First Order Logic

1. First Order Logic CS 171/271 (Chapters 8 and 9) Some text and images in these slides were drawn fromRussel & Norvig’s published material

2. Propositional Logic Limitations • Stating similar facts is cumbersome • Can’t make generalizations • The world (in propositional logic) contains facts, not objects • Natural language deals with objects (nouns) and relations (verbs), hence is more expressive • Ontological commitment of PL is limited

3. First Order Logic • World consists of • Objects • Relations • Functions • Sentences are made up of • Symbols: for constant objects, predicates, and functions • Connectives: as in PL • Quantifiers and variables: x, y

4. FOL Syntax • Sentence • Atomic Sentence • ( Sentence Connective Sentence ) • Sentence • Quantifier Variable Sentence • Atomic Sentence • Predicate-Symbol( Term, … ) • Term = Term

5. FOL Syntax • Term (refers to an object) • Function-Symbol( Term, … ) • Constant-Symbol • Variable • Connective: , , , , • Quantifier: ,  • Variable: x, y, z, …

6. Example

7. Symbols • Constants: John, Richard, C, L1, L2 • Predicates: • Person={John, Richard} • King={Richard} • Crown={C} • Brother={(John,Richard),(Richard,John)} • OnHead={(C,Richard)} • Functions: • LeftLeg={(Richard->L1),(John->L2)}(strictly speaking, the function should be total)

8. Models and Interpretations • A model in FOL consists of the objects (domain elements) and relations (including functions) • See example diagram • “conceptual” view of world • An interpretation associates the symbols to the objects, relations, and functions in the model • Number of interpretations for a given set of symbols is combinatorially explosive

9. Semantics • Truth of a sentence in FOL: • Determined with respect to a model and an interpretation • Analogous notions for entailment, validity, and satisfiability • Model enumeration is impractical in FOL

10. Sample Sentences • Person(John)  Person(Richard) • OnHead(C, John) • LeftLeg(John) = L1  LeftLeg(Richard) = L1 • Richard, and LeftLeg(John) are examples of terms(a term is an expression that refers to an object) • Atomic Sentences: constructed by equating terms (=) or by a predicate (with terms as arguments) • Complex Sentences: sentences with connectives

11. Quantifiers • Universal Quantification • x P is true in a model m iff P is true with x being each possible object in the model • A conjunction of instantiations • Existential Quantification • x P is true in a model m iff P is true with x being some object in the model • A disjunction of instantiations

12. Sample SentencesUsing Quantifiers • x King(x)  Person(x) • x Crown(x)  OnHead(x,John) • “John has a crown on his head” • xy Brother(x,y)  Brother(y,x) • xy Brother(x,Richard)  Brother(y,Richard)  (x=y) • “Richard has at least 2 brothers”

13. Properties of Quantifiers • Nested Quantifiers • x y P equivalent to y x P • y x P equivalent to y x P • Does not apply if quantifiers are different • De Morgan’s law for quantifiers • x P  x P • x P  x P • x P  x P • x P  x P

14. More About Quantifiers • Be careful when: • Using quantifiers (, ) in combination with ,  • The domain consists of multiple kinds of objects • In the quantified sentence(such as x P or x P), P would typically contain terms that are variables • Not just ground terms (terms that have no variables) • x P as a query: binding list more important than truth of the sentence

15. Axioms and Theorems • Axioms are sentences that represent first principles • Plain facts • Definitions • Theorems are sentences entailed by axioms

16. Some Useful Domains • Natural Numbers • Built from 0, successor function S, and Peano axioms • Sets • , , , , and element insertion • Lists • Nil, Cons, Append, First, Rest, Find, …

17. FOL and the Wumpus World • We can represent the Wumpus World in a more compact fashion • Less sentences needed to represent rules • We can include time and percept objects in the world • A percept is represented as a list of constant symbols • Predicates with time arguments capture the dynamic nature of the agent moving in this world

18. Knowledge Engineering • Identify the task • Assemble the relevant knowledge • Decide on a vocabulary • Encode general knowledge of the domain • Encode the specific problem instance • Pose queries to the inference procedure • Debug the knowledge base

19. Inference Algorithms in FOL • Reduction to Propositional Inference(Propositionalization) • Lifting and Unification • Resolution

20. Propositionalization • Strategy: convert KB to propositional logic and then use PL inference • Ground atomic sentences become propositional symbols • What about the quantifiers?

21. Example • KB in FOL: • x King(x)  Greedy(x)  Evil(x) • King(John) • Greedy(John) • Brother(Richard,John) • The last 3 sentences can be symbols in PL • Apply Universal Instantiation to the first sentence

22. Universal Instantiation • UI says that from a universally quantified sentence, we can infer any sentence obtained by substituting a ground term for the variable • Back to Example • From: x King(x)  Greedy(x)  Evil(x) • To: • King(John)  Greedy(John)  Evil(John) • King(Richard)  Greedy(Richard)  Evil(Richard) • …

23. Issue with UI • Ground terms: all symbols that refer to objects as well as function applications (recall that function applications return objects) • For example, suppose Father is a function: • Father(John) and Father(Richard) are also objects/ground terms • But so are Father(Father(John)) and Father(Father(Father(John))) • Infinitely many ground terms/instantiations

24. Existential Instantiation • Whenever there is a sentence, x P, introduce a new object symbol called the skolem constant and then add the unquantified sentence P, substituting the variable with that constant • Example: • From: x Crown(x)  OnHead(x, John) • To: Crown(Cnew)  OnHead(Cnew, John)

25. Substitution • UI and EI apply substitutions • A substitution is represented by a variable v and a ground term g; {v/g} • Can have sets of these pairs if there are more variables involved • Let  be a sentence (possibly containing v) • SUBST( {v/g},  ) stands for the sentence that applies the substitution to 

26. UI and EI Defined • UI:vα ___for any ground term gSUBST({v/g}, α) • EI:vα ___for some constant symbol k notSUBST({v/k}, α) yet in the knowledge base

27. Back to Propositionalization • Given a KB in FOL, convert KB to PL by • applying UI and EI to quantified sentences • converting atomic sentences to symbols • If there are no functions (Datalog KB), UI application does not result in infinitely many sentences • Regular PL Inference can now be carried out without problems • What if there are functions?

28. Dealing with Infinitely Many Ground Terms • Can set a depth-limit for ground terms • Depth specifies levels of function nesting allowed • Carry out reduction and inference process for depth 1, then 2, then 3, … • Stop when entailment can be concluded • This works if there is such a proof, but goes into an endless loop if there is not • The strategy is complete • The entailment problem in this sense is semidecidable

29. Inefficiencies in Propositionalization • An inordinate number of irrelevant sentences may be generated, resulting from UI • This motivates generating only those sentences that are important in entailment

30. Example • Suppose KB contains: • x King(x)  Greedy(x)  Evil(x) • y Greedy(y) • King(John) • Suppose we want to conclude Evil(John) • Because of the existence of objects other than John (such as Richard) and the existence of functions, UI will generate many sentences

31. Example, continued • It is sufficient to generate: • King(John)  Greedy(John)  Evil(John) • Greedy(John) • Which is just: • SUBST( {x/John}, King(x)  Greedy(x)  Evil(x) ) • SUBST( {y/John}, Greedy(y) ) • Applying the substitution matches the • Premises: King(x)  Greedy(x) • With other sentences in the KB:Greedy(y), King(John)

32. Lifted Modus Ponens • Lifting: Raising propositional inference rules to first order logic • Example: Generalized Modus Ponens If there is a substitution θ, such thatSUBST(θ, pi) = SUBST(θ, pi’) for all i, then p1', p2', … , pn’, ( p1 p2 …  pn q) _______________________________________________________________________________ SUBST(θ,q) • In our example,  = {x/John, y/John}

33. Unification • Process that makes logical expressions identical • Goal: match the premises of implications so that conclusions can be derived • UNIFY algorithm takes two sentences and returns a unifier (substitution) if it exists

34. Unification Algorithm

35. Unification Algorithm

36. About UNIFY • UNIFY returns a Most General Unifier (MGU) • There are efficiency issues withOCCUR-CHECK function • May need to standardize apart: rename variables to avoid name clashes • Unification is a key component of all first-order algorithms

37. Algorithms that use unification • Forward and backward chaining algorithms • Will not be discussed • Resolution-based theorem proving systems

38. PL Resolution Revisited • Recall PL Resolution algorithm: • Convert (KB  ) to CNF • Repeatedly get pairs of clauses and eliminate complementary literals • If an empty clause results, KB╞  • Resolution applies to FOL, but we need to refine definitions of: • CNF (for quantified sentences) • Resolution inference rule / complimentary literals

39. CNF Conversion in FOL • Eliminate biconditionals and implications • Move  inwards (De Morgan’s) • For quantifiers: x P  x P, x P x P • Standardize variables • Eliminate possible name clashes • Skolemize (Apply EI to existential sentences) • Introduce Skolem constants or functions • Drop universal quantifiers • Distribute  over 

40. Example Everyone who loves all animals is loved by someone: x [y Animal(y) Loves(x,y)]  [y Loves(y,x)] • Eliminate biconditionals and implications • x [y Animal(y) Loves(x,y)]  [yLoves(y,x)] • Move  inwards: • x [y (Animal(y) Loves(x,y))]  [y Loves(y,x)] • x [y Animal(y) Loves(x,y)]  [y Loves(y,x)] • x [y Animal(y) Loves(x,y)]  [y Loves(y,x)] • Standardize variables: each quantifier should use a different one • x [y Animal(y) Loves(x,y)]  [z Loves(z,x)]

41. Example, continued • Skolemize: a more general form of existential instantiation. • Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables: • x [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x) • Drop universal quantifiers: • [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x) • Distribute  over  : • [Animal(F(x)) Loves(G(x),x)] [Loves(x,F(x)) Loves(G(x),x)]

42. Resolution • Lifted version of resolution inference rule: l1··· lk, m1··· mn (l1··· li-1 li+1 ··· lk m1··· mj-1 mj+1··· mn)θ where Unify(li, mj) = θ. • The two clauses are assumed to be standardized apart so that they share no variables • For example, Rich(x) Unhappy(x) , Rich(Ken) Unhappy(Ken) with θ = {x/Ken}

44. Making ResolutionMore Efficient • Can favor particular clauses in the KB • Unit preference (unit clauses) • Sets of support • Input resolution (e.g. “single spine”) • Subsumption • Reduces size of KB by eliminating redundant sentences