First-Order Logic

1. First-Order Logic Chapter 8

2. Outline • Why FOL? • Syntax and semantics of FOL • Interacting with FOL KBs • Knowledge engineering in FOL

3. Pros and cons of propo. logic • Propositional logic is declarative • Its semantics is based on a truth relation between sentences and possible worlds.  Propositional logic allows partial information • Using disjunction and negation. • Propositional logic is compositional: • meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2  Meaning in propositional logic is context-independent • (unlike natural language, where meaning depends on context)  Propositional logic has very limited expressive power • (unlike natural language) • E.g., cannot say "pits cause breezes in adjacent squares“ • except by writing one sentence for each square

4. First-order logic • Whereas propositional logic assumes the world contains facts, • first-order logic (like natural language) assumes the world contains • Objects: people, houses, numbers, colors, baseball games, wars, … • Relations (verbs/properties): brother of, bigger than, part of, comes between, red, round, prime, … • Functions (one-to-one relation): father of, best friend, plus, …

5. Syntax of FOL: Basic elements • Constants (objects):KingJohn, 2,... • Predicates (relations):Brother, >,... • Functions (functions):Sqrt, LeftLegOf,... • Variables x, y, a, b,... • Connectives , , , ,  • Equality = • Quantifiers  (universal),  (existentail)

6. Atomic sentences Atomic sentence = predicate (term1,...,termn) Term = function (term1,...,termn) or constant or variable • E.g., Brother(John,Mike) Married(Father(John), Mother(Mike)) Brother(x,y) Married(Father(x), Mother(y))

7. Complex sentences • Complex sentences = atomic sentences + connectives S, S1 S2, S1  S2, S1 S2, S1S2, E.g. Brother(John,Mike)  Married(Father(John), Mother(Mike))

8. Truth in first-order logic • Sentences are true with respect to a model and an interpretation • Model contains objects (domainelements) and relations among them • Interpretation specifies referents for constantsymbols→objects predicatesymbols→relations functionsymbols→ functional relations • An atomic sentence predicate(term1,...,termn) is true iff the objects referred to by term1,...,termn are in the relation referred to by predicate • An complex sentence is true, depends on the truth value of atomic sentences and truth table of connectives.

9. Models for FOL: Example

10. Universal quantification • A quantifier is a determiner that expresses properties of entire collections of objects. • <variables> <sentence> (for all) • x P is true in a model m iff P is true with x being each possible object in the model

11. A common mistake to avoid • Typically,  is the main connective with  • Common mistake: using  as the main connective with .

12. Existential quantification • <variables> <sentence> (for some, there exists) • xP is true in a model m iff P is true with x being some possible object in the model

13. Another common mistake to avoid • Typically,  is the main connective with  • Common mistake: using  as the main connective with .

14. Properties of quantifiers • Quantifier duality: each can be expressed using the other • x P x P • x P xP

15. Properties of quantifiers • Nested quantifers • x y yx (x,y or y,x) x yyx(x,y or y,x) • x yyx x y loves(x,y) yx loves(x,y)

16. Equality • term1 = term2is true under a given interpretation iff term1and term2refer to the same object e.g. x=y, father(x) = y, (x=y) • E.g. John has two brothers • E.g., definition of Sibling in terms of Parent:

17. The Wumpus World KB • Perception • t,s,b Percept([s,b,Glitter],t)  Glitter(t) • t,b,g Percept([Stench,b,g],t)  Stench(t) • t,s,g Percept([s,Breeze,g],t)  Breeze(t) • Reflex action • t Glitter(t)  BestAction(Grab,t)

18. Inferring Hidden Properties Adjacent location: • x,y,a,b Adjacent([x,y],[a,b])  [a,b]  {[x+1,y], [x-1,y],[x,y+1],[x,y-1]} Properties of squares: • s,t At(Agent, L, t)  Breeze(t)  Breezy(L) Squares are breezy near a pit: • Diagnostic rule---infer cause from effect L Breezy(L)  [r Adjacent(r,L)  Pit(r)] • Causalrule---infer effect from cause r Pit(r)  [L Adjacent(r,L)  Breezy(L)]

19. Interacting with FOL KBs • Suppose a wumpus-world agent perceives a smell, a breeze and a glitter at t=5: Tell(KB,Percept([Stensh, Breeze, Glitter],5)) Ask(KB, a BestAction(a,5)) //does KB entail some best action at t=5? • Answer: Yes, {a/Grab} ← substitution (binding list): a set of variable/terms pairs • Given a sentence S and a substitution σ, Sσ denotes the result of plugging σ into S. e.g. S = Smarter(x,y) σ = {x/Hillary,y/Bill} Sσ = Smarter(Hillary,Bill) • Ask(KB,S) returns some/all σ that KB╞σ

20. Knowledge engineering in FOL • Identify the task • Assemble the relevant knowledge • Decide on a vocabulary of predicates, functions, and constants • Encode general knowledge about the domain • Encode a description of the specific problem instance • Pose queries to the inference procedure and get answers • Debug the knowledge base

21. Summary • First-order logic: • objects and relations are semantic primitives • syntax: constants, predicates, functions, equality, quantifiers • Increased expressive power: sufficient to define wumpus world