C LAUS B RABRAND Inference Systems J AN 27, 2009 I NFERENCE S YSTEMS Claus Brabrand IT University of Copenhagen [ ]

[ 2 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Monty Python "Monty Python and the Holy Grail" (1974) Scene V: "The Witch":

[ 3 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Deduction vs. Induction Deduction: whole  parts: (aka. “top-down reasoning”) abstract  concrete general  specific Induction: parts  whole: (aka. “bottom-up reasoning”) concrete  abstract specific  general Just two different ways of reasoning: Deduction  Induction (just swap directions of arrows) whole A B C A B C

[ 5 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Deductive Reasoning: witch(girl) "Deduction": witch(girl) burns(girl)female(girl) wooden(girl) floats(girl) floats(duck)sameweight(girl,duck) %- by observation %- by experiment -----%- King Arthur floats(X)  sameweight(X,Y)  floats(Y) witch(X)  female(X)  burns(X) burns(X)  wooden(X) wooden(X)  floats(X)  (aka. ”top-down reasoning”)

[ 6 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Inductive Reasoning: witch(girl) "Induction": witch(girl) burns(girl)female(girl) wooden(girl) floats(girl) floats(duck)sameweight(girl,duck) %- by observation %- by experiment -----%- King Arthur floats(X)  sameweight(X,Y)  floats(Y) witch(X)  female(X)  burns(X) burns(X)  wooden(X) wooden(X)  floats(X)  (aka. ”bottom-up reasoning”)

[ 7 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems  Deduction vs. Induction Deduction whole  parts: Induction parts  whole:  Just two different ways of reasoning: Deduction  Induction (just swap directions of arrows)

[ 8 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Hearing: Nomination of CIA Director, General Michael Hayden (USAF). LEVIN: "You in my office discussed, I think, a very interesting approach, which is the difference between starting with a conclusion and trying to prove it and instead starting with digging into all the facts and seeing where they take you. Would you just describe for us that difference and why [...]?" LEVIN: U.S. SENATOR CARL LEVIN (D-MI) HAYDEN: GENERAL MICHAEL B. HAYDEN (USAF), NOMINEE TO BE DIRECTOR OF CIA CQ Transcriptions Thursday, May 18, 2006; 11:41 AM "DEDUCTIVE vs. INDUCTIVE REASONING" 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."

C LAUS B RABRAND Inference Systems J AN 27, 2009 I NFERENCE S YSTEMS Keywords: relations, axioms, rules

[ 10 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Relations Example 1 : “even” relation: Written as: as a short-hand for: … and as: as a short-hand for: Example 2 : “equals” relation: Written as: as a short-hand for: … and as: as a short-hand for: Example 3 : “road” relation: Written as: as a s-h for: … and as: as a s-h for: | _ even  N | _ even 4 | _ even 5 4  | _ even 5  | _ even 2  3 (2,3)  ‘=’ ‘=’  N  N (2,2)  ‘=’ 2 = 2 ‘  ’  CITY  N  CITY KBH  Aarhus 305 (KBH, 305, Aarhus)  ‘  ’ KBH  NewYork x (KBH, x, New York)  ‘  ’

[ 11 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Inference System Inference System: is used for specifying relations consists of axioms and rules Example: Axiom: “0 (zero) is even”! Rule: “If n is even, then m is even (where m = n+2)” | _ even 0 | _ even n | _ even m m = n+2 | _ even  N

[ 12 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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 | _ even n | _ even m m = n+2 premise(s) conclusion side-condition(s)

[ 13 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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” | _ even n | _ even n+2 | _ even n | _ even m m = n+2 Even so; this is what we mean

[ 14 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Relation Membership? x  R Axiom: “0 (zero) is even”! Rule: “n+2 is even, if n is even” Is 6 even?!? The inference tree proves that: | _ even 0 | _ even n | _ even n+2 | _ even 0 | _ even 2 | _ even 4 | _ even 6 [rule 1 ] [axiom 1 ] inference tree 6  | _ even ? | _ even 6 written:

[ 15 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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: 0  0 n  m n  m+1 [rule 1 ] [axiom 1 ] ‘  ’  N  N n  m n+1  m+1 [rule 2 ] 0  0 0  1 1  2 [rule 2 ] [rule 1 ] [axiom 1 ] 0  0 1  1 1  2 [rule 1 ] [rule 2 ] [axiom 1 ]

[ 16 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Activation Exercise 1 Activation Exercise: 1. Specify the signature of the relation: ' << ' x << y " y is-double-that-of x " 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 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Activation Exercise 2 Activation Exercise: 1. Specify the signature of the relation: ' // ' x // y " x is-half-that-of y " 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 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Example: “add” Relation: Is ”2 + 2 = 4” ?!? Yes, because there exists an inf. tree for "+(2,2,4)": +(0,m,m) [axiom 1 ] ‘+’  N  N  N +(n,m,r) +(n+1,m,r+1) [rule 1 ] +(0,2,2) +(1,2,3) +(2,2,4) [rule 1 ] [axiom 1 ]

[ 19 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems Example: “add” (cont’d) Relation: Note: Many different inf. sys.’s may exist for same relation: +(0,m,m) [axiom 1 ] ‘+’  N  N  N +(n,m,r) +(n+1,m,r+1) [rule 1 ] +(0,m,m) [axiom 1 ] +(n,m,r) +(n+2,m,r+2) [rule 1 ] +(1,m,m+1) [axiom 2 ]

[ 20 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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 R f  A  B  a  A, b 1,b 2  B: R f (a,b 1 )  R f (a,b 2 ) => b 1 = b 2

[ 21 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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, r 1,r 2  N: R + (n,m,r 1 )  R + (n,m,r 2 ) => r1 = r2

[ 22 ] C LAUS B RABRAND J AN 27, 2009 Inference Systems 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