Artificial Intelligence

Artificial Intelligence Propositional Logic [1]

Boolean algebras by examples U X U U = {a} U = {a, b} U = {a, b, c} {a} {b} {a, b} {a, c} {b, c}... {a} {b} {c} {a, b} {a} The arrows represents proper inclusion {a} {a, b} (Hasse diagrams) X U 2 U X = 2 U \U < X,,, \U,, U > Propositional Logic [2]

Boolean algebras by examples U X U U = {a} U = {a, b} U = {a, b, c} {a} {b} {a, b} {a, c} {b, c}... {a} {b} {c} \U (Hasse diagrams) A A\U = U A = {a} A\U = {b, c} A A\U = {a, b, c} A (A B) = A A = {b} B = {c} A B = {b, c} A (A B) = {b} Propositional Logic [3]

De Morgan s laws Boolean algebras by examples U X U U = {a} U = {a, b} U = {a, b, c} {a} {b} {a, b} {a, c} {b, c}... {a} {b} {c} \U (Hasse diagrams) (A B)\U = A\U B\U A = {b} A\U = {a, c} B = {b, c} B\U = {a} A B = {b, c} (A B)\U = {a} A\U B\U = {a} (A B)\U = A\U B\U A = {b} A\U = {a, c} B = {b, c} B\U = {a} A B = {b} (A B)\U = {a, c} A\U B\U = {a, c} Propositional Logic [4]

Boolean algebras by examples U X U U = {a} U = {a, b} U = {a, b, c} {a} {b} {a, b} {a, c} {b, c}... {a} {b} {c} \U (Hasse diagrams) A\U B = U A = {a} A\U = {b, c} B = {b} A\U B = {b, c} * Ouch! This is NOT true in general It is only valid when A B Propositional Logic [5]

Abstract Boolean Algebras < X,,, \U,, U > A, B, C X A A = A A = A A B = B A, A B = B A A (B C) = (A B) C, A (B C) = (A B) C A (A B) = A, A (A B) = A A (B C) = (A B) (A C), A (B C) = (A B) (A C) A = A, A =, U A = U, U A = A A (A \U) = U, A (A \U) = Propositional Logic [6]

Which Boolean algebra for logic? < {0,1}, OR, AND, NOT, 0, 1> X {U, } {nothing, everything} {false, true} {, } or {0, 1} AND OR NOT f : {0, 1} n {0, 1} A B OR 0 0 0 0 1 1 1 0 1 1 1 1 A B AND 0 0 0 0 1 0 1 0 0 1 1 1 A NOT 0 1 1 0 Propositional Logic [7]

Composite functions These columns are identical De Morgan s laws A B NOT A NOT B A OR B NOT(A OR B) NOT A AND NOT B 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 Propositional Logic [8]

2 n rows Adequate basis A 1 A 2... A n f(a 1, A 2,..., A n ) 0 0... 0 f 1 0 0... 1 f 2.............................. 1 1... 1 f 2 n OR AND NOT f = 1 AND n A i i 1 A i i 0 OR Propositional Logic [9]

Other adequate basis {OR, NOT} {AND, NOT} NOR NAND A B A NOR B 0 0 1 0 1 0 1 0 0 1 1 0 {IMP, NOT} A B A IMP B 0 0 1 0 1 1 1 0 0 1 1 1 A B A NAND B 0 0 1 0 1 1 1 0 1 1 1 0 A B A EQU B 0 0 1 0 1 0 1 0 0 1 1 1 Identities: A IMP B = NOT A OR B A EQU B = (A IMP B) AND (B IMP A) Propositional Logic [10]

Propositional logic false true Propositional Logic [11]

The class of propositional, semantic structures <{0,1}, P, v> {0,1} P v P {0,1} P P P A B C D <{0,1}, P, v> <{0,1}, P, v> <{0,1}, P, v>... P v P {0,1} 2 P Propositional Logic [12]

Propositional language L P P P = {A, B, C,...},,, (, ) L P (L P ) A P A (L P ) (L P ) () (L P ), (L P ) ( ) (L P ), (L P ) ( ) (L P ), ( ) (() ), (L P ) ( ) (L P ), ( ) (( ())), (L P ) ( ) (L P ), ( ) (( ) ( )) Propositional Logic [13]

Semantics: interpretations <{0,1}, P, v> v : P {0,1} v() = NOT(v()) v( ) = AND(v(), v()) v( ) = OR(v(), v()) v( ) = OR(NOT(v()), v()) IMP(v(), v()) v( ) = AND(OR(NOT(v()), v()), OR(NOT(v()), v())) v (L P ) v : (L P ) {0,1} v L P v <{0,1}, P, v> P Propositional Logic [14]

Satisfaction, models = (A B) C (L P ) A B C A B (A B) C 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 v() = 1 <{0,1}, P, v> v v 1, 2,..., n <{0,1}, P, v> = { 1, 2,..., n } v Propositional Logic [15]

Tautologies, contradictions A A A A A 0 0 1 1 0 1 A B (A B) (B A) 0 0 1 0 1 1 1 0 1 1 1 1 A B ((A B) (B A)) 0 0 0 0 1 0 1 0 0 1 1 0 Propositional Logic [16]

Formulae and subsets W L P W {v : v() = 1} {v : v } W The set of all possible worlds W Propositional Logic [17]

Formulae and subsets W L P W {v : v() = 1} {v : v } W The set of all possible worlds W is a tautology any possible world in W is a model of is (logically) valid Furthermore: is satisfiable is not falsifiable Propositional Logic [18]

Formulae and subsets W L P W {v : v() = 1} {v : v } W The set of all possible worlds W is a contradiction none of the possible worlds in W is a model of is not (logically) valid Furthermore: is not satisfiable is falsifiable Propositional Logic [19]

Formulae and subsets W L P W {v : v() = 1} {v : v } W The set of all possible worlds W is neither a contradiction nor a tautology some possible worlds in W are model of, others are not is not (logically) valid Furthermore: is satisfiable is falsifiable Propositional Logic [20]

About formulae and their hidden relations 1 = B D (A C) 2 = B C 3 = A D 4 = B = D Is there any logical relation between hypothesis and thesis? And among the propositions in the hypothesis? Propositional Logic [21]

Logical consequence 1 = B D (A C) 2 = B C 3 = A D 4 = B = D { 1, 2, 3, 4 } A B C D 1 2 3 4 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1, 2, 3, 4 (Pay attention to notation!) Propositional Logic [22]

Formulae, subsets and entailment W All possible worlds W All possible worlds that are model of Propositional Logic [23]

Formulae, subsets and entailment W All possible worlds W 1 All possible worlds that are model of 1 { 1 } because the set of models of { 1 } is not contained in the set of models of Propositional Logic [24]

Formulae, subsets and entailment W All possible worlds W 2 1 All possible worlds that are models of 2 { 1, 2 } because the set of models of { 1, 2 } (i.e. the intersection of the two subsets) is not contained in the set of models of Propositional Logic [25]

Formulae, subsets and entailment W All possible worlds V 2 1 3 All possible worlds that are models of 3 { 1, 2, 3 } because the set of models of { 1, 2, 3 } is not contained in the set of models of Propositional Logic [26]

Formulae, subsets and entailment W All possible worlds V 4 2 1 3 All possible worlds that are models of 4 { 1, 2, 3, 4 } Because the set of models of { 1, 2, 3, 4 } is contained in the set of models of Propositional Logic [27]

Formulae, subsets and entailment W All possible worlds V 4 2 1 3 All possible worlds that are models of 4 { 1, 2, 3, 4 } Because the set of models of { 1, 2, 3, 4 } is contained in the set of models of In this case, all the s 1, 2, 3, 4 are needed for the relation of entailment to hold Propositional Logic [28]

Symmetric entailment = logical equivalence { 1, 2, 3, 4 } 1 = B D (A C) 2 = B C 3 = A D 4 = B = D 1 = B D (A C) 2 = B C 3 = A D 4 = B = D Propositional Logic [29]

Implication {, } 1 = C (B (A D)) 2 = B C 3 = A D 4 = B = D 1 = B D (A C) 2 = B C 3 = A D 4 = B = D,, Propositional Logic [30]

Modern formal logic: fundamentals 1 {1, 0} Propositional Logic [31]

What we have seen so far language meaning semantics v() entailment semantics v() Propositional Logic [32]

Subtleties: object language and metalanguage L P P,,,,,, (, ), There are a few more symbols in the metalanguage, to be introduced during the course Propositional Logic [33]