1 / 18

Lecture 03 First-Order Predicate Logic

Lecture 03 First-Order Predicate Logic. Topics Syntax Formal Semantics Denotational Semantics Formal Inference Resolution. Syntax. Atomic Sentence Predicate(term 1 , term 2 , …, term n ) Term Constant Variable Function Predicate must be constant Classmate(Jack, x, Brother(Allen))

Download Presentation

Lecture 03 First-Order Predicate Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  2. Syntax • Atomic Sentence • Predicate(term1, term2, …, termn) • Term • Constant • Variable • Function • Predicate must be constant • Classmate(Jack, x, Brother(Allen)) • Function • Fun-name(term1, term2, …, termn) • Fun-name:function name must be constant • Cardinality • Classmate(Jack, x) vs Classmate(Jack, x, Brother(Allen))

  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. Formal Semantics • Atomic sentence • True (T)/ False (F) • Example: Classmate(x, Jack)= T • Connectives • Truth tables • Identity • Example: S1→S2 ≡ S1S2 • Quantifiers • "xS(x)=TIF S(x1)S(x2)…S(xn)=T • $xS(x)=TIF S(x1)S(x2)…S(xn)=T • Truth functional: The formal semantics of a sentence can be determined by the formal semantics of its components

  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. Denotational Semantics • Connectives • S denotes that the fact denoted by Sisn’t existent • S1S2 denotes that the fact denoted by S1 and the fact denoted by S2are co-existent • S1S2 denotes that one or both of the facts denoted by S1 and by S2are existent • S1→S2 denotes that ifthe fact denoted by S1 exists, the fact denoted by S2 will exist

  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 ofdenotational mappingsofits constituents.

  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. 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 amodel of KB={S|S:wff}, denoted as MKB, if M makes the formal semantics of KB true.

  10. Formal Inference • Definition of “Entailment” • Given KB={S|S:wff} and a is a wff, if every MKB is also Ma, 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 MKB are also M1 and M2 KB={S1S2} S1 S2 {MKB}: T {M1}: T {M2}: T F {M10}: T F F F {M20}: T F F F

  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 iis sound, iff {b}{a}, iis complete, iff{b}{a}, and iis sound and complete, iff{b} = {a}

  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. Formal Inference • Example of formal inference mechanisms • ae, aformal inference, defined as {S1S2…Sn}├aeSi, i =1,2…, n • Example: • KB┝ {KB={S1S2}, 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. Resolution • Canonical form • Clause • l1 … lj… lm, where Liis a literal • Literal: positive or negative atomic sentence • CNF (Conjunctive Normal Form) • KB={l1 … lj… lm, L1 …Lk… Ln} • Horn Clause: at most one positive literal in a sentence • First-Order Definite Clause: exactly one positive literal in a sentence

  15. Resolution • Resolution, denoted by res, as a formal inference mechanism on CNF • {l1…lj…lm, L1…Lk…Ln} ├ress(, l1…lj-1lj+1…lmL1… Lk-1Lk+1 …Ln) •  = Unify(lj, Lk), a substitution • s is a substitution application function

  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. s({x/Adam},Like(x, Joyce))= Like(Adam, Joyce)

  17. Resolution • ├resis 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. 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

More Related