![Propositional logic
• Logical constants: true, false
• Propositional symbols: P, Q,... (atomic sentences)
• Wrapping parentheses: ( … )
• Sentences are combined by connectives:
and [conjunction]
or [disjunction]
implies [implication / conditional]
is equivalent[biconditional]
not [negation]
• Literal: atomic sentence or negated atomic sentence
P, P](https://image.slidesharecdn.com/predicatelogic-230427174746-8c57df2f/85/predicateLogic-ppt-4-320.jpg)
predicateLogic.ppt
- 1. Propositional Logic and predicate logic
- 2. Big Ideas • Logic is a great knowledge representation language for many AI problems • Propositional logic is the simple foundation and fine for some AI problems • First order logic (FOL) is much more expressive as a KR language and more commonly used in AI • There are many variations: horn logic, higher order logic, three-valued logic, probabilistic logics, etc.
- 3. Propositional logic (PL) • Simple language for showing key ideas and definitions • User defines set of propositional symbols, like P and Q • User defines semantics of each propositional symbol: – P means “It is hot”, Q means “It is humid”, etc. • A sentence (well formed formula) is defined as follows: – A symbol is a sentence – If S is a sentence, then S is a sentence – If S is a sentence, then (S) is a sentence – If S and T are sentences, then (S T), (S T), (S T), and (S ↔ T) are sentences – A sentence results from a finite number of applications of the rules
- 4. Propositional logic • Logical constants: true, false • Propositional symbols: P, Q,... (atomic sentences) • Wrapping parentheses: ( … ) • Sentences are combined by connectives: and [conjunction] or [disjunction] implies [implication / conditional] is equivalent[biconditional] not [negation] • Literal: atomic sentence or negated atomic sentence P, P
- 5. Examples of PL sentences • (P Q) R “If it is hot and humid, then it is raining” • Q P “If it is humid, then it is hot” • Q “It is humid.” • We’re free to choose better symbols, btw: Ho = “It is hot” Hu = “It is humid” R = “It is raining”
- 6. Some terms • The meaning or semantics of a sentence determines its interpretation • Given the truth values of all symbols in a sentence, it can be “evaluated” to determine its truth value (True or False) • A model for a KB is a possible world – an assignment of truth values to propositional symbols that makes each sentence in the KB True
- 7. Truth tables Truth tables for the five logical connectives Example of a truth table used for a complex sentence • Truth tables are used to define logical connectives • and to determine when a complex sentence is true given the values of the symbols in it
- 8. On the implies connective: P Q • Note that is a logical connective • So PQ is a logical sentence and has a truth value, i.e., is either true or false • If we add this sentence to the KB, it can be used by an inference rule, Modes Ponens, to derive/infer/prove Q if P is also in the KB • Given a KB where P=True and Q=True, we can also derive/infer/prove that PQ is True
- 9. Propositional Logic can’t say • If X is married to Y, then Y is married to X. • If X is west of Y, and Y is west of Z, then X is west of Z. • And a million other simple things. • Fix: – extend representation: add predicates – Extend operator(resolution): add unification
- 10. Syntax • See text for formal rules. • All of propositional + quantifiers, predicates, functions, and constants. • Variables can take on values of constants or terms. • Term = reference to object • Variables not allowed to be predicates. – E.G. What is the relationship between Bill and Hillary? • Text Notation: variables lower case, constants upper • Prolog Notation: variables are upper case, etc
- 11. Semantics • Validity = true in every model and every interpretation. • Interpretation = mapping of constants, predicates, functions into objects, relations, and functions.
- 12. Advance Artificial Intelligence 12 Predicate Calculus So it is a better idea to use predicates instead of propositions. This leads us to predicate calculus. Predicate calculus has symbols called • object constants, • relation constants, and • function constants These symbols will be used to refer to objects in the world and to propositions about the word.
- 13. Advance Artificial Intelligence 13 Quantification Introducing the universal quantifier and the existential quantifier facilitates the translation of world knowledge into predicate calculus. Examples: Paul beats up all professors who fail him. x(Professor(x) Fails(x, Paul) BeatsUp(Paul, x)) There is at least one intelligent UMB professor. x(UMBProf(x) Intelligent(x))
- 14. Representing World in FOL • All kings are persons. goes to? • for all x, King(x) & Person(x). • for all x, King(x) => Person(x).
- 15. Representing World in FOL • All kings are persons. • for all x, King(x) => Person(x). OK. • for all x, King(x) & Person(x). Not OK. – this says every object is a king and a person. • In Prolog: person(X) :- king(X). • Everyone Likes icecream. • for all x, Likes(x, icecream).
- 16. Negating Quantifiers • ~ there exist x, P(x) • ~ for all x, P(x) For all x, Likes(x,Icecream) No one likes liver. For all x, not Likes(x,Liver) • For all x, ~P(x) • There exists x, ~P(x) • There does not exist an x, not Likes(x,Icecream) • Not there exists x, Iikes(x,Liver).
- 17. More Translations • Everyone loves someone. • There is someone that everyone loves. • Everyone loves their father. • See text. • For all x, there is a y(x) such that Loves(x,y(x)). • There is an M such that for all x, Loves(x,M). • M is skolem constant • For all x, Loves(x,Father(x)). • Father(x) is skolem function.
- 18. Occurs Checking • When unifying a variable against a complex term, the complex term should not contain the same variable, else non-match. • Prolog doesn’t check this. • Ex. f(X,X) and f(Y,g(Y)) should not unify.
- 19. Modeling with Definite Clauses: at most one positive literal 1. It is a crime for an american to sell weapons to a hostile country. 1’. American(x)&Weapons(y)&Hostile(z) & Sell(x,y,z) => Criminal (x). 2. The country Nono has some missiles. There exists x Owns(Nono,x)&Missile(x). 2’. Missile(M1). … Skolem Constant introduction 2’’. Owns(Nono,M1).
- 20. Resolution gives forward chaining • Enemy(x,America) =>Hostile(x) • Enemy(Nono,America) • |- Hostile(Nono) • Not Enemy(x,America) or Hostile(x) • Enemy(Nono,America) • Resolve by {x/Nono} • To Hostile(Nono)
- 21. FOL -> Conjuctive Normal Form • Similar to process for propositional logic, but • Use negations rules for quantifiers • Standarize variables apart • Universal quantification is implicit. • Skolemization: introduction of constants and functions to remove existential quantifiers.
- 22. Time • Before (x,y) implies After(y,x) • After(x,y) imples Before(y,x) • Before(x,y) and Before(y,z) implies Before(x,z) etc. • When are you done? • What about during?
- 23. Space and more • In(x,y) and In(y,z) implies in(x,z). • Infront, behind, etc • Frame problem: you turn, some predicates change and some don’t. • etc. And lots more: heat, wind, hitting, physical objects versus thoughts, knowing,