Kolmogorov Complexity of Categories Complexity Programing Language - PowerPoint PPT Presentation

Kolmogorov Complexity of Categories Noson S. Yanofsky Kolmogorov Kolmogorov Complexity of Categories Complexity Programing Language Kolmogorov Noson S. Yanofsky Complexity of Categories Complexity Brooklyn College, CUNY Computability Algebra May 28, 2013 Future Directions

Outline of Talk Kolmogorov Complexity of Categories Classical Kolmogorov Complexity 1 Noson S. Yanofsky A Programing Language for Categorical Structures 2 Kolmogorov Complexity Kolmogorov Complexity of Categories 3 Programing Language Kolmogorov Complexity with Categorical Structures 4 Complexity of Categories Complexity Computability with Categorical Structures 5 Computability Algebra Kolmogorov Complexity of Algebraic Structure 6 Future Directions Future Directions 7

Classical Motivation Consider the following three strings: Kolmogorov Complexity of 1. 00000000000000000000000000000000000000000000000 Categories 2. 11011101111101111111011111111111011111111111110 Noson S. Yanofsky 3. 01010010110110101011011101111001100000111111010 Kolmogorov Complexity Programing Language Kolmogorov Complexity of Categories Complexity Computability Algebra Future Directions

Classical Motivation Consider the following three strings: Kolmogorov Complexity of 1. 00000000000000000000000000000000000000000000000 Categories 2. 11011101111101111111011111111111011111111111110 Noson S. Yanofsky 3. 01010010110110101011011101111001100000111111010 Kolmogorov Complexity 1. Print 45 0’s Programing Language 2. Print the first 6 primes Kolmogorov 3. Print 010100101101101010110111011110011000001111110 Complexity of Categories Complexity Computability Algebra Future Directions

Classical Motivation Consider the following three strings: Kolmogorov Complexity of 1. 00000000000000000000000000000000000000000000000 Categories 2. 11011101111101111111011111111111011111111111110 Noson S. Yanofsky 3. 01010010110110101011011101111001100000111111010 Kolmogorov Complexity 1. Print 45 0’s Programing Language 2. Print the first 6 primes Kolmogorov 3. Print 010100101101101010110111011110011000001111110 Complexity of Categories Complexity Let U be a universal Turing machine, then Computability K ( s ) = min {| p | : U ( p , λ ) = s } . Algebra Future Relative Kolmogorov complexity: Directions K ( s | t ) = min {| p | : U ( p , t ) = s } . If K ( s ) > | s | then s is “incompressible” or “random.”

The Sammy Programming Language Kolmogorov Constant Categories: 0 = ∅ ; 1 = ⋆ ; 2 = ⋆ − → ⋆ ; Cat . Complexity of Categories Constant Functors: s : 1 − → 2 ; t : 1 − → 2 . Noson S. Yanofsky If C = Source ( F : A − → B ), then C = A . If C = Target ( F : A − → B ), then C = B . Kolmogorov Complexity If F = Ident ( A ) then F = Id A . Programing Language If C = Op ( A ) then C = A op . The Op operation also acts Kolmogorov on functors. Complexity of Categories α = Hcomp ( β, γ ). Complexity α = Vcomp ( β, γ ). Computability Algebra Regular composition of functors is a special case of Future horizontal composition. Directions For categories A and B , we have C = Pow ( A , B ) be the category of all functors and natural transformations from A to B .

� � � � � � The Sammy Programming Language Kolmogorov For functors G : A − → B and F : A − → C , a right Kan Complexity of Categories extension is a pair ( R , α ) = KanEx ( G , F ) where Noson S. R : B − → C and α : R ◦ G − → F . Yanofsky R B C ❴ ❴ ❴ ❴ ❴ ❴ ❴ Kolmogorov � ❄ Complexity ❄ ⑧ ❄ ⑧ ❄ ⑧ ❄ ⑧ ❄ ⑧ Programing ❄ ⑧ G F ❄ ⑧ Language ⑧ A Kolmogorov Complexity of For every H : B − → C and β : H ◦ G − → F there is a Categories unique γ = KanInd ( F , G ; H , β ) where γ : H − → R and Complexity satisfies α · γ G = β . Computability Algebra H ❍ Future ❍ � ✈✈✈✈✈✈✈✈✈✈ ❍ β 0 β 1 Directions ❍ ❍ ! γ ❍ ❍ ❍ ❍ ❍ F 0 F 0 × F 1 α 1 F 1 α 0 .

The Sammy Programming Language Kolmogorov Complexity of Categories Left Kan extensions are made with the Op operation. Noson S. Yanofsky Using Kan extensions, one can derive products, Kolmogorov coproducts, pushouts, pullbacks, equalizers, coequalizers, Complexity (and constructible) limits, colimits, ends, coends, etc. Programing Language If G : A − → B is a right adjoint (left adjoint, equivalence, Kolmogorov isomorphism), then its left adjoint (right adjoint, Complexity of Categories quasi-inverse, inverse) G ∗ : B − → A can be found as a Complexity simple Kan extension of the identity Id A along G , that is, Computability G ∗ = KanEx ( G , Id A ). Algebra There are also Kan liftings operations. Future Directions Other operations...

Remarks About Sammy Kolmogorov Not the first programing language for Categories Complexity of Categories Rydeheard and Burstall: Computational Category Theory Noson S. Tatsuya Hagino: A Categorical Programming Language Yanofsky Kolmogorov Complexity Programing Language Kolmogorov Complexity of Categories Complexity Computability Algebra Future Directions

Remarks About Sammy Kolmogorov Not the first programing language for Categories Complexity of Categories Rydeheard and Burstall: Computational Category Theory Noson S. Tatsuya Hagino: A Categorical Programming Language Yanofsky Not the best programing language for Categories Kolmogorov e.g. Target from Source and Op Complexity Programing Language Kolmogorov Complexity of Categories Complexity Computability Algebra Future Directions

Remarks About Sammy Kolmogorov Not the first programing language for Categories Complexity of Categories Rydeheard and Burstall: Computational Category Theory Noson S. Tatsuya Hagino: A Categorical Programming Language Yanofsky Not the best programing language for Categories Kolmogorov e.g. Target from Source and Op Complexity Notice that numbers, strings, trees, graphs, arrays, and Programing Language other typical data types are not mentioned in Sammy. Kolmogorov They can be derived. Categories and algorithms are more Complexity of Categories “primitive” than numbers, strings, trees, etc. Complexity Computability Algebra Future Directions

Remarks About Sammy Kolmogorov Not the first programing language for Categories Complexity of Categories Rydeheard and Burstall: Computational Category Theory Noson S. Tatsuya Hagino: A Categorical Programming Language Yanofsky Not the best programing language for Categories Kolmogorov e.g. Target from Source and Op Complexity Notice that numbers, strings, trees, graphs, arrays, and Programing Language other typical data types are not mentioned in Sammy. Kolmogorov They can be derived. Categories and algorithms are more Complexity of Categories “primitive” than numbers, strings, trees, etc. Complexity In need of a Church-Turing type thesis that says that Computability anything that can be described by category theory can be Algebra described by Sammy. Future Directions No discussion of “self-delimiting.” Easily encode and decode Sammy programs as a number... or as a functor P : 1 − → N . Self-Reference!

Basic Definitions and Theorems Kolmogorov Complexity of Categories K Sammy ( C ) = K ( C ) = The smallest number of operations Noson S. Yanofsky needed to describe C . Kolmogorov An invariance theorem . The Kolmogorov complexity does not Complexity depend on which programing language is used. Programing Language Theorem Kolmogorov Complexity of There exists a constant c such that for all categorical Categories structures X we have | K Sammy ( X ) − K Saunders ( X ) | ≤ c. Complexity Computability Algebra Future Directions

Basic Definitions and Theorems Kolmogorov Complexity of Categories K Sammy ( C ) = K ( C ) = The smallest number of operations Noson S. Yanofsky needed to describe C . Kolmogorov An invariance theorem . The Kolmogorov complexity does not Complexity depend on which programing language is used. Programing Language Theorem Kolmogorov Complexity of There exists a constant c such that for all categorical Categories structures X we have | K Sammy ( X ) − K Saunders ( X ) | ≤ c. Complexity Computability Algebra Theorem Future There exists a constant c Kan such that for all G : A − → B and Directions F : A − → C if ( Lan G ( F ) , α ) is the left Kan extension, then K (( Lan G ( F ) , α )) ≤ K ( F ) + K ( G | F ) + c Kan

Basic Theorems Kolmogorov Complexity of Categories Noson S. Yanofsky Theorem Kolmogorov Complexity If A and B are two equivalent categories, then Programing Language K Sammy ( A ) ≈ K Sammy ( B ) . Kolmogorov Complexity of Categories Complexity Computability Algebra Future Directions

Basic Theorems Kolmogorov Complexity of Categories Noson S. Yanofsky Theorem Kolmogorov Complexity If A and B are two equivalent categories, then Programing Language K Sammy ( A ) ≈ K Sammy ( B ) . Kolmogorov Complexity of Categories Conclusion: Complexity Kolmogorov complexity is an invariant of categorical structure. Computability Algebra Future Directions