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I NFERENCE S YSTEMS

I NFERENCE S YSTEMS. Claus Brabrand brabrand@itu.dk IT University of Copenhagen [ http://www.itu.dk/people/brabrand/ ]. Monty Python. "Monty Python and the Holy Grail" (1974) Scene V: " The Witch " :. The Monty Python Reasoning:. "Axioms" (aka. "Facts") : "Rules":.

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I NFERENCE S YSTEMS

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  1. INFERENCE SYSTEMS Claus Brabrand brabrand@itu.dk IT University of Copenhagen [http://www.itu.dk/people/brabrand/]

  2. Monty Python • "Monty Python and the Holy Grail" (1974) • Scene V: "The Witch":

  3. The Monty Python Reasoning: • "Axioms" (aka. "Facts"): • "Rules": female(girl) %- by observation ----- floats(duck) %- King Arthur ----- sameweight(girl,duck) %- by experiment ----- witch(X)  female(X)  burns(X) burns(X)  wooden(X) wooden(X)  floats(X) floats(X)  sameweight(X,Y)  floats(Y)

  4. Deduction: whole  parts: (aka. “top-down reasoning”) abstract  concrete general  specific Induction: parts  whole: (aka. “bottom-up reasoning”) concrete  abstract specific  general whole whole A B C A B C Deduction vs. Induction • Just two different ways of reasoning: • Deduction  Induction (just swap directions of arrows)

  5. Deductive Reasoning: witch(girl) (aka. ”top-down reasoning”) • "Deduction": witch(girl) witch(X)  female(X)  burns(X)  female(girl) burns(girl) %- by observation ----- burns(X)  wooden(X) wooden(girl) wooden(X)  floats(X) floats(girl) floats(X)  sameweight(X,Y)  floats(Y) sameweight(girl,duck) floats(duck) %- by experiment ----- %- King Arthur -----

  6. Inductive Reasoning: witch(girl) (aka. ”bottom-up reasoning”) • "Induction": witch(girl) witch(X) female(X)  burns(X)  female(girl) burns(girl) %- by observation ----- burns(X)  wooden(X) wooden(girl) wooden(X)  floats(X) floats(girl) floats(X)  sameweight(X,Y)  floats(Y) sameweight(girl,duck) floats(duck) %- by experiment ----- %- King Arthur -----

  7. Deduction whole  parts: Induction parts  whole: Deduction vs. Induction   • Just two different ways of reasoning: • Deduction  Induction (just swap directions of arrows)

  8. Hearing: Nomination of CIA Director, General Michael Hayden (USAF). LEVIN: U.S. SENATOR CARL LEVIN (D-MI) HAYDEN: GENERAL MICHAEL B. HAYDEN (USAF), NOMINEE TO BE DIRECTOR OF CIA CQ TranscriptionsThursday, May 18, 2006; 11:41 AM "DEDUCTIVEvs. INDUCTIVE REASONING" LEVIN: "You in my office discussed, I think, a very interesting approach, which is the difference betweenstarting with a conclusion and trying to prove itand insteadstarting with digging into all the facts and seeing where they take you. Would you just describe for us that difference and why[...]?" HAYDEN: "Yes, sir. And I actually think I prefaced that with both of these are legitimate forms of reasoning,  that you've got deductive[...] in which you begin with, first, [general] principles and then you work your way down the specifics.  And then there's an inductive approach to the world in which you start out there with all the data and work yourself up to general principles. They are both legitimate."

  9. INFERENCE SYSTEMS Keywords: relations, axioms, rules

  10. Relations • Example1: “even” relation: • Written as: as a short-hand for:… and as: as a short-hand for: • Example2: “equals” relation: • Written as: as a short-hand for:… and as: as a short-hand for: • Example3: “road” relation: • Written as: as a s-h for:… and as: as a s-h for: |_even N |_even 4 4  |_even |_even 5 5  |_even ‘=’  N  N (2,2)  ‘=’ 2 = 2 (2,3)  ‘=’ 2  3 ‘’  CITY  N  CITY 305 KBH  Aarhus (KBH, 305, Aarhus)  ‘’ x KBH  NewYork (KBH, x, New York)  ‘’

  11. Inference System • Inference System: • is used for specifying relations • consists of axiomsand rules • Example: • Axiom: • “0(zero) is even”! • Rule: • “If n is even, then m is even (where m = n+2)” |_even N |_even0 |_evenn |_evenm m = n+2

  12. Terminology • Interpretation: • Deductive:“m is even, if n is even (where m = n+2)” • Inductive:“If n is even, then m is even (where m = n+2)”; or premise(s) |_evenn |_evenm side-condition(s) m = n+2 conclusion

  13. Abbreviation • Often, rules are abbreviated: • Rule: • “m is even, if n is even (where m = n+2)” • Abbreviated rule: • “n+2 is even, if n is even” |_evenn |_evenm m = n+2 Even so; this is what we mean |_evenn |_evenn+2

  14. [axiom1] |_even0 |_even2 |_even4 |_even6 [rule1] inference tree [rule1] [rule1] Relation Membership?xR ? • Axiom: • “0(zero) is even”! • Rule: • “n+2 is even, if n is even” • Is 6 even?!? • The inference treeproves that: |_even0 |_evenn |_evenn+2 written: 6 |_even |_even6

  15. Example: “less-than-or-equal-to” • Relation: • Is ”1  2”? (why/why not)!? [activation exercise] • Yes, because there exists an inference tree: • In fact, it has two inference trees: ‘’ N  N n  m n  m+1 n  m n+1  m+1 [axiom1] 0 0 [rule1] [rule2] [axiom1] [axiom1] 0  0 0 1 1 2 0 0 1 1 1 2 [rule1] [rule2] [rule2] [rule1]

  16. Activation Exercise 1 • Activation Exercise: • 1. Specify the signature of the relation: '<<' • x << y"yis-double-that-ofx" • 2. Specify the relation via an inference system • i.e. axioms and rules • 3. Prove that indeed: • 3 << 6"6 is-double-that-of 3"

  17. Activation Exercise 2 • Activation Exercise: • 1. Specify the signature of the relation: '//' • x // y"xis-half-that-ofy" • 2. Specify the relation via an inference system • i.e. axioms and rules • 3. Prove that indeed: • 3 // 6"3 is-half-that-of 6" Syntactically different: ‘<<‘ vs. ‘//’ Semantically the same relation: ‘<<‘ = ‘//’ = {(1,2), (2,4), …}

  18. Example: “add” • Relation: • Is ”2 +2 = 4” ?!? • Yes, because there exists an inf. tree for "+(2,2,4)": ‘+’ N  N  N +(n,m,r) +(n+1,m,r+1) [axiom1] +(0,m,m) [rule1] [axiom1] +(0,2,2) +(1,2,3) +(2,2,4) [rule1] [rule1]

  19. [axiom1] +(0,m,m) +(n,m,r) +(n+2,m,r+2) [rule1] [axiom2] +(1,m,m+1) Example: “add” (cont’d) • Relation: • Note: • Many different inf. sys.’s may exist for same relation: ‘+’ N  N  N +(n,m,r) +(n+1,m,r+1) [axiom1] +(0,m,m) [rule1]

  20. Relation vs. Function • A function... ...is a relation ...with the special requirement: • i.e., "the result", b, is uniquely determined from "the argument", a. f : AB Rf AB aA,b1,b2B: Rf(a,b1) Rf(a,b2) => b1 = b2

  21. Relation vs. Function (Example) • The (2-argument) function '+'... ...induces a (3-argument)relation ...that obeys: • i.e., "the result", r, is uniquely determined from "the arguments", n and m + : N N N R+ N N N n,mN,r1,r2N: R+(n,m,r1) R+(n,m,r2) => r1 = r2

  22. Exercises • For each of the following relations…: • a) Determine signature of the odd relation • b) Define the relation formally using an inf.sys. • c) Use the inference system to etablish that… • 1) “Odd” (written “|-odd”): • c) …5 is-odd; i.e., “|-odd 5” • 2) “Double-or-more” (written “x >2> y”): • c) …5 is-double-or-more-than 2; i.e., “5 >2> 2” • 3) “Sum-from-zero” (wr. “i = y” or “x  y”): • c) …4 sum-from-zero-is 10; i.e., “4  10” x i=0

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