Logics for Data and Knowledge Representation

Logics for Data and KnowledgeRepresentation Introduction to Algebra Chiara Ghidini, Luciano Serafini, Fausto Giunchiglia and Vincenzo Maltese

Roadmap • Set theory • Basic notions • Operations • Properties • Relations • Functions

Describing the world individuals Cita Monkey sets relations Eats Hunts Kimba Simba Lion Near 3

Sets SETS :: RELATIONS :: FUNCTIONS • A set is a collection of elements • The description of a set must be unambiguous and unique: it must be possible to decide whether an element belongs to the set or not. 3 1 5 7 9 The set of odd numbers < 10 The set of students in this room The set of lions in a certain zoo

Describing sets SETS :: RELATIONS :: FUNCTIONS A = {1, 3, 5, 7, 9} A = { x | x is an odd number < 10} A • Listing: the set is described by listing all its elements • Abstraction: the set is described through a common property of its elements • Venn Diagrams: graphical representation that supports the formal description 3 1 5 7 9

Basic notions on sets SETS :: RELATIONS :: FUNCTIONS • Empty Set: the set with no elements; A = { } A =  • Membership: element a belongs to the set A; A = {a, b, c} a  A • Non membership: element a doesn't belong to the set A A = {b, c} a  A • Equality: the sets A and B contain the same elements; A = {b, c}; B = {b, c} A = B

Basic notions on sets (cont.) SETS :: RELATIONS :: FUNCTIONS • Inequality: the sets A and B contain the same elements; A = {c}; B = {b, c} A ≠ B • Subset: all elements of A belong to B; A = {c}; B = {b, c} A B • Proper subset: all elements of A belong to B and they are not the same A B and A ≠ B then A  B • Power set: the set of all the subsets of A A = {a, b} P(A) = {, {a}, {b}, {a, b}} |A| = n then |P(A)| = 2n

Operations on sets SETS :: RELATIONS :: FUNCTIONS • Union: the set containing the the members of A or B • Intersection: the set containing the members of both A and B A B A  B a c d b A B A  B a c d b

Operations on sets (cont.) SETS :: RELATIONS :: FUNCTIONS • Difference: the set containing the members of A and not of B • Complement: given a universal set U, the complement of A is the set whose members are the members of U - A. A B A - B a c d b U _ A A

Exercises SETS :: RELATIONS :: FUNCTIONS • Given A = {t, z} and B = {v, z, t}, say whether the following statements are true or false: • A  B • A  B • z  A B • v  B • {v}  B • v A- B • Given A = {a, b, c, d} and B = {c, d, f} • Find a set X such that A  B = B  X. Is this set unique? • Is there any set Y such that A  Y = B ?

Properties of sets SETS :: RELATIONS :: FUNCTIONS • A A = A A A = A • A  =  A  = A • A B = B  A AB = B  A (commutative) • (A B)  C = A (B  C) (AB) C = A(B C) (associative) • A (B  C) = (A B)  (A  C) A (B  C) = (A B)  (A  C) (distributive) _____ _ _ • A B = A  B _____ _ _ A  B = A  B (De Morgan laws)

Cartesian product SETS :: RELATIONS :: FUNCTIONS • Cartesian product of A and B: the set of ordered couples (a, b) where a is a member of A and b a member of B A x B = {(a, b) : a  A and b  B} • Notice that A x B ≠ B x A • Example: A = {a, b, c}, B = {s, t} A x B = {(a, s), (a, t), (b, s), (b, t), (c, s), (c, t)}

Relations SETS :: RELATIONS :: FUNCTIONS • A (binary) relation R from set A to set B is a subset of A x BR  A x B xRy indicates that (x, y)  R • The domain of R is the set Dom(R) = {a  A | ∃ b  B s.t. aRb} • The co-domain of R is the set Cod(R) = {b  B | ∃ a  A s.t. aRb} A B a b (a,b) ∈ R

Relations (cont.) SETS :: RELATIONS :: FUNCTIONS • An n-ary relation Rn is a subset of A1 x … x An n is the arity of the relation • The inverse relation of R  A x B is the relation R-1B x A where: R-1 = {(b, a) | (a, b)  R} A B a b (b, a) ∈ R-1

Properties of relations SETS :: RELATIONS :: FUNCTIONS Let R be a binary relation on A, i.e. R  A x A. R is said to be: • reflexiveiffaRa ∀ a  A; • symmetriciffaRb implies bRa∀ a, b  A; • transitiveiffaRb and bRc imply aRc ∀ a, b, c  A; • anti-symmetric iffaRb and bRa imply a = b ∀ a, b  A;

Equivalence relations SETS :: RELATIONS :: FUNCTIONS • Given R  A x A, R is an equivalence relation iffit is reflexive, symmetric and transitive. • A partitionof a set A is a family F of non-empty subsets of A s.t.: • the subsets are pairwise disjoint • the union of all the subsets is the set A Notice that each element of A belongs to exactly one subset in F. • Given ≡ equivalence relation on A and a  A, the equivalence classof a is the set [a] = {x | a ≡ x} Notice that if x  [a] then [x] = [a] • The quotient set of A w.r.t. ≡ is the set {[x] | x  A} which defines a partition of A.

Order relations SETS :: RELATIONS :: FUNCTIONS • Given R  A x A, R is a (partial) order relation iffit is reflexive, anti-symmetric and transitive. • If the relation holds ∀ a, b  A then it is a total order • If ∀ a, b  A either aRbor bRaor a = b then it is a strict order

Functions SETS :: RELATIONS :: FUNCTIONS • A functionf from A to B is a binary relation that associates to each element a in A exactly one element b in B. f : A  B • The image of an element a  A is denoted with f(a)  B Notice that it can be the case that the same element in B is theimage of several elements in A.

Functions (cont.) SETS :: RELATIONS :: FUNCTIONS • f: A  B is injective if for distinct elements in A there is a distinct element in B: ∀ a, b  A and a ≠ b then f(a) ≠ f(b) • f: A  B is surjective if for each element in B there is at least one element in A: ∀ b B ∃ a A s.t.f(a) = b • f: A  B is bijectiveif it is injective and surjective.

Functions (cont.) SETS :: RELATIONS :: FUNCTIONS • If f: A  B is bijective we can define its inverse functionf-1: B  A • Given two functions f: A  B and g: B  C, the compositionof f and g is the function g ○f :  C such that: g ○f = {(a, g(f(a)) | a  A}

Logics for Data and Knowledge Representation