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Statistical properties of lambda terms Maciej Bendkowski Olivier - PowerPoint PPT Presentation

Lambda terms Statistical properties of lambda terms Maciej Bendkowski Olivier Bodini Sergey Dovgal CLA, Jussieu, Paris, 24/05/2018 Lambda terms Chapter 1. Historical overview Lambda terms Historical overview Lambda terms Historical


  1. Lambda terms Historical overview Natural counting Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

  2. Lambda terms Historical overview Natural counting Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

  3. Lambda terms Historical overview Natural counting Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

  4. Lambda terms Historical overview Natural counting Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

  5. Lambda terms Historical overview Why consider a simpler model?

  6. Lambda terms Historical overview Why consider a simpler model? Main reason : advantages in random generation

  7. Lambda terms Historical overview Why consider a simpler model? Marking variables imply control over expectations

  8. Lambda terms Historical overview Why consider a simpler model? Another reason : parameter study

  9. Lambda terms Historical overview Why consider a simpler model? Marking variables allow to study distributions

  10. Lambda terms Statistical properties Chapter 2. Statistical properties

  11. Lambda terms Statistical properties

  12. Lambda terms Statistical properties Plain lambda terms

  13. Lambda terms Statistical properties Generating function for plain lambda terms λ @ L = + + D L L L z L ( z ) = zL ( z ) + zL 2 ( z ) + 1 − z

  14. Lambda terms Statistical properties Abstractions in plain terms? λ @ = D L + + L L L z L ( z , u ) = zuL ( z , u ) + zL ( z , u ) + 1 − z � z L ( z , u ) ∼ a ( z , u ) − b ( z , u ) 1 − ρ ( u )

  15. Lambda terms Statistical properties Multivariate Central Limit Theorem

  16. Lambda terms Statistical properties Step 1 . Extract coefficient of L ( z , u )

  17. Lambda terms Statistical properties � z L ( z , u ) ∼ a ( z , u ) − b ( z , u ) 1 − ρ ( u )

  18. Lambda terms Statistical properties Step 2 . Asymptotic behaviour of probability generating function

  19. Lambda terms Statistical properties p n ( u ) ∼ A ( u ) B ( u ) n

  20. Lambda terms Statistical properties Step 3. Gaussian approximation from A ( u ) B ( u ) n .

  21. Lambda terms Statistical properties Applying multivariate CLT Theorem Joint Gaussian distribution for number of abstractions number of variables number of redexes in plain lambda terms

  22. Lambda terms Statistical properties Discrete distributions in plain terms

  23. Lambda terms Statistical properties Head abstractions H H L ∞ = + D @ L ∞ | u = 1 L ∞ | u = 1 � � 1 z 1 − z + zL ( z , 1 ) 2 L ( z , u ) = 1 − zu

  24. Lambda terms Statistical properties Theorem The following statistics follow discrete (geometric) limiting distributions The number of head abstractions Randomly chosed value of de Bruijn index

  25. Lambda terms Infinite systems Chapter 3. Infinite systems

  26. Lambda terms Infinite systems Drmota–Lalley–Woods theorem

  27. Lambda terms Infinite systems Drmota–Lalley–Woods theorem Let F ( z ) be a generating function Suppose it satisfies F ( z ) = Φ( F ( z ) , z ) with Φ having combinatorial origin Then, � 1 − z F ( z ) ∼ a − b ρ

  28. Lambda terms Infinite systems Drmota–Lalley–Woods theorem Let F ( z ) be a generating function Suppose it satisfies F ( z ) = Φ( F ( z ) , z ) with Φ having combinatorial origin Then, � 1 − z F ( z ) ∼ a − b ρ

  29. Lambda terms Infinite systems Drmota–Lalley–Woods theorem Let F ( z ) be a generating function Suppose it satisfies F ( z ) = Φ( F ( z ) , z ) with Φ having combinatorial origin Then, � 1 − z F ( z ) ∼ a − b ρ

  30. Lambda terms Infinite systems What happens with infinite systems?

  31. Lambda terms Infinite systems [Drmota, Gitenberger, Morgenbesser ’12] If Jacobian is a sum of identity matrix and a compact operator And specification is strongly connected Then Infinite-dimensional version holds

  32. Lambda terms Infinite systems Closed lambda terms satisfy an infinite system

  33. Lambda terms Infinite systems L 0 ( z ) = zL 1 ( z ) + zL 0 ( z ) 2 , L 1 ( z ) = zL 2 ( z ) + zL 1 ( z ) 2 + z , L 2 ( z ) = zL 3 ( z ) + zL 2 ( z ) 2 + z + z 2 , . . . z L ∞ ( z ) = zL ∞ ( z ) + zL ∞ ( z ) + 1 − z .

  34. Lambda terms Infinite systems Why?

  35. Lambda terms Infinite systems Adding m abstractions on the top of L m makes the term closed

  36. Lambda terms Infinite systems Example.

  37. Lambda terms Infinite systems Example. λ @ 0 λ @ @ @ 1 0 0 3

  38. Lambda terms Infinite systems λ @ L m D m = + + L m + 1 L m L m m ( z ) + z 1 − z m L m ( z ) = zL m + 1 ( z ) + zL 2 1 − z

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