1. 03 -1 Lecture 03 First-Order Predicate Logic Topics –Syntax –Formal Semantics –Denotational Semantics –Formal Inference –Resolution

2. 03 -2 Syntax Atomic Sentence –Predicate(term 1, term 2, …, term n ) –Term Constant Variable Function –Predicate must be constant –Classmate(Jack, x, Brother(Allen)) Function –Fun-name(term 1, term 2, …, term n ) –Fun-name : function name must be constant Cardinality –Classmate(Jack, x) vs Classmate(Jack, x, Brother(Allen))

3. 03 -3 Syntax Connectives –NOT  / AND  / OR  / Imply → –Example:  Classmate(x, Allen)  Classmate(x, Jack)  Classmate(x, Andy) → Classmate(x, Aho) Quantifiers –Universal quantifier  ForAll  –Existential quantifier  ThereExist  –Example:  x Classmate(Adam, x) →  y Like(x, y) Well formed sentence (wff, or Sentence)

4. 03 -4 Formal Semantics Atomic sentence –True (T)/ False (F) –Example: Classmate(x, Jack)= T Connectives –Truth tables –Identity Example: S1→S2 ≡  S1  S2 Quantifiers –  x S(x)=T IF S(x 1 )  S(x 2 )  …  S(x n )=T –  x S(x)=T IF S(x 1 )  S(x 2 )  …  S(x n )=T Truth functional: The formal semantics of a sentence can be determined by the formal semantics of its components

5. 03 -5 Denotational Semantics Denotational mappings to objects and relationships (Physical meaning) Atomic sentence –Constant denotes a named object –Variable denotes some unnamed object –Function indirectly denotes an object –Predicate denote a relationship –Atomic sentence denotes a fact –Example: Classmate(x, Jack) Denotes the fact that some unnamed man denoted by x is a classmate of an object named Jack

6. 03 -6 Denotational Semantics Connectives –  S denotes that the fact denoted by S isn’t existent –S1  S2 denotes that the fact denoted by S1 and the fact denoted by S2 are co-existent –S1  S2 denotes that one or both of the facts denoted by S1 and by S2 are existent –S1 → S2 denotes that if the fact denoted by S1 exists, the fact denoted by S2 will exist

7. 03 -7 Denotational Semantics Quantifiers – –  x S denotes the fact that every object in the system can make the fact of S existent –  x S denotes the fact that there is at least one object in the system which can make the fact of S existent The denotational semantics of a sentence contains the set of denotational mappings of its constituents.

8. 03 -8 Formal Inference Reason about the formal semantics of a new sentence only according to syntactical structure –From KB={Classmate(Adam, Allen)  Classmate(Allen, Andy)}= T we derive Classmate(Adam, Allen) = T without consulting the underlying physical meanings Problem: How can we guarantee that under all denotational semantics, the above inference is correct? Or the denotational semantics of the derived sentence holds?

9. 03 -9 Formal Inference Key: Make the inference independent of denotation semantics How: Make the inference sound and complete Definition of “Model” –Give a denotational semantics M, M is a model of KB={S|S:wff}, denoted as M KB, if M makes the formal semantics of KB true.

10. 03 -10 Formal Inference Definition of “Entailment” –Given KB={S|S:wff} and a is a wff, if every M KB is also M a, then we say a is entailed by KB (or KB entails a ), denoted as KB ┝ a –Example: KB={S1=Classmate(Adam, Allen)  S2=Classmate(Allen, Andy)} then KB ┝ S1; KB ┝ S2; KB ┝ KB All M KB are also M 1 and M 2 KB={S1  S2}S2S1 F F F F {M 20 }: T F {M 10 }: T F F {M KB }: T{M 1 }: T{M 2 }: T

11. 03 -11 Formal Inference Definitions of Soundness and Completeness –Suppose KB ┝ a. Given i a formal inference mechanism, if i can derive b from KB, denoted as KB├ i b, then i is sound, iff {b}  {a}, i is complete, iff {b}  {a}, and i is sound and complete, iff {b} = {a}

12. 03 -12 Formal Inference Sound and complete inference mechanisms –A sound inference mechanism only derives wff’s that are entailed by the original KB; that is, no matter what models are used to interpret the derived wff’s they are CORRECT. –A complete inference mechanism can derive all entailed wff’s.

13. 03 -13 Formal Inference Example of formal inference mechanisms –ae, a formal inference, defined as  {S 1  S 2  …  S n }├ ae S i, i =1,2…, n Example: –KB ┝ {KB={S 1  S 2 }, S1, S2} (P. 9) –KB├ ae {S1, S2}  {KB, S1, S2} –ae is sound Is ae complete? –In general, NO, if KB contains other connectives than  Find a sound and complete formal inference mechanism for First-Order Logic?

14. 03 -14 Resolution Canonical form –Clause l 1  …  l j  …  l m, where L i is a literal Literal: positive or negative atomic sentence –CNF (Conjunctive Normal Form) KB={l 1  …  l j  …  l m, L 1  …   L k  …  L n } Horn Clause: at most one positive literal in a sentence First-Order Definite Clause: exactly one positive literal in a sentence

15. 03 -15 Resolution Resolution, denoted by res, as a formal inference mechanism on CNF –{l 1  …  l j  …  l m, L 1  …  L k  …  L n } ├ res  ( , l 1  …  l j-1  l j+1  …  l m  L 1  …  L k-1  L k+1  …  L n )  = Unify(l j, L k ), a substitution  is a substitution application function

16. 03 -16 Resolution Illustration of ├ res –KB={  Classmate(x, Allen)  Like(x, Joyce), Classmate(Adam, Allen)} –Resolution procedure 1.  = Unify(Classmate(x, Allen), Classmate(Adam, Allen))={x/Adam} 2. KB={  Classmate(x, Allen)  Like(x, Joyce), Classmate(Adam, Allen)} 3.  ({x/Adam}, Like(x, Joyce))= Like(Adam, Joyce)

17. 03 -17 Resolution ├ res is sound on CNF All First-Order Logic KBs can be converted to CNF ├ res is a sound formal inference mechanism for First-Order Logic ├ res is refutationally complete on CNF and First-Order Logic –Given any C with KB ┝ C, resolution can prove KB  C contains contradiction –Proof by contradiction

18. 03 -18 Resolution Application –Conversion of wffs to CNF –Control strategies Set-of-support resolution strategy with unit preference –Automated theorem prover –System verification Related languages –Horn clause/ First-order definite clause/ Prolog/ Rule/ Attribute-based language/ Planning language/ Frame/ Description Logic