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Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning For when I am presented with a false theorem, I do not need to examine or even to know the demonstration,

  1. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning For when I am presented with a false theorem, I do not need to examine or even to know the demonstration, since I shall discover its falsity a posteriori by means of an easy experiment, that is, by a calculation, costing no more than paper and ink, which will show the error no matter how small it is. . . And if someone would doubt my results, I should say to him: ”Let us calculate, Sir,” and thus by taking to pen and ink, we should soon settle the question. —Gottfried Wilhelm Leibniz [1677] � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 1

  2. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Learning Objectives At the end of the class you should be able to: explain how symbols can have meaning represent a problem in a simple logic prove soundness and completeness of a proof procedure debug a logic program without knowing the how inference works use negation-as-failure where appropriate use assumption-based reasoning for a simple domain � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 2

  3. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Outline Propositions and Semantics Proofs Bottom-up Proof Procedure Top-down Proof Procedure Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Proof by Contradiction Abduction � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 3

  4. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Why propositions? A proposition is a statement that is either true or false. Propositions can be built using logical connectives. Specifying proposition is often a natural specification Correctness can be checked locally The answer depends on the semantics, not how it is implemented Debugging can use the semantics of propositions We choose inference method to be efficient It provides a language for asking queries It is easy to incrementally add formulae It can be extended to infinite domains (using quantification) � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 4

  5. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Simple language: propositional definite clauses An atom is a symbol starting with a lower case letter A body is an atom or is of the form b 1 ∧ b 2 where b 1 and b 2 are bodies. A definite clause is an atom or is a rule of the form h ← b where h is an atom and b is a body. A knowledge base is a set of definite clauses � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 5

  6. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Definite Clauses Which of the following are definite clauses? (a) happy ← sad (b) blimsy (c) old ∧ wise ← teenager (d) happy ∧ sad (e) glad ← happy ∧ sad (f) green ∨ blue ← ¬ red (g) glad ← happy ∧ sad ∧ mad ∧ bad (h) glad ← happy ∧ rad ← sad ∧ mad ∧ bad (i) happy ← happy � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 6

  7. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Human’s view of semantics Step 1 Begin with a task domain. Step 2 Choose atoms in the computer to denote propositions. — These atoms have meaning to the KB designer. Step 3 Tell the system knowledge about the domain. Step 4 Ask the system questions. — The system will answer whether the question is a logical consequence. Step 4 Interpret the answers with the meaning associated with the atoms. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 7

  8. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Role of semantics In computer: In user’s mind: light 1 broken : light #1 light 1 broken ← sw up is broken ∧ power ∧ unlit light 1 . sw up : switch is up sw up . power : there is power in power ← lit light 2 . the building unlit light 1 . unlit light 1: light #1 lit light 2 . isn’t lit lit light 2: light #2 is lit Conclusion: light 1 broken The computer doesn’t know the meaning of the symbols The user can interpret the symbol using their meaning � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 8

  9. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Semantics An interpretation I assigns a truth value to each atom. A body b 1 ∧ b 2 is true in I if b 1 is true in I and b 2 is true in I , and is false otherwise. A rule h ← b is false in I if b is true in I and h is false in I . The rule is true otherwise. A knowledge base KB is true in I if and only if every clause in KB is true in I . � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 9

  10. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Models and Logical Consequence A model of a set of clauses is an interpretation in which all the clauses are true . If KB is a set of clauses and g is a body, g is a logical consequence of KB , written KB | = g , if g is true in every model of KB . That is, KB | = g if there is no interpretation in which ... � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 10

  11. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Simple Example  p ← q .  KB = q . r ← s .  p q r s model? I 1 true true true true I 2 false false false false I 3 true true false false I 4 true true true false I 5 true true false true Which of p , q , r , s , t logically follow from KB? � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 11

  12. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning User’s view of Semantics 1. Choose a task domain: intended interpretation. 2. Associate an atom with each proposition you want to represent. 3. Tell the system clauses that are true in the intended interpretation: axiomatizing the domain. 4. Ask questions about the intended interpretation. 5. If KB | = g , then g must be true in the intended interpretation. 6. Users can interpret the answer using their intended interpretation of the symbols. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 12

  13. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Computer’s view of semantics The computer doesn’t have access to the intended interpretation. All it knows is the knowledge base. The computer can determine if a formula is a logical consequence of KB. If KB | = g then g must be true in the intended interpretation. If KB �| = g then there is a model of KB in which g is false. This could be the intended interpretation. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 13

  14. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Electrical Environment outside power cb1 w5 s1 w1 circuit breaker cb2 s2 w2 w3 off s3 w0 switch on w6 w4 two-way p2 switch l1 light p1 l2 power outlet � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 14

  15. Propositions and Semantics Proofs Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Representing the Electrical Environment lit l 1 ← live w 0 ∧ ok l 1 light l 1 . live w 0 ← live w 1 ∧ up s 2 . light l 2 . live w 0 ← live w 2 ∧ down s 2 . down s 1 . live w 1 ← live w 3 ∧ up s 1 . up s 2 . live w 2 ← live w 3 ∧ down s 1 . up s 3 . lit l 2 ← live w 4 ∧ ok l 2 . ok l 1 . live w 4 ← live w 3 ∧ up s 3 . ok l 2 . live p 1 ← live w 3 . ok cb 1 . live w 3 ← live w 5 ∧ ok cb 1 . ok cb 2 . live p 2 ← live w 6 . live outside . live w 6 ← live w 5 ∧ ok cb 2 . � D. Poole and A. Mackworth 2010 c live w 5 ← live outside . Artificial Intelligence, Lecture 5., Page 15

  16. Propositions and Semantics Proofs Bottom-up Proof Procedure Ask-the-user and Knowledge-level Debugging Top-down Proof Procedure Complete Knowledge Assumption Assumption-based Reasoning Outline Propositions and Semantics Proofs Bottom-up Proof Procedure Top-down Proof Procedure Ask-the-user and Knowledge-level Debugging Complete Knowledge Assumption Assumption-based Reasoning Proof by Contradiction Abduction � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 5., Page 16

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