Logic: First Order Logic (Part I)

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Logic: First Order Logic (Part I) Alessandro Artale Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof. Enrico Franconi for provoding the slides

Motivation We can already do a lot with propositional logic. But it is unpleasant that we cannot access the structure of atomic sentences. Atomic formulas of propositional logic are too atomic they are just statement which may be true or false but which have no internal structure. In First Order Logic (FOL) the atomic formulas are interpreted as statements about relationships between objects.

Predicates and Constants Let s consider the statements: Mary is female John is male Mary and John are siblings In propositional logic the above statements are atomic propositions: Mary-is-female John-is-male Mary-and-John-are-siblings In FOL atomic statements use predicates, with constants as argument: Female(mary) Male(john) Siblings(mary,john)

Variables and Quantifiers Let s consider the statements: Everybody is male or female A male is not a female In FOL, predicates may have variables as arguments, whose value is bounded by quantifiers: x. Male(x) Female(x) x. Male(x) Female(x) Deduction (why?): Mary is not male Male(mary)

Functions Let s consider the statement: The father of a person is male In FOL objects of the domain may be denoted by functions applied to (other) objects: x. Male(father(x))

Syntax of FOL: Terms and Atomic Sentences Countably infinite supply of symbols (signature): variable symbols: x, y, z,... n-ary function symbols: f, g, h,... individual constants: a, b, c,... n-ary predicate symbols: P, Q, R,... Terms: t x variable a constant f (t 1,..., t n ) function application Ground terms: terms that do not contain variables Formulas: φ P(t 1,..., t n ) atomic formulas E.g., Brother(kingJohn, richardthelionheart) > (length(leftlegof (richard)), length(leftlegof (kingjohn)))

Syntax of FOL Formulas Formulas: φ, ψ P(t 1,..., t n ) atomic formulas false true φ negation φ ψ conjunction φ ψ disjunction φ ψ implication φ ψ equivalence x. φ universal quantification x. φ existential quantification E.g. Everyone in England is smart: x. In(x, england) Smart(x) Someone in France is smart: x. In(x, france) Smart(x)

Grounding FOL Formulas A ground term is a term which does not contain any variable. E.g., succ(1, 2) is a ground function. A ground atomic formula is an atomic formula, all of whose terms are ground. E.g., Sibling(kingJohn, richard) is a ground atom. A ground literal is a ground atomic formula or the negation of one. A ground formula is a quantifier-free formula all of whose atomic formulas are ground. E.g., Sibling(kingJohn, richard) Sibling(richard, kingjohn).

Summary of Syntax of FOL Terms variables constants functions Literals atomic formulas relation (predicate) negation of atomic formulas Well formed formulas truth-functional connectives existential and universal quantifiers

Semantics of FOL: Intuitions Just like in propositional logic, a (complex) FOL formula may be true (or false) with respect to a given interpretation. An interpretation specifies referents for constant symbols objects predicate symbols relations function symbols functional relations An atomic sentence P(t 1,..., t n ) is true in a given interpretation iff the objects referred to by t 1,..., t n are in the relation referred to by the predicate P. An interpretation in which a formula is true is called a model for the formula.

Models for FOL: Example { { objects relations: < sets of tuples of objects < < {,,,,... < functional < relations: all tuples of objects < + "value" object < {,,,,... <

Semantic of FOL: Interpretations Interpretation: I = D, I where D is an arbitrary non-empty set and I is a function that maps individual constants to elements of D: a I D n-ary function symbols to functions over D: f I [D n D] n-ary predicate symbols to relation over D: P I D n

Semantic of FOL: Ground Terms and Atoms Interpretation of ground terms: f (t 1,..., t n ) I = f I (t 1 I,..., t n I ) ( D) Satisfaction of ground atoms P(t 1,..., t n ): I = P(t 1,..., t n ) iff t 1 I,..., t n I P I

Examples D = {d 1,..., d n, n > 1} a I = d 1 b I = d 2 Block I = {d 1 } Red I = D

Examples D = {d 1,..., d n, n > 1} a I = d 1 b I = d 2 Block I = {d 1 } Red I = D I = Red(b) I = Block(b)

Examples D = {d 1,..., d n, n > 1} a I = d 1 b I = d 2 Block I = {d 1 } Red I = D I = Red(b) I = Block(b) D = {1, 2, 3,...} 1 I = 1 2 I = 2. Even I = {2, 4, 6,...} succ I = {(1 2), (2 3),...}

Examples D = {d 1,..., d n, n > 1} a I = d 1 b I = d 2 Block I = {d 1 } Red I = D I = Red(b) I = Block(b) D = {1, 2, 3,...} 1 I = 1 2 I = 2. Even I = {2, 4, 6,...} succ I = {(1 2), (2 3),...} I = Even(3) I = Even(succ(3))

Semantics of FOL: Variable Assignments Let V be the set of all variables. A Variable Assignment is a function α : V D. Notation: α[x/d] is a variable assignment identical to α except for the variable x mapped to d. Interpretation of terms under I, α: x I,α = α(x) a I,α = a I f (t 1,..., t n ) I,α = f I (t 1 I,α,..., t n I,α ) Satisfiability of atomic formulas: I, α = P(t 1,..., t n ) iff t 1 I,α,..., t n I,α P I

Variable Assignment example D = {d 1,..., d n, n > 1} a I = d 1 b I = d 2 Block I = {d 1 } Red I = D α = {(x d 1 ), (y d 2 )} I, α = Red(x) I, α[y/d 1 ] = Block(y)

Semantics of FOL: Satisfiability of formulas A formula φ is satisfied by (is true in) an interpretation I under a variable assignment α, in symbols I, α = φ I, α = P(t 1,..., t n ) iff t 1 I,α,..., t n I,α P I I, α = φ iff I, α = φ I, α = φ ψ iff I, α = φ and I, α = ψ I, α = φ ψ iff I, α = φ or I, α = ψ I, α = x. φ iff for all d D : I, α[x/d] = φ I, α = x. φ iff there exists a d D : I, α[x/d] = φ

Examples D = {d 1,..., d n, } n > 1 a I = d 1 b I = d 1 c I = d 2 Block I = {d 1 } Red I = D α = {(x d 1 ), (y d 2 )} 1 I, α = Block(c) Block(c)?

Examples D = {d 1,..., d n, } n > 1 a I = d 1 b I = d 1 c I = d 2 Block I = {d 1 } Red I = D α = {(x d 1 ), (y d 2 )} 1 I, α = Block(c) Block(c)? 2 I, α = Block(x) Block(x) Block(y)?

Examples D = {d 1,..., d n, } n > 1 a I = d 1 b I = d 1 c I = d 2 Block I = {d 1 } Red I = D α = {(x d 1 ), (y d 2 )} 1 I, α = Block(c) Block(c)? 2 I, α = Block(x) Block(x) Block(y)? 3 I, α = x. Block(x) Red(x)?

Examples D = {d 1,..., d n, } n > 1 a I = d 1 b I = d 1 c I = d 2 Block I = {d 1 } Red I = D α = {(x d 1 ), (y d 2 )} 1 I, α = Block(c) Block(c)? 2 I, α = Block(x) Block(x) Block(y)? 3 I, α = x. Block(x) Red(x)? { Block(a), Block(b) 4 Θ = x (Block(x) Red(x)) I, α = Θ? }

Example Find a model of the formula: y. [ P(y) Q(y) ] z. [ P(z) Q(z) ]

Example Find a model of the formula: y. [ P(y) Q(y) ] z. [ P(z) Q(z) ] Possible Solution. = {a, b} P I = {a} Q I = {b}

Satisfiability and Validity An interpretation I is a model of φ under α, if I, α = φ. Similarly as in propositional logic, a formula φ can be satisfiable, unsatisfiable, falsifiable or valid the definition is in terms of the pair (I, α). A formula φ is satisfiable, if there is some (I, α) that satisfies φ; unsatisfiable, if φ is not satisfiable; valid (i.e., a tautology), if every (I, α) is a model of φ; falsifiable, if there is some (I, α) that does not satisfy φ.

Equivalence Analogously, two formulas are logically equivalent (φ ψ), if for all (I, α) we have: Note: P(x) P(y)! I, α = φ iff I, α = ψ

Free and Bound Variables x. (R( y, z ) y. ( P(y,x) R(y, z ))) Variables in boxes are free; other variables are bound. Definition. The free variables of a formula are inductively defined over the structure of formulas (structural induction): free(x) = {x} free(a) = free(f (t 1,..., t n )) = free(t 1 )... free(t n ) free(p(t 1,..., t n )) = free(t 1 )... free(t n ) free( φ) = free(φ) free(φ ψ) = free(φ) free(ψ), =,,... free( x. φ) = free(φ) {x} free( x. φ) = free(φ) {x}

Open and Closed Formulas A formula is closed or a sentence if no free variables occurs in it. Viceversa, the formula is said open. Note: For closed formulas, the properties logical equivalence, satisfiability, entailment etc. do not depend on variable assignments: If the property holds for one variable assignment then it holds for all of them. Thus, For closed formulas, the symbol α on the left hand side of the = sign is omitted: I = φ Note: Unless specified, in the following we consider closed formulas.

Entailment Entailment is defined similarly as in propositional logic. Definition. The formula φ is logically implied by a formula ψ, if φ is true in all models of ψ (symbolically, ψ = φ): ψ = φ iff I = φ, for all models I of ψ

More Exercises = x. (P(x) P(x))

More Exercises = x. (P(x) P(x)) x. [ P(x) (P(x) Q(x)) ] = x. Q(x)

More Exercises = x. (P(x) P(x)) x. [ P(x) (P(x) Q(x)) ] = x. Q(x) = [ x. y. ( P(x) Q(y) ) ]

More Exercises = x. (P(x) P(x)) x. [ P(x) (P(x) Q(x)) ] = x. Q(x) = [ x. y. ( P(x) Q(y) ) ] y. [ P(y) Q(y) ] z. [ P(z) Q(z) ] satisfiable

Equality Equality is a special predicate. Definition. Given two terms, t 1, t 2, t 1 = t 2 is true under a given interpretation, I, α = t 1 = t 2, if and only if t 1 and t 2 refer to the same object: t 1 I,α = t 2 I,α Consider the following examples: x. ( (sqrt(x), sqrt(x)) = x), is satisfiable 2 = 2, is valid Definition of (full) Sibling in terms of Parent: x, y. Sibling(x, y) ( (x = y) m, f. (m = f ) Parent(m, x) Parent(f, x) Parent(m, y) Parent(f, y) )

Universal quantification Everyone in England is smart: x. LivesIn(x, england) Smart(x) ( x. φ) is equivalent to the conjunction of all possible instantiations of x in φ: LivesIn(kingJohn, england) Smart(kingJohn) LivesIn(richard, england) Smart(richard) LivesIn(england, england) Smart(england)... Note. Typically, is the main connective with. Common mistake: using as the main connective with : x. LivesIn(x, england) Smart(x) means Everyone lives in England and everyone is smart

Existential quantification Someone in France is smart: x. LivesIn(x, france) Smart(x) ( x. φ) is equivalent to the disjunction of all possible instantiations of x in φ: LivesIn(kingJohn, france) Smart(kingJohn) LivesIn(richard, france) Smart(richard) LivesIn(france, france) Smart(france)... Note. Typically, is the main connective with. Common mistake: using as the main connective with : x. LivesIn(x, france) Smart(x) is true if there is anyone who is not in France!

Logical Equivalences in FOL Commutativity ( x. y. φ) ( y. x. φ) ( x. y. φ) ( y. x. φ) ( x. y. φ) ( y. x. φ) and commute only in one direction = ( x. y. φ) ( y. x. φ) x. y. Loves(x, y) There is a person who loves everyone in the world, then y. x. Loves(x, y) Everyone in the world is loved by at least one person Quantifier duality: each can be expressed using the other. x. Likes(x, icecream) x. Likes(x, icecream) x. Likes(x, broccoli) x. Likes(x, broccoli)

Logical Equivalences in FOL (cont.) Quantification distributes if the variable is not free. ( x. φ) ψ x. (φ ψ) if x not free in ψ ( x. φ) ψ x. (φ ψ) if x not free in ψ ( x. φ) ψ x. (φ ψ) if x not free in ψ ( x. φ) ψ x. (φ ψ) if x not free in ψ distributes over - distributes over x. (φ ψ) x. φ x. ψ x. (φ ψ) x. φ x. ψ

Logical Equivalences in FOL (cont.) Quantification over Implication. x. (φ ψ(x)) φ x. ψ(x) if x is not free in φ x. (φ(x) ψ) ( x. φ(x)) ψ if x is not free in ψ x. (φ(x) ψ(x)) ( x. φ(x) x. ψ(x))

Exercises Show the following: x. φ x. φ (De Morgan) x. φ x. φ (De Morgan) = ( y. x. φ) ( x. y. φ) = x. φ x. ψ x(φ ψ) = x(φ ψ) x. φ x. ψ = x(φ ψ) x. φ x. ψ = x. φ x. ψ x(φ ψ) = ( x. φ(x) x. ψ(x)) x(φ(x) ψ(x)) = x(φ(x) ψ(x)) ( x. φ(x) x. ψ(x)) = x(φ(x) ψ(x)) ( x. φ(x) x. ψ(x))

The Prenex Normal Form Quantifier prefix + (quantifier free) matrix x 1 x 2 x 3... x n φ 1 Elimination of and 2 push inwards 3 pull quantifiers outwards E.g. x. (( x. p(x)) q(x)) x. ( ( x. p(x)) q(x)) x. (( x. p(x)) q(x)) and now? Definition: renaming of variables. Let φ[x/t] be the formula φ where all occurrences of x have been replaced by the term t.

The Prenex Normal Form: Theorem Lemma. Let y be a variable that does not occur in φ. Then we have xφ ( xφ)[x/y] and xφ ( xφ)[x/y]. Theorem. There is an algorithm that computes for every formula its equivalent prenex normal form: 1 Rename bound variables; 2 Eliminate and ; 3 Push inwards; 4 Extract quantifiers outwards.

The Prenex Normal Form: Example Original formula x y. p(x, y) y x. p(x, y) Rename bound variables x y. p(x, y) w z. p(z, w) Eliminate and x y. p(x, y) w z. p(z, w) Push inwards x y. p(x, y) w z. p(z, w) Extract quantifiers outwards x y w z. p(x, y) p(z, w)

FOL at work: reasoning by cases Θ = FRIEND(john,susan) FRIEND(john,andrea) LOVES(susan,andrea) LOVES(andrea,bill) Female(susan) Female(bill) FRIEND FRIEND andrea LOVES susan: Female bill: LOVES john Female

FOL at work: reasoning by cases (cont.) FRIEND FRIEND andrea LOVES susan: Female bill: LOVES john Female Entailment: Does John have a female friend loving a male (i.e., not female) person?

FOL at work: reasoning by cases (cont.) FRIEND FRIEND andrea LOVES susan: Female bill: LOVES john Female Entailment: Does John have a female friend loving a male (i.e., not female) person? YES! Θ = X, Y. FRIEND(john, X ) Female(X ) LOVES(X, Y ) Female(Y )

FOL at work: reasoning by cases (cont.) In all models where andrea is not a Female, then: FRIEND LOVES [ Female] andrea bill: LOVES john Female FRIEND susan: Female FRIEND(john,susan), Female(susan), LOVES(susan,andrea), Female(andrea)

FOL at work: reasoning by cases (cont.) In all models where andrea is a Female, then: john FRIEND FRIEND [Female] andrea LOVES susan: Female LOVES bill: Female FRIEND(john,andrea), Female(andrea), LOVES(andrea,bill), Female(bill)

Theories and Models Θ 1 = FRIEND(john, susan) FRIEND(john, andrea) LOVES(susan, andrea) LOVES(andrea, bill) Female(susan) Male(bill) X. Male(X ) Female(X ) john FRIEND FRIEND andrea LOVES susan: Female LOVES bill: Male Male =. Female

Theories and Models (cont.) john FRIEND FRIEND andrea LOVES susan: Female LOVES bill: Male Male =. Female Entailment: Does John have a female friend loving a male person? Θ 1 = X, Y. FRIEND(john, X ) Female(X ) LOVES(X, Y ) Male(Y )

Theories and Models (cont.) Θ = FRIEND(john,susan) FRIEND(john,andrea) LOVES(susan,andrea) LOVES(andrea,bill) Female(susan) Female(bill) Θ 1 = FRIEND(john, susan) FRIEND(john, andrea) LOVES(susan, andrea) LOVES(andrea, bill) Female(susan) Male(bill) X. Male(X ) Female(X )

Theories and Models (cont.) Θ = FRIEND(john,susan) FRIEND(john,andrea) LOVES(susan,andrea) LOVES(andrea,bill) Female(susan) Female(bill) Θ 1 = FRIEND(john, susan) FRIEND(john, andrea) LOVES(susan, andrea) LOVES(andrea, bill) Female(susan) Male(bill) X. Male(X ) Female(X ) = {john, susan, andrea, bill} Female I = {susan}

Theories and Models (cont.) Θ = FRIEND(john,susan) FRIEND(john,andrea) LOVES(susan,andrea) LOVES(andrea,bill) Female(susan) Female(bill) Θ 1 = FRIEND(john, susan) FRIEND(john, andrea) LOVES(susan, andrea) LOVES(andrea, bill) Female(susan) Male(bill) X. Male(X ) Female(X ) = {john, susan, andrea, bill} Female I = {susan} I 1 = {john, susan, andrea, bill} Female I 1 = {susan, andrea} Male I 1 = {bill, john} I 2 = {john, susan, andrea, bill} Female I 2 = {susan} Male I 2 = {bill, andrea, john} I 1 = {john, susan, andrea, bill} Female I 1 = {susan, andrea, john} Male I 1 = {bill} I 2 = {john, susan, andrea, bill} Female I 2 = {susan, john} Male I 2 = {bill, andrea}

Theories and Models (cont.) The following entailments hold: Θ = Female(andrea) Θ = Female(andrea) Θ 1 = Female(andrea) Θ 1 = Female(andrea) Θ 1 = Male(andrea) Θ 1 = Male(andrea)

Exercise? 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 Is it true that the top block is on a white block touching a black block?

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