sb fol iii - rensselaer polytechnic institutekryten.mm.rpi.edu/courses/intlogw/sb_fol_iii.pdf ·...

FOL III

Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive Science

Department of Computer Science Lally School of Management & Technology

Rensselaer Polytechnic Institute (RPI)Troy, New York 12180 USA

Intro to Logic3/5/2018

Selmer Bringsjord

Live-action on HyperGrader redux …

ThxForThePCOracle

Interlude re Formal Logic & Games …

The Entscheidungsproblem

Praiseworthy AI simplicter, perhaps.


But certainly not AI = HI!


But certainly not AI = HI!

“AlphaGo, from the perspective of South, how many majuscule Roman letters are in black? Why do you say that?”

Logico-Mathematical Landscape that Has Them Turning in Their Graves

Polynomial Hierarchy

Arithmetical Hierarchy

Analytical HierarchySuper-Serious Human Cognitive Power

Serious Human Cognitive Power

Mere Calculative Cognitive Power Entscheidungsproblem






Mere Calculative Cognitive Power

Descartes

Entscheidungsproblem







LeibnizDescartes








Leibniz ChurchDescartes








Leibniz ChurchDescartes Gödel








Leibniz Church TuringDescartes Gödel





Analytical Hierarchy









Leibniz Church Turing

P ✓ NP ✓ PSPACE = NPSPACE ✓ EXPTIME ✓ NEXPTIME ✓ EXPSPACE

Descartes Gödel








⌃0

⌃1

⇧1

⌃2

⇧2

...Descartes Gödel







Go: AlphaGo


⌃0

⌃1

⇧1

⌃2

⇧2

...Descartes Gödel







Go: AlphaGoJeopardy!: Watson


⌃0

⌃1

⇧1

⌃2

⇧2

...Descartes Gödel







Go: AlphaGoJeopardy!: WatsonChess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Descartes Gödel







Checkers: ChinookGo: AlphaGoJeopardy!: Watson

Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Descartes Gödel







Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Leibniz Church TuringDescartes Gödel







Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Leibniz Church TuringDescartes







Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Leibniz TuringDescartes







Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Leibniz Turing







Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...Leibniz







Chess: Deep Blue


⌃0

⌃1

⇧1

⌃2

⇧2

...


Rengo Kriegspiel

Rengo Kriegspiel

“One of the authors has personally played this game, and it’s intriguing to think that it’s possible he has played the hardest game in the world, which cannot even in principle be played by any algorithm. (Hearn & Domaine 2009, sect 3.4.2, para. 2)

But starting simpler …

Tabular “Deduction” (Example)

13


14


15


15

In Slate …


16


17

Tabular “Deduction”: It’s Taught!

18


18

X


18

XIMHO very bad idea—if before real learning of deduction to answer “Why, exactly? Prove it!”

Tabular “deduction” not the skill that’s needed.

19

Back to FOL …

Our Final New Inference Rule in FOL


• existential elimination (intuitively put):


• existential elimination (intuitively put):

• If we know that (i) there’s something x which is an R, and (ii) on the supposition that a is an arbitrary representative (a “witness”) of such an x we can prove P, then we are permitted to deduce P from (i) alone.

existential elimination, precise version:

Starting State of Workspace(Assumes a domain of e.g. players on a

March-madness basketball court.)

Starting State of Workspace

“Each and every thing is either a player or a referee.”

(Assumes a domain of e.g. players on a March-madness basketball court.)



“All players are tall.”




“All players are tall.”

“There’s a non-tall thing.”


Step 1

Step 1

The witness!

Step 2

The witness!

Step 3

The witness!

Step 4

The witness!

Step 4

The witness!

We earned our stripes!

Step 5

The witness!


Step 6

The witness!


Practice Problems in Slate!

{8x(Scared(x) $ Small(x)), 9x¬Scared(x)} ` 9x¬Small(x)

{9x, yContiguous(x, y), 8x, y(Contiguous(x, y) ! ¬SameCountry(x, y))} ` 9x, y¬SameCountry(x, y)

sb fol iii - rensselaer polytechnic institutekryten.mm.rpi.edu/courses/intlogw/sb_fol_iii.pdf ·...

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