74.419 Artificial Intelligence 2004 - First-Order Predicate Logic -

1. 74.419 Artificial Intelligence 2004 - First-Order Predicate Logic - • First-Order Predicate Logic (FOL or FOPL), also called First-Order Predicate Calculus • Formal Language • Semantics through Interpretation Function • Axioms • Inference System

2. FOPL- Formal Language / Syntax -

3. Formal Language A Formal Language is specified as L = (NT, T, P, S) NT Set of Non-Terminal Symbols T Set of Terminal Symbols P Set of Production or Grammar Rules S Start Symbol (top-level node in syntax tree / parse tree) A formal language specifies the syntactically correct or well-formed expressions of a language.

4. Terminals and Non-Terminals • NT Non-Terminals • wff (well-formed formula), atomic-formula; Predicate, Term, Function, Constant, Variable; Quantifier, Connective • T Terminals • Predicate (Symbols) P, Q, married, ..., T, F • Function (Symbols) f, g, father-of, ... • Variables x, y, z, ... • Constants Sally, block-1, c • Connectives , , ,  • Negation Symbol  • Quantifiers ,  • Equality Symbol = • Parentheses ( , ) • Other Symbols : Domain Specifc General

5. Production / Grammar Rules Non-terminal Rules wff ::= atomic-formula |(wff)|  wff | wff Connective wff|Quantifier Variablewff atomic-formula ::= Predicate (Term, ...)*| Term = Term Term ::= Function (Term, ...)*| Variable | Constant Terminal Rules Connective ::=  |  |  |  Quantifier ::=  |  *Note: n-ary functions and predicates go with n terms

6. Domain-Specific Terminal Rules • Terminal Rules for the specific Domain • Predicate ::= on(_,_) |near(_,_)| ... • Function ::= distance(_,_) | location(_)| ... • Variable ::= x|y | ... • Constant ::= Flakey|John-Bear | Karen| Alan-Alder | The-File | Kurt

7. Quantifiers and Binding A variable in a formula can be bound by a quantifier. bound variable x: married (Sally, x) open formula: a variable in the formula is not bound by a quantifier x: married (Sally, x)  happy (y) closed formula: all variables in the formula are bound by quantifiers: x y: married (x, y) Most authors regard quantified formulas only as wffs if • all quantified variablesappear in the formula. Some authors regard quantified formulas only as wffs if • all variables arebound by quantifiers.

8. FOPL- Semantics / Interpretation -

9. Semantics - Overview • Define the Semantics of FOPL expressions (formulae): • Interpretation – Maps symbols of the formal language (predicates, functions, variables, constants) onto objects, relations, and functions of the “world” (formally: Domain, relational Structure, or Universe) • Valuation - Assigns domain objects to variables* • Constructive Semantics – Determines the semantics of complex expressions inductively, starting with basic expressions • * The Valuation function can be used for describing value assignments and constraints in case of nested quantifiers. • The Valuation function otherwise determines the satisfaction of a formula only in case of open formulae.

10. Semantics – Domain, Interpretation I Domain, relational Structure, Universe D set of Objects R, S, ... set of Relations over D f, g, ... set of Functions in D Basic Mapping / Interpretation constants I [c] = dD functions I [f]= F: Dn →DFunction predicates I [P] = RDn Relation (set of n-tuples) Valuation V variables V(x) = dD

11. Semantics –Interpretation Next, determine the semantics for complex terms and formulae constructively - regarding the syntax - from the basic interpretation above.

12. Semantics - Interpretation II • Term with function • I [f(t1,...,tn)) = I [f] (I [t1],..., I [tn]) = F(I [t1],..., I [tn]) D • atomic Formula • I [P(t1,...,tn)] true if(I [t1],..., I [tn]) I [P] = R • negated Formula • I []true if I []is not true • complex Formula • I [] true ifI [] or I []true • I []true if I [] and I []true • I []ifI [] not true or I []true

13. Semantics - Interpretation III quantified Formula (relative to Valuation function) I [x.]true if istrue with V’(x)=d for some dDwhere V’ is otherwise identical to the prior V. I [x.] true if  is true with V’(x)=d for all dD and where V’ is otherwise identical to the prior V. Note: The order of quantifiers plays a role for the semantic interpretation and evaluation: x y. is different from y x.  In the first case, we go through all values for x, and for each value of x we pick a suitable value for y. In the second case, we have to find one value for y which is good for all values of x.

14. Semantics - Model • Model • Given a set of formulaeand a domain D with an interpretation I. Then • D is a model of if • I[]is truefor all * • That means the interpretation I into the domain D makes every formula  in  true. • * for every possible valuation, in case  has open formulae.

15. Semantics – Logical Consequence • Logical Consequence • Given a set of formulaeand aformulaα. • αis a logical consequence of if • αis true in every model of . • Notation: • |=α • That means that for every model (interpretation into a domain) in which  is true, α must also be true.

16. FOPL- Inference System -Axioms & Inference Rules

17. FOPL Axioms A1      A2      A3        A4 (  )  ((  )  (  )) A5 x: (x)  (y) A6 (x)  y: (y)

18. Formal Inference - Overview • Derive new formulae by syntactic manipulation of existing formulae: • given set of formulae  •  describes your KB, or a Theory, ... (FOPL axioms + your own "proper" axioms) • apply inference rule (based on  ) • new formula αis derived • add new formula to KB or Theory • new KB or Theory is α

19. Formal Inference Formal Inference, Theorem Given a set of formulae  and a set of inference rules IR. A new formula αcan begenerated based on using inference rules in IR. We say that α is formally inferred or derived from  or αis a Theorem (of ) Notation: |– α

20. IR Modus Ponens Modus Ponens   ,   States that  can be concluded provided we know that the formulae    and  are true in our knowledge base.

21. IR Universal Instantiation Universal Instantiation x: (x) (c) where (x) is any formula containing the variable x, and (c) is the formula (x) where every occurrence of the quantified variable x is substituted with the arbitrary constant c.

22. IR Existential Generalization Existential Generalization (c) x: (x) where (c) is any formula containing the arbitrary constant c, and (x) is the same formula as (c) but with every occurrence of the constant c replaced by a variable x.

23. IR Replacement Rules Replacement Rules                (  ) (  )   

24. FOPL Inference System • The Axioms and the Inference Rules above constitute a formal inference system for FOPL. • This system - we call it FS1 - is complete and sound.

25. Soundness and Completeness Soundness An Inference System is sound iff  |– α |= α every formula which can be derived by formal inference from  is a also logical consequence of . Completeness An Inference System is complete iff  |= α |– α every formula which is a logical consequence of  can be derived by formal inference from  .

26. FOPL - Sound and Complete 2 The above inference system for FOPL is sound and complete. Thus, every formula which can be derived in FOPL using FS1 (|– α) is also a logical consequence of the given axioms (|= α) : |–αiff |= α Thus, there is a correspondence between formal Inference and semantic Interpretation.

27. Semantics - Example A1 Predicate Logic Language constants Bill-1, John-3, Sally-1, Mary-1, Mary-2 predicates happy-together, hate-each-other Structure D objects: Uncle-Bill, Uncle-John, Aunt-Sally, The-woman-I-don't-like, Mary relations: Married, Divorced (Uncle-Bill, Aunt-Sally)  Married, (Uncle-John, Mary)  Married (Uncle-John, The-woman-I-don't-like)  Divorced Interpretation I(Bill-1)=Uncle-Bill, I(John-3)=Uncle-John, I(Sally-1)=Aunt-Sally, I(Mary-1)=The-woman-I-don't-like, I(Mary-2)=Mary I(happy-together)=Married, I(hate-each-other)=Divorced True or false? hate-each-other (Bill-1, John-3) happy-together(Bill-1, Sally-1) hate-each-other(John-3, Mary-1) happy-together(John-3, Mary-2)

28. Semantics -Example A2 Structure D objects: Uncle-Bill, Uncle-John, Aunt-Sally, The-woman-I-don't-like, Mary relations: Married, Divorced (Uncle-John, The-woman-I-don't-like)  Divorced (Uncle-Bill, Aunt-Sally)  Married, (Uncle-John, Mary)  Married (or: {(Uncle-Bill, Aunt-Sally), (Uncle-John, Mary)} = Married) Interpretation I I(Bill-1) = Uncle-Bill, I(John-3) = Uncle-John, I(Sally-1) = Aunt-Sally, I(Mary-1) = Mary, I(Mary-2) = The-woman-I-don't-like I(happy-together) = Married, I(hate-each-other) = Divorced True or false? hate-each-other (Bill-1, John-3) hate-each-other (John-3, Mary-1) happy-together (Bill-1, Sally-1)  happy-together (John-3, Mary-2) x: happy-together(Uncle-Bill, x)) x,y,z: happy-together(x,y)  hate-each-other (x,z) What if you want to add a formula? x,y: happy-together(x,y)  happy-together(y,x)

29. Additional References • Frost, Richard: Introduction to Knowledge Base Systems. Collins Professional and Technical Books, William Collins Sons & Co. Ltd, London, 1986. • Nilsson, Nils J.: Artificial Intelligence - A new synthesis. Morgan Kaufmann Publishers, San Francisco, CA, 1998.