Chapter 5 Logic and Inference: Rules Grigoris Antoniou Frank van Harmelen 1 Chapter 5 A Semantic Web Primer
1 Chapter 5 A Semantic Web Primer Chapter 5 Logic and Inference: Rules Grigoris Antoniou Frank van Harmelen
Lecture Outline Introduction 2. Monotonic Rules: EXample 3. Monotonic Rules: Syntax Semantics 4. Nonmonotonic Rules: Syntax 5. Nonmonotonic Rules: Example 6. A dTd For Monotonic rules 7. Adtd For nonmonotonic rules Chapter 5 A Semantic Web primer
2 Chapter 5 A Semantic Web Primer Lecture Outline 1. Introduction 2. Monotonic Rules: Example 3. Monotonic Rules: Syntax & Semantics 4. Nonmonotonic Rules: Syntax 5. Nonmonotonic Rules: Example 6. A DTD For Monotonic Rules 7. A DTD For Nonmonotonic Rules
Knowledge Representation o The subjects presented so far were related to the representation of knowledge o Knowledge Representation was studied long before the emergence of Ww in Al o Logic is still the foundation of KR, particularly in the form of predicate logic first-order logic) 3 Chapter 5 A Semantic Web primer
3 Chapter 5 A Semantic Web Primer Knowledge Representation ⚫ The subjects presented so far were related to the representation of knowledge ⚫ Knowledge Representation was studied long before the emergence of WWW in AI ⚫ Logic is still the foundation of KR, particularly in the form of predicate logic (first-order logic)
The Importance of Logic e High-level language for expressing knowledge High expressive power Well-understood formal semantics Precise notion of logical consequence statements syntactically from a set of erive Proof systems that can automatically de premises Chapter 5 A Semantic Web primer
4 Chapter 5 A Semantic Web Primer The Importance of Logic ⚫ High-level language for expressing knowledge ⚫ High expressive power ⚫ Well-understood formal semantics ⚫ Precise notion of logical consequence ⚫ Proof systems that can automatically derive statements syntactically from a set of premises
The Importance of Logic (2) There exist proof systems for which semantic logical consequence coincides with syntactic derivation within the proof system Soundness completeness Predicate logic is unique in the sense that sound and complete proof systems do exist Not for more expressive logics(higher-order logics) trace the proof that leads to a logical consequence Logic can provide explanations for answers By tracing a proof 5 Chapter 5 A Semantic Web primer
5 Chapter 5 A Semantic Web Primer The Importance of Logic (2) ⚫ There exist proof systems for which semantic logical consequence coincides with syntactic derivation within the proof system – Soundness & completeness ⚫ Predicate logic is unique in the sense that sound and complete proof systems do exist. – Not for more expressive logics (higher-order logics) ⚫ trace the proof that leads to a logical consequence. ⚫ Logic can provide explanations for answers – By tracing a proof
Specializations of Predicate Logic. RDF and owl RDF/S and OWL Lite and DL)are specializations of predicate logic correspond roughly to a description logic o They define reasonable subsets of logic o Trade-off between the expressive power and the computational complexity The more expressive the language, the less efficient the corresponding proof systems 6 Chapter 5 A Semantic Web primer
6 Chapter 5 A Semantic Web Primer Specializations of Predicate Logic: RDF and OWL ⚫ RDF/S and OWL (Lite and DL) are specializations of predicate logic – correspond roughly to a description logic ⚫ They define reasonable subsets of logic ⚫ Trade-off between the expressive power and the computational complexity: – The more expressive the language, the less efficient the corresponding proof systems
Specializations of Predicate Logic Horn Logic ● a rule has the form:A1,,,,An→>B Ai and b are atomic formulas o There are 2 ways of reading such a rule Deductive rules If a1. an are known to be true then b is also true Reactive rules: If the conditions Al. An are true, then carry out the action B 7 Chapter 5 A Semantic Web primer
7 Chapter 5 A Semantic Web Primer Specializations of Predicate Logic: Horn Logic ⚫ A rule has the form: A1, . . ., An → B – Ai and B are atomic formulas ⚫ There are 2 ways of reading such a rule: – Deductive rules: If A1,..., An are known to be true, then B is also true – Reactive rules: If the conditions A1,..., An are true, then carry out the action B
Description Logics VS Horn Logic e Neither of them is a subset of the other e It is impossible to assert that persons who study and live in the same city are home students"in OWL This can be done easily using rules studies(X, Y), lives(X, 2), loc(Y,U), loc(z,U)) home Student(X Rules cannot assert the information that a person is either a man or a woman This information is easily expressed in OWL using disjoint union 8 Chapter 5 A Semantic Web primer
8 Chapter 5 A Semantic Web Primer Description Logics vs. Horn Logic ⚫ Neither of them is a subset of the other ⚫ It is impossible to assert that persons who study and live in the same city are “home students” in OWL – This can be done easily using rules: studies(X,Y), lives(X,Z), loc(Y,U), loc(Z,U) → homeStudent(X) ⚫ Rules cannot assert the information that a person is either a man or a woman – This information is easily expressed in OWL using disjoint union
Monotonic vs, non-monotonic rules Example: An online vendor wants to give a special discount if it is a customer's birthday Solution 1 R1: If birthday, then special discount But what happens if a customer refuses t R2: If not birthday, then not special discot provide his birthday due to privacy concerns? 9 Chapter 5 A Semantic Web primer
9 Chapter 5 A Semantic Web Primer Monotonic vs. Non-monotonic Rules ⚫ Example: An online vendor wants to give a special discount if it is a customer’s birthday Solution 1 R1: If birthday, then special discount R2: If not birthday, then not special discount ⚫ But what happens if a customer refuses to provide his birthday due to privacy concerns?
Monotonic vs, non -monotonic Rules (2) Solution 2 R1: If birthday then special discount R2: If birthday is not known, then not special discount Solves the problem but The premise of rule R2 is not within the expressive power of predicate logic We need a new kind of rule system 10 Chapter 5 A Semantic Web primer
10 Chapter 5 A Semantic Web Primer Monotonic vs. Non-monotonic Rules (2) Solution 2 R1: If birthday, then special discount R2’: If birthday is not known, then not special discount ⚫ Solves the problem but: – The premise of rule R2' is not within the expressive power of predicate logic – We need a new kind of rule system