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“High Dimension CNF To DNF Conversion in Grid Computing”

“High Dimension CNF To DNF Conversion in Grid Computing”. Presented By: Mayuresh S. Pardeshi SY M.Tech , CSE – 23.

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“High Dimension CNF To DNF Conversion in Grid Computing”

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  1. “High Dimension CNF To DNF Conversion in Grid Computing” Presented By: Mayuresh S. Pardeshi SY M.Tech, CSE – 23 Department of Computer Science and Engineering. Walchand College of Engineering, Sangli.

  2. Contents • Introduction • Schedule • Objectives • Literature survey • Methodology • Study of logic parallel algorithms • Results

  3. Problem Statement Design and implementation of High dimension optimal conjunctive normal form to optimal(prime implicants) disjunctive normal form conversion which is an “NP hard problem conversion to an NP complete”.

  4. Introduction • Broad Area Conversion of NP hard problem to NP complete • Need for research To achieve maximum possible performance for simplification • Significance Vast field Impact • Concepts conjunctive normal form disjunctive normal form

  5. Schedule:

  6. Objectives: Achieve maximum simplification of Boolean functions. Operations on high dimension data as much as possible. Improve performance by utilizing grid resources in parallel.

  7. Literature Survey: On converting CNF to DNF [1] References Wegener, I. , “The Complexity of Boolean Functions” , John Wiley & Sons Ltd, and B. G. Teubner, Stuttgart..References Peter Bro Miltersen, JaikumarRadhakrishnan and Ingo Wegener “ On Converting CNF To DNF” Electronic Colloquium on Computational Complexity, Report No. 17 (2003) [2] References Hisayuki TATSUMI , Masahiro MIYAKAWA Masao MUKAIDONO , ”Upper and Lower Bounds on the Number of Disjunctive Forms” , Proceedings of the 36th International Symposium on Multiple-Valued Logic (ISMVL’06), IEEE computer society.

  8. Methodology:

  9. Multi-Line Boole’s algorithm:

  10. Example: We save the first step and define f ∈ B4 by Q4 , the set of implicants of length 4. Q4: Q4,4 = ø , Q4,3 = {a b c d ;a b c d },Q4,2 = {a b c d ;a b c d ; a b c d }, Q4,1 = {a b c d ;a b c d ; a b c d},Q4,0 = {a b c d} . Q3: Q3,3 = ø , Q3,2 = {a b c ;a b c ; b c d } Q3,1 = {a b d ; a c d ; a b c ; a c d ; b c d }, Q3,0 = {a b c ; a c d ; b c d }, P4 = ø. Q2: Q2,2 = ø , Q2;1 = {b c} , Q2;0 = {c d ; a c} P3 = {a b c; a b d} . Q1 = ø . P2 = Q2 .  PI(f) = { a b ; ac ; a d }:

  11. Results for multi-line boolean algorithm:

  12. CNF algorithm:

  13. Result of CNF conversion: Problem statement taken is: ∀x [∀y Animal(y) ⇒ Loves(x, y)] ⇒ [∃y Loves(y, x)] Solution achieved using coding: Eliminate implications: ∀x [¬∀y ¬Animal(y) ∨ Loves(x, y)] ∨ [∃y Loves(y, x)] 2. Move ¬ inwards • ∀x [∃y ¬(¬Animal(y) ∨ Loves(x, y))] ∨ [∃y Loves(y, x)] • ∀x [∃y ¬¬Animal(y) ∧ ¬Loves(x, y)] ∨ [∃y Loves(y, x)] (De Morgan) • ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃y Loves(y, x)] (double negation) 3. Standardize variables: ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃z Loves(z, x)] 4. Skolemization: ∀x [Animal(F(x)) ∧ ¬Loves(x, F(x))] ∨ [Loves(G(x), x)] 5. Drop universal quantifiers: [Animal(F(x)) ∧ ¬Loves(x, F(x))] ∨ [Loves(G(x), x)] 6. Distribute ∨ over ∧: [Animal(F(x)) ∨ Loves(G(x), x)] ∧ [¬Loves(x, F(x)) ∨ Loves(G(x), x)]

  14. DNF algorithm:

  15. Various parallel algorithms for logic implementation : GridSAT ManySAT SArTagnan: A Parallel portfolio solver Plingeling

  16. GridSAT Sequential algorithm

  17. GridSAT Parallel Version:

  18. SArTagnan: A Parallel portfolio solver

  19. Plingeling : Multi Threaded version For each worker a separate duplicated instance thread is generated Send/Receive generated clause. Function: Produce, Consume and Terminate Strategies used are differentiated around the amount of pre-processing, random seeds and variables branching.

  20. Shared Clause architecture:

  21. REFERENCES: [1] References Wegener, I. , “The Complexity of Boolean Functions” , John Wiley & Sons Ltd, and B. G. Teubner, Stuttgart..References Peter Bro Miltersen, JaikumarRadhakrishnan and Ingo Wegener “ On Converting CNF To DNF” Electronic Colloquium on Computational Complexity, Report No. 17 (2003) [2] References Hisayuki TATSUMI , Masahiro MIYAKAWA Masao MUKAIDONO , ”Upper and Lower Bounds on the Number of Disjunctive Forms” , Proceedings of the 36th International Symposium on Multiple-Valued Logic (ISMVL’06), IEEE computer society. [3] References Paul Beame, “A Switching Lemma Primer”, University of Toronto. [4] References RajatAuora and Michael S. Hsiao, “CNF Formula Simplification Using Implication Reasoning" 0-7803-8714-7104 2004 IEEE Transactions. [5] References Kenneth H. Rosen, “Discrete Mathematics and It's Applications” , 6th Edition, CRC Press. [6] References Wahid Chrabakh , Rich Wolski , “GrADSAT: A Parallel SAT Solver for the Grid”, UCSB Computer Science Technical Report Number 2003-05, Department of Computer Science, University of California Santa Barbara [7] References R. Zuim, J.T. de Sousa and C.N. Coelho, “Decision heuristic for Davis Putnam, Loveland and Logemann algorithm satisfiability solving based on cube subtraction”, IET Comput. Digit. Tech., 2008, 2, (1), pp. 30–39

  22. Thank You!

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