ynA

is ∀y1 · · · ∀ynA[c1/x1, . . . , ck/xk ],where c1, . . . , ck are new (Skolem) constants.

Goranko

Skolemization II: Skolem functions

I The result of Skolemization of ∀y∃zP(y , z)

is ∀yP(y , f (y)),where f is a new unary function, called Skolem function.

I More generally, the result of Skolemization of∀y∃x1 · · · ∃xk∀y1 · · · ∀ynA is ∀y∀y1 · · · ∀ynA[f1(y)/x1, . . . , fk(y)/xk ],where f1, . . . , fk are new Skolem functions.

I The result of Skolemization of

∀x∃y∀z∃uA(x , y , z , u)

is∀x∀zA[x , f (x)/y , z , g(x , z)/u),

where f is a new unary Skolem function and g is a new binary Skolemfunction.

Goranko

I The result of Skolemization of ∀y∃zP(y , z) is ∀yP(y , f (y)),where f is a new unary function, called Skolem function.

∀x∃y∀z∃uA(x , y , z , u)

is∀x∀zA[x , f (x)/y , z , g(x , z)/u),

Goranko

I More generally, the result of Skolemization of∀y∃x1 · · · ∃xk∀y1 · · · ∀ynA

is ∀y∀y1 · · · ∀ynA[f1(y)/x1, . . . , fk(y)/xk ],where f1, . . . , fk are new Skolem functions.

∀x∃y∀z∃uA(x , y , z , u)

is∀x∀zA[x , f (x)/y , z , g(x , z)/u),

Goranko

∀x∃y∀z∃uA(x , y , z , u)

is∀x∀zA[x , f (x)/y , z , g(x , z)/u),

Goranko

∀x∃y∀z∃uA(x , y , z , u)

is∀x∀zA[x , f (x)/y , z , g(x , z)/u),

Goranko

∀x∃y∀z∃uA(x , y , z , u)

is∀x∀zA[x , f (x)/y , z , g(x , z)/u),

Goranko

Skolemization III: the general case

I In the general case of Skolemization, the existential quantifiers areeliminated one by one, from left to right, by introducing at every step aSkolem function depending on all existentially quantified variables to theleft of the existential quantifier that is being eliminated:

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

where f is a new k-ary Skolem function.

Thus, eventually, all existential quantifiers are eliminated.

Again, the resulting formula after Skolemization is generally notequivalent to the original one, but is equally satisfiable with it.

Goranko

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

Goranko

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

Goranko

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

Goranko

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

Goranko

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

Goranko

∀x1 . . . ∀xk∃yA(x1, . . . , xk , y , . . .)

is transformed to

∀x1 . . . ∀xkA(x1, . . . , xk , y , . . .)[f (x1, . . . , xk)/y ]

Goranko

Clausal form of first-order formulae

A literal is an atomic formula or a negation of an atomic formula.

Examples: P(x), ¬P(f (c , g(y))), ¬Q (f (x , g(c)), g(g(g(y)))).

A clause is a set of literals.

Example: {P(x),¬P(f (c , g(y))),¬Q(f (x , g(c)), g(g(g(y))))}.

A clausal form is a set of clauses.

Example:{{P(x)},{¬P(f (c)),¬Q(g(x , x), y)},{¬P(f (y)),P(f (c)),Q(y , f (x))}}.

Goranko

Examples: P(x),

¬P(f (c , g(y))), ¬Q (f (x , g(c)), g(g(g(y)))).

Goranko

Examples: P(x), ¬P(f (c , g(y))),

¬Q (f (x , g(c)), g(g(g(y)))).

Goranko

Example:{

{P(x)},{¬P(f (c)),¬Q(g(x , x), y)},{¬P(f (y)),P(f (c)),Q(y , f (x))}}.

Goranko

Example:{{P(x)},

{¬P(f (c)),¬Q(g(x , x), y)},{¬P(f (y)),P(f (c)),Q(y , f (x))}}.

Goranko

Example:{{P(x)},{¬P(f (c)),¬Q(g(x , x), y)},

{¬P(f (y)),P(f (c)),Q(y , f (x))}}.

Goranko

Example:{{P(x)},{¬P(f (c)),¬Q(g(x , x), y)},{¬P(f (y)),P(f (c)),Q(y , f (x))}

Goranko

The logical meaning of first-order clauses

All variables in a clause are assumed to be universally quantified.

Thus, a clause represents the universal closure of the disjunction ofliterals in it.

Example: {P(x),¬P(f (c , g(y))),¬Q(f (x , g(c)), g(g(g(y))))}

represents

∀x∀y(P(x) ∨ ¬P(f (c , g(y))) ∨ ¬Q(f (x , g(c)), g(g(g(y))))

)The universal quantifiers will hereafter be omitted.

Goranko

represents

Goranko

represents

Goranko

represents

Goranko

represents

Goranko

represents

The universal quantifiers will hereafter be omitted.

Goranko

represents

Goranko

The logical meaning of sets of first-order clauses

A set of clauses represents the conjunction of the (formulae representedby the) clauses contained in it.

Example: the set{{P(x)}, {¬P(f (c)),¬Q(x , y)}, {¬P(f (y)),Q(y , f (x))}

}represents the formula

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

Hereafter we will assume that no two clauses in a clausal form sharecommon variables, which can always be achieved by means of renaming.Thus, the clausal form above can be re-written as:{{P(x)}, {¬P(f (c)),¬Q(x1, y1)}, {¬P(f (y2)),Q(y2, f (x2))}

}representing the formula

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

Because ∀ distributes over ∧, after the renaming the clausal form alsorepresent the universal closure of the conjunction of the disjunctionsrepresented by the clauses contained in it:

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

The logical meaning of sets of first-order clausesA set of clauses represents the conjunction of the (formulae representedby the) clauses contained in it.

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

represents the formula

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

Hereafter we will assume that no two clauses in a clausal form sharecommon variables, which can always be achieved by means of renaming.

Thus, the clausal form above can be re-written as:{{P(x)}, {¬P(f (c)),¬Q(x1, y1)}, {¬P(f (y2)),Q(y2, f (x2))}

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

∀xP(x) ∧ ∀x∀y(¬P(f (c)) ∨ ¬Q(x , y)

)∧ ∀x∀y

(¬P(f (y)) ∨ Q(y , f (x))

∀xP(x) ∧ ∀x1∀y1(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧ ∀x2∀y2

(¬P(f (y2)) ∨ Q(y2, f (x2))

∀x∀x1∀y1∀x2∀y2(P(x) ∧

(¬P(f (c)) ∨ ¬Q(x1, y1)

)∧(¬P(f (y2)) ∨ Q(y2, f (x2))

Goranko

Transformation of first-order formulae to clausal form

Theorem: Every set of first-order formulae {A1, . . . ,An} can betransformed to a set of clauses {C1, . . . ,Ck} where no two clauses sharecommon variables, such that {A1, . . . ,An} is equally satisfiable with theuniversal closure (C1 ∧ · · · ∧ Ck) of the conjunction of all clauses, eachtaken as disjunction of its literals.

The algorithm applies to each formula A ∈ {A1, . . . ,An} as follows:

1. Transform A to a prenex CNF.

2. Eliminate all existential quantifiers by introducing Skolem constantsor functions.

3. Remove all universal quantifiers.

4. Write the matrix (which is in CNF) as a set of clauses.

Finally, apply in the union of all sets of clauses produced as aboverenaming of variables occurring in more than one clause.

Goranko

Logic as a Tool 3mm Chapter 4: Deductive Reasoning in ......October 2016. Goranko Revision: CNF and...

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