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The Semantic Web – WEEK 7: Logic, Reasoning and Proof

The Semantic Web – WEEK 7: Logic, Reasoning and Proof

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The Semantic Web – WEEK 7: Logic, Reasoning and Proof

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  1. The Semantic Web –WEEK 7: Logic, Reasoning and Proof You are here! The “Layer Cake” Model – [From Rector & Horrocks Semantic Web cuurse]

  2. Recap • A good deal of the data on the Semantic Web will be in the form of ‘Conceptual Knowledge’. This has traditionally been captured by diagrams in software engineering but for the web we need a precise, symbolic language such as a logic language. • Fundamental to logic languages is the idea of INTERPRETATIONS - mapping predicates and constants to some conceptualization of the world. • A well formed sentence in Logic is called a Wff. • Wff2 LOGICALLY FOLLOWS from Wff1 if and only if every interpretation that makes Wff1 true also makes Wff2 true. • Wff2 is LOGICALLY EQUIVALENT to Wff1 if and only if every interpretation that makes Wff1 true also makes Wff2 true AND vice-versa. The Semantic Web

  3. Reasoning • Processes (agents) working in the Semantic Web need to be a good deal more flexible than normal programs - they need to be able to negotiate, reason, plan etc • As well as being suited to capturing conceptual knowledge, FOL is well known for its AUTOMATED REASONING capabilities In the next couple of weeks I will show you methods that do automated reasoning with FOL The Semantic Web

  4. Interpretations revisited These 2 Interpretations SATISFY this WFF Ax Ey R(y,x) Greater_than Mother_of persons numbers “Given any person there is Someone who is their mother” “Given any number there Is some number greater than it” WFF = WFF = NB Ax Ey … =/= Ey Ax The Semantic Web

  5. Meaning of Quantifiers Consider a Universe with individuals a,b,c,… Ax P(x) = P(a) & P(b) & P(c) & …. Ex P(x) = P(a) V P(b) V P(c) V …. Ax Ay R(x,y) = R(a,a) & R(a,b) & R(a,c) &… & R(b,a) & R(b,b) & R(b,c) & … Ax Ey R(y,x) = Ey R(y,a) & Ey R(y,b) & Ey R(y,c) &… = (R(a,a) V R(b,a) V R(c,a) V …) & (R(a,b) V R(b,b) V R(c,b) V …) & (R(a,c) V R(b,c) V R(c,c) V …) & …. The Semantic Web

  6. Meaning of Connectives • The Connectives &, V, ¬, =>, , <= Get their meaning via propositional truth tables – P Q P V Q ETC T T T T F T F F F F T T The Semantic Web

  7. “Laws” These are some well known equivalent FORMS in FOL called laws ( De Morgans laws etc) ¬ ( P & Q ) = ¬P V ¬Q ¬ ( P V Q ) = ¬P & ¬Q P=>Q = ¬P V Q ¬ ¬ P = P P  Q = (P=>Q)&(Q=>P) etc The Semantic Web

  8. Quantifiers + Negation LAWS • ¬ Ax P(x) = Ex ¬ P(x) • ¬ Ex P(x) = Ax ¬ P(x) Similary (and abstractly) ¬ A E = E A ¬ The Semantic Web

  9. Example Revisited “Every student is an academic. Everybody who teaches an academic is an academic. Jeff teaches Fred who is a student.” What can we say about the statement “Jeff is an academic” Translate to FOL: S = student, D = academic, T = teaches Ax S(x)=>D(x) Ax (Ey T(x,y) & D(y)) => D(x) S(Fred) T(Jeff,Fred) Goal: D(Jeff) How can we get agents to automatically deduce such facts?? The Semantic Web

  10. Another Example: The Remote Agent Experiment On May 17th and 21st 1999 NASA’s Deep Space 1 spacecraft was controlled completely by its own REASONING SYSTEM! The Semantic Web

  11. Another Example Imagine Deep Space 1 travels to Mars and observes many things about the Martians, including the fact that some seem very hostile towards humans. Concrete observations are as follows: (a) All green Martians have antennae. (b) A Martian is friendly to humans if all of its children have antennae. (c) A Martian is green if at least one of its parents is green. On its way back from Mars the robot is hotly pursued by a spacecraft containing green Martians only. Should the robot suspect it is being attacked? Or can the robot reason with its observations to answer the question: `Are all green Martians friendly?'' and hence avert an inter-planetary conflict. The Semantic Web

  12. Automated Deduction Roughly, agents can be equipped with the ability to deduce knowledge from observations by automating Laws of Inference A Law of Inference • is a method for deducing wffs • is SOUND if it only ever deduces wffs that also logically follow. The Semantic Web

  13. Natural Deduction The most famous Laws of Inference is known by its Latin name “Modus Ponens” From wffs OF THE FORM… P(a) Ax P(x)=>Q(x) We can deduce the following Wff Q(a) Example: Socrates is a Man, All Men are Mortal Deduce: Socrates is Mortal The Semantic Web

  14. Natural Deduction Another is called “Modus Tollens” From wffs OF THE FORM… ¬Q(a) Ax P(x)=>Q(x) We can deduce the following Wff ¬P(a) Example: If a thing is smoking then it is on fire. I am not on fire. Deduce: I am not smoking The Semantic Web

  15. Unsound Deduction Example: If a person is the murderer then that person must have bloody hands. The Butler has bloody hands. Deduce: The Butler is the murderer This is UNSOUND!!! BH(butler) Ax M(x)=>BH(x) We can’t deduce anything from this!! The Semantic Web

  16. Summary FOL is equipped with a form of reasoning called deduction that can be automated Next week I will cover ONE very efficient way to automate deduction. The Semantic Web