day40 - Dec 2 UCF(Charles E 44 Encoding Use(p p as form of...

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 11 Diadic PIPC Diadic limits us to two variables PIPC means Partial Implicational Propositional Calculus, and limits us to implication as only connective Partial just means we get a fragment Problems Is fragment complete? Can F be derived by substitution and

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 22 Living without Associativity Consider a two-stack model of a TM Could somehow use one variable for left stack and other for right Must find a way to encode a sequence as a composition of forms – that’s the key to this simulation
10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 33 Composition Encoding Consider (p ° p), (p ° (p ± p) ), (p ° (p “ (p ° p) ) ), … No form is a substitution instance of any of the other, so they can’t be confused All are tautologies Consider ((X p Y) ° Y) This is just X ° Y

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E.

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Unformatted text preview: Dec 2, UCF (Charles E. 44 Encoding Use (p p) as form of bottom of stack Use (p (p p)) as form for letter 0 Use (p (p (p p))) as form for 1 Etc. String 01 (reading top to bottom of stack) is ( ( (p &amp;amp; p) ( (p &amp;amp; p) ( (p &amp;amp; p) (p &amp;amp; p) ) ) ) 1 ( ( (p &amp;amp; p) ( (p p) ( (p &amp;amp; p) (p &amp;amp; p) ) ) ) 1 ( (p &amp;amp; p) 1 ( (p &amp;amp; p) ( (p &amp;amp; p) (p &amp;amp; p) ) ) ) ) ) 10/4/11 Dec 2, UCF (Charles E. 55 Encodings 10/4/11 Dec 2, UCF (Charles E. 66 Creating Terminal IDs 10/4/11 Dec 2, UCF (Charles E. 77 Reversing Print and Left 10/4/11 Dec 2, UCF (Charles E. 88 Reversing Right 10/4/11 Dec 2, UCF (Charles E. 99 The Rest of the Story Its in the paper Result is that word decision problem for membership in the theorems of a diadic pipc is undecidable...
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