First Order Logic (FOL)
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1 First Order Logic (FOL) CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2015 Soleymani Artificial Intelligence: A Modern Approach, 3 rd Edition, Chapter 8
2 Why FOL? To represent knowledge of complex environments concisely. Expressive enough to represent a good deal of of our knowledge E.g., pits cause breezes in adjacent squares needs many rules in propositional logic but one rule in FOL 2
3 Natural language elements & FOL elements Some basic elements of natural language (included also in FOL): Nouns and noun phrases referring to objects (squares, pits, wumpus) Some of objects are defined as functions of other objects Verbs and verb phrases referring to relation among objects (is breezy, is adjacent to, shoot) Examples: Objects: people, houses, numbers, baseball games, Relations Unary relation or property: red, round, prime, n-ary: brother of, bigger than, inside, part of, owns, comes between, Functions: father of, best friend, one more than, beginning of, FOL does not include all elements of natural language. 3
4 Symbols & interpretations Types of symbols Constant: objects Predicate: relations Function: functional relations 4
5 Example Brother(Richard, John) Father(John) = Henry Married(Father(Richard), Mother(John)) x King x Person x x Crown x OnHead x, John x, y Brother x, Richard Brother y, Richard (x = y) King(Richard) King(John) x, y Sibling(x, y) [ x = y m, f (m = f) Parent(m, x) Parent(f, x) Parent(m, y) Parent(f, y)] 5
6 Syntax of FOL: Basic elements Logical elements Connectives,,,, Quantifiers, Domain specific elements Constants John, Richard, Predicates Brother,... Functions LeftLegOf,... Non-logical general elements Variables x, y, a, b, Equality = 6
7 Syntax of FOL: Atomic sentences Atomic Sentence Predicate Predicate (Term,, Term) Term=Term Term Constant variable Function(Term, ) Object Relation 1) Brother(Richard, John) Constant Richard A X 1 Variable a x s Function Mother LeftLeg 2) Father(John) = Henry 3) Married(Father(Richard), Mother(John)) Predicate True False After Loves Hascolor. 7
8 Syntax of FOL (BNF Grammar) Sentence Atomic Sentence ComplexSentence Atomic Sentence Predicate Predicate (Term,, Term) Term=Term ComplexSentence ( Sentence ) Sentence Sentence Sentence Sentence Sentence Sentence Sentence Sentence Sentence Quantifier variable, Sentence Term Function(Term, ) Constant variable Quantifier Constant A X 1 Variable a x s Predicate True False After Loves Hascolor. Function Mother LeftLeg 8 Operator Precedence:, =,,,,
9 Universal quantification x P(x) is true in a model m iff P(x) is true with x being each possible object in the model conjunction of instantiations of P(x) 9
10 Existential quantification x P(x) is true in a model m iff P(x) is true with x being some possible object in the model disjunction of instantiations of P(x) 10
11 Properties of quantifiers x y P is the same as y x P x y P is the same as y x P x y P is not the same as y x P x y Mother(x, y) There is a person who is mother of everyone in the world y x Mother(x, y) Everyone in the world has a mother Quantifier duality x P xp x P x P 11
12 FOL: Kinship domain example Objects people Functions Predicates 12
13 FOL: Kinship domain example Objects: people Functions: Mother, Father Predicates: Unary: Male, Female Binary:Parent, Sibling, Brother, Sister, Child, Daughter, Son, Spouse, Wife, Husband, Grandparent, Grandchild, Cousin, Aunt, Uncle 13
14 FOL: Kinship domain example Sample sentences: m, c Mother c = m (Female(m) Parent(m, c)) p, c Parent p, c Child(c, p) g, c Grandparent g, c p Parent g, p Parent p, c x, y Sibiling x, y x y p Parent p, x Parent p, x 14
15 Assertions & queries in FOL KBs Assertions: sentences added to KB using TELL Tell(KB, King(John)) Tell(KB, ( x King x Person x )) Queries or goals: questions asked from KB using ASK ASK KB, King John ASK KB, x King x Substitution or binding list: AskVars 15 In KB of Horn clauses, every way of making the query true will bind the variables to specific values AskVars(KB, King(x)): answer {x/john} If KB contains King John King(Richard), there is no binding although ASK KB, x King x is true
16 FOL: Set domain example Objects: sets, elements Functions: s 1 s 2, s 1 s 2, {x s} Predicates: Unary: Set Binary: x s, s 1 s 2 s Set(s) (s = {}) ( x, s 2 Set(s 2 ) s = {x s 2 }) x, s {x s} = {} x, s x s s = {x s} x, s x s y, s 2 (s = {y s 2 } (x = y x s 2 ))] s 1, s 2 s 1 s 2 (x xs 1 x s 2 ) s 1, s 2 (s 1 = s 2 ) (s 1 s 2 s 2 s 1 ) x, s 1, s 2 x (s 1 s 2 ) (x s 1 x s 2 ) x, s 1, s 2 x (s 1 s 2 ) (x s 1 x s 2 ) 16
17 FOL: Wumpus world example Environment Objects: pairs identifying squares [i, j], Agent, Wumpus Relations: Pit, Adjacent, Breezy, Stenchy, Percept, Action, At, HaveArrow, 17
18 FOL: Wumpus world example Perceptions: perceives a smell, a breeze, and glitter at t = 5: Percept Stench, Breeze, Glitter, None, None, 5 Actions: Turn(Left), Turn(Right), Forward, Shoot, Grab, Climb Perceptions implies facts about current state t, s, g, m, c Percept( s, Breeze, g, m, c, t) Breeze(t) Simple reflex behavior t Glitter(t) BestAction(Grab, t) AskVars( a BestAction(a, 5)) Binding list: e.g., {a/grab} 18
19 FOL: Wumpus world example Samples of rules: x, y, a, b Adj x, y, a, b x = a y = b 1 y = b 1 At Agent, s, t x, y t At(Wumpus, [x, y], t) s, t At Agent, s, t Breeze t Breezy s y = b x = a 1 x = a + 1 x, s 1, s 2, t At Agent, s 1, t At Agent, s 2, t s 1 = s 2 s, t Breezy s r Adj r, s Pit r One successor-state axiom for each predicate t HaveArrow t + 1 ( HaveArrow t Action Shoot, t ) 19
20 Knowledge Engineering (KE) in FOL 1) Identify the task 2) Assemble the relevant knowledge 3) Decide on a vocabulary of predicates, functions, and constants (Ontology) 4) Encode general knowledge about the domain 5) Encode a description of the specific problem instance 6) Pose queries to the inference procedure and get answers 7) Debug the knowledge base 20
21 KE: electronic circuits example One-bit full adder XOR XOR Carry AND AND OR 21
22 KE: electronic circuits example (steps) 1) Identify the task o Does the circuit add properly? (circuit verification) 2) Assemble the relevant knowledge o Signals, input and output terminals, gates, wires connecting gates, types of gates (AND, OR, XOR, NOT) 3) Decide on a vocabulary (ontology) o o types of gates, terminals of gates, connections between gates, signal on terminals of gates e.g., alternatives for determining the type of a gate: o Type(X 1 ) = XOR o XOR(X 1 ) o Type(X 1, XOR) 22
23 KE: electronic circuits example (steps) 4) Encode general knowledge of the domain t 1, t 2 Connected(t 1, t 2 ) Signal(t 1 ) = Signal(t 2 ) t Signal t = 1 Signal t = 0 t 1, t 2 Connected t 1, t 2 Connected t 2, t 1 g Type g = OR (Signal Out 1, g = 1 n Signal In n, g = 1) g Type g = AND (Signal Out 1, g = 0 n Signal In n, g = 0) g Type g = XOR (Signal Out 1, g = 0 Signal In 1, g = Signal In 2, g ) g Type g = NOT (Signal Out 1, g Signal In 1, g ) [You can see a more accurate encoding in 3 rd edition of AIMA]
24 KE: electronic circuits example (steps) 5) Encode the specific problem instance Type(X 1 ) = XOR Type(A 1 ) = AND Type(O 1 ) = OR Type(X 2 ) = XOR Type(A 2 ) = AND Connected Out 1, X 1, In 1, X 2 Connected(In(1, C 1 ), In(1, X 1 )) Connected Out 1, X 1, In 2, A 2 Connected(In(1, C 1 ), In(1, A 1 )) Connected Out 1, A 2, In 1, O 1 Connected(In(2, C 1 ), In(2, X 1 )) Connected Out 1, A 1, In 2, O 1 Connected(In(2, C 1 ), In(2, A 1 )) Connected Out 1, X 2, Out 1, C 1 Connected(In(3, C 1 ), In(2, X 2 )) Connected Out 1, O 1, Out 2, C 1 Connected(In(3, C 1 ), In(1, A 2 )) 24
25 KE: electronic circuits example (steps) 6) Pose queries to the inference procedure i 1, i 2, i 3 Signal In 1, C 1 = i 1 Signal In 2, C 1 = i 2 Signal In 3, C 1 = i 3 Signal Out 1, C 1 = 0 Signal(Out(2, C 1 )) = 1 Bindings= {i 1 /1, i 2 /1, i 3 /0}, {i 1 /1, i 2 /0, i 3 /1}, {i 1 /0, i 2 /1, i 3 /1} 25
26 KE: electronic circuits example (steps) 6) Pose queries to the inference procedure i 1, i 2, i 3, o 1, o 2 Signal In 1, C 1 = i 1 Signal In 2, C 1 = i 2 Signal In 3, C 1 = i 3 Signal Out 1, C 1 = o 1 Signal(Out(2, C 1 )) = o 2 Bindings = whole input-output table 7) Debug the knowledge base Has 1 0 been asserted? 26
27 Summary First-order logic: objects and relations are semantic primitives syntax: constants, variables, functions, predicates, quantifiers Knowledge engineering 27
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