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159.302 LECTURE

159.302 LECTURE. 1. Propositional Logic Syntax. Source: MIT OpenCourseWare. What is a logic?. A formal language. Propositional Logic. What is a logic?. A formal language. Propositional Logic. Syntax – what expressions are legal. What is a logic?. A formal language.

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159.302 LECTURE

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  1. 159.302LECTURE 1 Propositional Logic Syntax Source: MIT OpenCourseWare

  2. What is a logic? A formal language Propositional Logic

  3. What is a logic? A formal language Propositional Logic • Syntax – what expressions are legal

  4. What is a logic? A formal language Propositional Logic • Syntax – what expressions are legal • Semantics – what legal expressions mean

  5. What is a logic? A formal language Propositional Logic • Syntax – what expressions are legal • Semantics – what legal expressions mean • ProofSystem – a way of manipulating syntactic expressions to get other syntactic expressions (which will tell us something new)

  6. What is a logic? A formal language Propositional Logic • Syntax – what expressions are legal • Semantics – what legal expressions mean • ProofSystem – a way of manipulating syntactic expressions to get other syntactic expressions (which will tell us something new) Why proofs? • Two kinds of inferences an agent might want to make:

  7. What is a logic? A formal language Propositional Logic • Syntax – what expressions are legal • Semantics – what legal expressions mean • ProofSystem – a way of manipulating syntactic expressions to get other sytactic expressions (which will tell us something new) Why proofs? • Two kinds of inferences an agent might want to make: • Multiplepercepts => conclusions about the world

  8. What is a logic? A formal language Propositional Logic • Syntax – what expressions are legal • Semantics – what legal expressions mean • ProofSystem – a way of manipulating syntactic expressions to get other sytactic expressions (which will tell us something new) Why proofs? • Two kinds of inferences an agent might want to make: • Multiplepercepts => conclusions about the world • Currentstate & operator => properties of next state

  9. Syntax: what you’re allowed to write Propositional Logic • for( thing t=fizz; t == fuzz; t++ ){…}

  10. Syntax: what you’re allowed to write Propositional Logic • for(thing t=fizz; t == fuzz; t++){…} • Colorless green ideas sleep furiously.

  11. Syntax: what you’re allowed to write Propositional Logic • for(thing t=fizz; t == fuzz; t++){…} • Colorless green ideas sleep furiously. Sentences (wwfs: well-formed formulas)

  12. Syntax: what you’re allowed to write Propositional Logic • for(thing t=fizz; t == fuzz; t++){…} • Colorless green ideas sleep furiously. Sentences (wwfs: well-formed formulas) • true and false are sentences

  13. Syntax: what you’re allowed to write Propositional Logic • for(thing t=fizz; t == fuzz; t++){…} • Colorless green ideas sleep furiously. Sentences (wwfs: well-formed formulas) • true and false are sentences • Propositional variables are sentences: P, Q, R, Z

  14. Syntax: what you’re allowed to write Propositional Logic • for(thing t=fizz; t == fuzz; t++){…} • Colorless green ideas sleep furiously. Sentences (wwfs: well-formed formulas) • true and false are sentences • Propositional variables are sentences: P, Q, R, Z • If φ and ψ are sentences, then so are

  15. Syntax: what you’re allowed to write Propositional Logic • for(thing t=fizz; t == fuzz; t++){…} • Colorless green ideas sleep furiously. Sentences (wwfs: well-formed formulas) • true and false are sentences • Propositional variables are sentences: P, Q, R, Z • If φ and ψ are sentences, then so are • Nothing else is a sentence

  16. Precedence

  17. 159.302LECTURE 2 Propositional Logic Semantics Source: MIT OpenCourseWare

  18. Semantics • Meaning of a sentence is truth value {t, f} Propositional Logic

  19. Semantics • Meaning of a sentence is truth value {t, f} Propositional Logic • Interpretation is an assignment of truth values to the propositional variables holds( , i) [Sentence is t in interpretation i]

  20. Semantics • Meaning of a sentence is truth value {t, f} Propositional Logic • Interpretation is an assignment of truth values to the propositional variables holds( , i) [Sentece is t in interpretation i] fails( , i) [Sentence is f in interpretation i]

  21. Semantic Rules holds(true, i) for all i Propositional Logic

  22. Semantic Rules holds(true, i) for all i Propositional Logic fails(false, i) for all i

  23. Semantic Rules holds(true, i) for all i Propositional Logic fails(false, i) for all i holds( , i) if and only if fails( , i)(negation)

  24. Semantic Rules holds(true, i) for all i Propositional Logic fails(false, i) for all i holds( , i) if and only if fails( , i)(negation) holds( , i) iff holds( ,i) and holds( , i)(conjunction)

  25. Semantic Rules holds(true, i) for all i Propositional Logic fails(false, i) for all i holds( , i) if and only if fails( , i)(negation) holds( , i) iff holds( ,i) and holds( , i)(conjunction) holds( , i) iff holds( ,i) or holds( , i)(disjunction)

  26. Semantic Rules holds(true, i) for all i Propositional Logic fails(false, i) for all i holds( , i) if and only if fails( , i)(negation) holds( , i) iff holds( ,i) and holds( , i)(conjunction) holds( , i) iff holds( ,i) or holds( , i)(disjunction) holds( P, i) iff i(P) = t fails( P, i) iff i(P) = f

  27. Some important shorthand (antecedent consequent) Propositional Logic

  28. Some important shorthand (antecedent consequent) Propositional Logic

  29. Some important shorthand (antecedent consequent) Propositional Logic

  30. Some important shorthand (antecedent consequent) Propositional Logic Note that implication is not “causality”, if P is f then is t Implication doesn't mean "is related" in any kind of real-world way

  31. Terminology A sentence is valid iff its truth value is t in all interpretations Propositional Logic A sentence is satisfiable iff its truth value is t in at least one interpretation A sentence is unsatisfiable iff its truth value is f in all interpretations All are finitely decidable.

  32. Examples contrapositive

  33. Examples contrapositive

  34. Examples contrapositive

  35. Examples contrapositive

  36. Examples contrapositive

  37. Related to constraint satisfaction • Given a sentence S, try to find an interpretation i such that holds(S,i) Satisfiability • Analogous to finding an assignment of values to variables such that the constraints hold

  38. Related to constraint satisfaction • Given a sentence S, try to find an interpretation i such that holds(S,i) Satisfiability • Analogous to finding an assignment of values to variables such that the constraints hold • Brute force method: enumerate all interpretations and check

  39. Related to constraint satisfaction • Given a sentence S, try to find an interpretation i such that holds(S,i) Satisfiability • Analogous to finding an assignment of values to variables such that the constraints hold • Brute force method: enumerate all interpretations and check • Better methods: Heuristic search Constraint propagation Stochastic search

  40. Scheduling nurses to work in a hospital • propositional variables represent for example, that Pat is working on Tuesday at 2pm Satisfiability Problems • constraints on the schedule are represented using logical expressions over the variables • Finding bugs in software • propositional variables represent state of the program • use logic to describe how the program works and to assert there is a bug • if the sentence is satisfiable, you’ve found the bug!

  41. 159.302LECTURE 3 Propositional Logic Proof Source: MIT OpenCourseWare

  42. Imagine we knew that: • If today is sunny, then Tomas will be happy A Good lecture? • If Tomas is happy, the lecture will be good • Today is sunny Should we conclude that the lecture will be good?

  43. Checking Interpretations Good lecture!

  44. Adding a variable

  45. A knowledge base (KB) entails a sentence S iff every interpretation that makes KB true also makes S true Entailment

  46. enumerate all interpretations • select those in which all elements of KB are true Computing Entailment • check to see if S is true in all of those interpretations

  47. enumerate all interpretations • select those in which all elements of KB are true Computing Entailment • check to see if S is true in all of those interpretations • Way too many interpretations, in general!!

  48. A proof is a way to test whether a KB entails a sentence, without enumerating all possible interpretations Entailment and Proof

  49. A proof is a sequence of sentences • First ones are premises (KB) Proof • Then, you can write down on the next line the result of applying an inference rule to previous lines • When S is on a line, you have provedS from KB • If inference rules are sound, then any S you can prove from KB is entailed by KB • If inference rules are complete, then any S that is entailed by KBcan be proven from KB

  50. Some inference rules: Natural Deduction Modus ponens

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