First Order Logic (FOL)

Transcription

1 First Order Logic (FOL) CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2013 Soleymani Course material: 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, wumpuses) Verbs and verb phrases referring to relation among objects (is breezy, is adjacent to, shoot) Some of them are functions (relations with only one value for each input) Examples: Objects: people, houses, numbers, baseball games, Relations Unary relation or property:red,round,prime, -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: objects, relations, and functions 5

6 Syntax of FOL: Basic elements Logical elements Connectives,,,, Quantifiers, Domain specific elements Constants h,h, Predicates h,... Functions,... Non-logical general elements Variables,,,, Equality = 6

7 Example h(h, h) h(h) = (h(h), h(h)), h, h, h h, h ( = ) (h) (h), (, ) [ =, ( = ) (, ) (, ) (, ) (, )] 7

8 Syntax of FOL: Atomic sentences Atomic Sentence Predicate Predicate (Term,, Term) Term=Term Term Constant variable Function(Term, ) Object 1) h(h, h) Constant Richard A X 1 Variable a x s Function Mother LeftLeg 2) h(h) = 3) (h(h), h(h)) Relation Predicate True False After Loves Hascolor. 8

9 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 9 :, =,,,,

10 Universal quantification () is true in a model iff () is true with being each possible object in the model conjunction of instantiations of () 10

11 Existential quantification () is true in a model iff () is true with being some possible object in the model disjunction of instantiations of () 11

12 Properties of quantifiers is the same as is the same as is not the same as h(, ) There is a person who is mother of everyone in the world h(, ) Everyone in the world has a mother Quantifier duality 12

13 Using FOL: Kinship domain example Objects people Functions Predicates 13

14 Using FOL: Kinship domain example Objects: people Functions: h, h Predicates: Unary:, Binary:,, h,, h, h,,,,,, h,,, 14

15 Using FOL: Kinship domain example Sample sentences:, h = (() (, )),, h(, ),,,,,,,, 15

16 Assertions & queries in FOL KBs Assertions: sentences added to using (, (h)) (, ( )) Queries or goals: questions asked from using, h, Substitution or binding list: AskVars 16 In KB of Horn clauses, every way of making the query true will bind the variables to specific values AskVars(, ()): answer {/h} If KB contains h (h), there is no binding while, is true

17 Using FOL: Set domain example Objects: sets, elements Functions:,,{ } Predicates: Unary: Set Binary:, () ( = {}) (, ( 2 2) = { 2 }), { } = {}, = { },, 2 ( = { 2 } (= 2 ))] 1, ( 1 2 ) 1, 2 ( 1 = 2 ) ( ), 1, 2 ( 1 2 ) ( 1 2 ), 1, 2 ( 1 2 ) ( 1 2 ) 17

18 Using FOL: Wumpus world example Environment Objects: pairs identifying squares [, ],, Relations:,,, h,,,,, 18

19 Using FOL: Wumpus world example Perceptions: perceives a smell, a breeze, and glitter at =5: h,,,,, 5 Actions: (), (h),, h,, Perceptions implies facts about current state,,,, (,,,,,) () Simple reflex behavior () (, ) AskVars( (, 5)) Binding list: e.g., {/} 19

20 Using FOL: Wumpus world example Samples of rules:,,,,,, = = 1 = 1 = = 1 =+1,,, (, [, ], ),,,,,,,,,, =,, One successor-state axiom for each predicate + 1 ( h, ) 20

21 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 21

22 KE: electronic circuits example One-bit full adder XOR XOR Carry AND AND OR 22

23 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 ( )= o ( 1 ) o (,) 23

24 KE: electronic circuits example (steps) 4) Encode general knowledge of the domain 1, 2 ( 1, 2 ) ( 1 ) = ( 2 ) = 1 = 0 1, 2 1, 2, = ( 1, = 1, = 1) = ( 1, = 0, = 0) = ( 1, = 0 1, = 2, ) = ( 1, 1, ) [You can see a more accurate encoding in 3 rd edition of AIMA]

25 KE: electronic circuits example (steps) 5) Encode the specific problem instance ( 1 ) = ( 1 ) = ( 1 ) = ( 2 ) = ( 2 ) = 1, 1, 1, 2 ((1, 1 ), (1, 1 )) 1, 1, 2, 2 ((1, 1 ), (1, 1 )) 1, 2, 1, 1 ((2, 1 ), (2, 1 )) 1, 1, 2, 1 ((2, 1 ), (2, 1 )) 1, 2, 1, 1 ((3, 1 ), (2, 2 )) 1, 1, 2, 1 ((3, 1 ), (1, 2 )) 25

26 KE: electronic circuits example (steps) 6) Pose queries to the inference procedure 1, 2, 3 1, = 1 2, 1 = 2 3, 1 = 3 1, 1 =0 ((2, 1 )) = 1 Bindings= { /1, /1, /0}, { /1, /0, /1}, { /0, /1, /1} 7) Debug the knowledge base Has 1 0 been asserted? 26

27 KE: electronic circuits example (steps) 6) Pose queries to the inference procedure 1, 2, 3,, 2 1, = 1 2, 1 = 2 3, 1 = 3 1, 1 = ((2, 1 )) = Bindings = whole input-output table 7) Debug the knowledge base Has 1 0 been asserted? 27

28 Summary First-order logic: objects and relations are semantic primitives syntax: constants, variables, functions, predicates, equality, quantifiers Knowledge engineering 28

29 Review only Propositional logic (BNF Grammar) () :,,,, 30