Logic as a Tool Chapter 4: Deductive Reasoning in First-Order Logic - - PowerPoint PPT Presentation

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Logic as a Tool Chapter 4: Deductive Reasoning in First-Order Logic 4.4 Prenex normal form.

• Skolemization. Clausal form.

Valentin Goranko Stockholm University October 2016

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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 elementary

conjunctions.

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

disjunctions.

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Conjunctive and disjunctive normal forms

• f 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 propositional formula in DNF (resp. CNF), obtained by uniform substitution of atomic formulae 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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Prenex normal forms

A first-order formula Q1x1...QnxnA, where Q1, ..., Qn are quantifiers and A 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 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 conjunctive normal 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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Transformation to prenex normal forms

THEOREM: Every first-order formula is equivalent to a formula in a prenex disjunctive normal form (PDNF) and to a formula in a prenex conjunctive 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 bound variables if necessary.

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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 into

a 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 a

propositional formula.

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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))).

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Skolemization I: Skolem constants

Skolemization: procedure for systematic elimination of the existential quantifiers in a first-order formula in a prenex form, by introducing new constant and functional symbols, called Skolem constants and Skolem functions, in the formula. ◮ Simple case: the result of Skolemization of the formula ∃x∀y∀zA is the 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)). ◮ 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, but is equally satisfiable with it.

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Skolemization II: Skolem functions

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

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Skolemization III: the general case

◮ In the general case of Skolemization, the existential quantifiers are eliminated one by one, from left to right, by introducing at every step a Skolem function depending on all existentially quantified variables to the left 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 not equivalent to the original one, but is equally satisfiable with it.

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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))} }.

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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 of literals 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.

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The logical meaning of sets of first-order clauses

A set of clauses represents the conjunction of the (formulae represented by 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 share common 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 also represent the universal closure of the conjunction of the disjunctions represented 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))
• .

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Transformation of first-order formulae to clausal form

Theorem: Every set of first-order formulae {A1, . . . , An} can be transformed to a set of clauses {C1, . . . , Ck} where no two clauses share common variables, such that {A1, . . . , An} is equally satisfiable with the universal closure (C1 ∧ · · · ∧ Ck) of the conjunction of all clauses, each taken 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 constants
• r 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 above renaming of variables occurring in more than one clause.