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Conjunctive Normal Form Lecture 16: Normal Forms • Can be helpful if interested in validity: • A formula is in conjunctive normal form if it is generated by the following grammar. CSCI 81 Spring, 2012 • CNF ::= C | C ∧ CNF C ::=L | L ∨ C L ::=⊥ | ⊤ | P | ¬ P P ::=p|q|r|... Kim Bruce ca!ed literals • For example, (¬q ∨ p ∨ r) ∧ (¬p ∨ r) ∧ q CNF • Any formula can be transformed into CNF: • See algo in book p q r ψ Τ Τ T T F F F F Τ Τ F F T T F F Τ F T F T F T F Τ F T T F T F T • Can also read oﬀ truth table • Take false lines in truth table and negate prop letters: ψ ≈ (¬p∨¬q∨r) ∧(p ∨ ¬q∨¬r)∧(p∨q∨¬r) Note ψ fails iﬀ one of conjuncts fails. Validity Horn Formulas • A Horn formula is one that is constructed from • Disjunction of literals valid iﬀ contains some p and its negation • ... p ∨...∨ ¬p ... • To determine if fmla in CNF valid, check all conjuncts. the following grammar, with start symbol H. • H ::= C | C ∧ H C ::=A→P A ::= P | P ∧ A P ::=⊥ | ⊤ p | q | r |... • Note that ⊤ → p is equivalent to p, and p → ⊥ to ¬p. • Example: (p ∧ q ∧ s → p) ∧ (q ∧ r → p) ∧ (p ∧ s → s) Satisfiability for Horn Prolog • Suppose that F is a Horn formula. Let P be the collection of all subformulas of F. • Mark ⊤ if it occurs in the list P. • If P0 ∧P1 ∧ ... ∧Pk →Q is a clause in F, and all of the Pi’s are marked, then mark Q. Repeat until there are no more Q’s to mark. • If ⊥ is marked, declare F to be unsatisfiable. If ⊥ is not marked, then declare F to be satisfiable. • Get V by making all marked symbols T and others F. • Revise syntax from P0 ∧ P1 ∧ ... ∧ Pk → Q to Q, if P0 ∧ P1 ∧ ... ∧ Pk to Q :- P0 , P1 , ... , Pk • Do more after predicate logic. Predicate Logic Syntax • Symbols needed include • variables: x, y, ... • • constant symbols: c, d, ... Propositional Logic not expressive enough. • Definition of limx→a f(x) = L iﬀ for all ε > 0 there is a δ >0 s.t. for all x s.t. |x-a| < δ, then |f(x)-L| < ε. • Statement of pumping lemma, etc. • k-ary function symbols: fk, gk, ... for all k • k-ary predicate symbols: Pk, Qk, ... for all k • parentheses: (, ) • truth constants ⊥, ⊤ • quantifiers: ∃, ∀ • logical connectives: ¬, ∧, ∨, → Terms & Formulas • A term is a variable, a constant symbol, or a term of the form f(Term1,...,Termk) for f a k-ary function symbol. • Atomic formulas are of the form P(Term1,...,Termk) for P a k-ary relation • Formulas: • An atomic formula is a formula • If φ, ψ are formulas, so are (¬φ), (φ∧ψ), (φ∨ψ), and (φ→ψ) • If φ is a formula and x is a variable, so are (∀xφ), (∃xφ) Examples • Let L(x,y) stand for x loves y. • “Everybody loves somebody” • ∀x.∃y.L(x,y) • “Someone loves everyone” • Ambiguous: ∃x.∀y.L(x,y) or ∀y. ∃x.L(x,y) • “Jane’s mother loves her” • L(mother(Jane),Jane) where mother() is a function Limiting Domains Free & Bound Variables • Historically confusing! • Every person hates a wall • ∀x. (P(x) → ∃y. H(x,y) ∧ W(y)) • There is a wall that is hated by all people. • ∃y. W(y) ∧ ∀x. (P(x) → H(x,y)) • In Love(x,y) the variables x and y are free • the meaning of the wﬀ depends on the meaning of x,y • In ∀x.∃y.L(x,y) occurrences of x and y are bound by the quantifiers. • Meaning does not depend on meanings of x, y. Free Variables Examples • An occurrence of x in φ is free in φ if it is a leaf node in the parse tree of φ such that there is no path upwards from that node x to a node ∀x or ∃x. • Otherwise, that occurrence of x is called bound. • For ∀xφ, or ∃xφ, we say that φ – minus any of φ’s subformulas ∃x ψ, or ∀x ψ – is the scope of ∀x, respectively ∃x. • (∀x.∃y.L(x,y)) ∧ H(x,y) • Some occurrences free and some bound. Substitution • Define φ[t/x] to be the formula obtained by replacing each free occurrence of variable x in φ with t. • Expect ∀x.φ(x) ⇒ φ[t/x] for every term t • What about ∀x.∃y.L(x,y) ⇒ ∃y.L(y,y) ? • Say that t is free for x in φ if no free x leaf in φ occurs in the scope of ∀y or ∃y for any variable y occurring in t. • y not free for x in ∃y.L(x,y) • Only allow substitution φ[t/x] if t free for x in φ