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From Corecursive Algebras to Corecursive Monads

From Corecursive Algebras to Corecursive Monads. Stefan Milius joint work with Jiři Adámek und Mahdieh Haddadi. Why corecursive algebras ?. No general recursion in: languages of total functional programming type- theoretic proof assistants dependently typed languages. Gerhard Osius.

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From Corecursive Algebras to Corecursive Monads

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  1. FromCorecursiveAlgebrastoCorecursiveMonads Stefan MiliusjointworkwithJiřiAdámek und MahdiehHaddadi

  2. Whycorecursivealgebras? • Nogeneralrecursion in: • languagesof total functionalprogramming • type-theoreticproofassistants • dependentlytypedlanguages Gerhard Osius Paul Taylor Instead: • Structured recursionandcorecursionschemesforinductiveandcoinductivetypes General structuredrecursionandinduction: • G. Osius (1974): Recursiveand well-foundedcoalgebrasfor • P. Taylor (1999): Recursiveand well-foundedcoalgebrasforpreimagepreservingsetfunctors General structuredcorecursionandcoinduction: • V. Capretta, T. Uustalu, V. Vene (2009): corecursivealgebras Ourgoal: systematicmathematicalstudyof (free) corecursivealgebras

  3. Whycorecursivealgebras? • E. Nelson (1983): • iterative algebrasas an easy approachto Calvin Elgot‘siterative theories S. Milius (2005): • completelyiterative algebras(cia) andcompletely iterative theories(cim) • Corecursivealgebras/monadsare a naturalweakeningofcia/cim theycallfor a systematicstudy

  4. Main result Example.

  5. Rest ofthe Talk • Definitions • Details andresults • Examples • Conclusions

  6. Whatarecorecursivealgebras? Definition. Facts.

  7. Example: Streams (Capretta, Uustalu, Vene ’09)

  8. Binary corecursivealgebras - characterization a b c

  9. Constructionoffreecorecursivealgebras Assumption. Construction 1. Recall theFree Algebra Chain (Adámek 1974) Theorem. • Construction 2. Free Corecursive Algebra Chain Theorem.

  10. Free corecursivealgebra Corollary. Questions.

  11. Ad question 1: Bloom algebras Definition. Theorem.

  12. Characterizationof Bloom algebras Theorem.

  13. Ad question 2: CorecursiveMonads Definition. Theorem. Theorem.

  14. Conclusions + Future work • CorecursiveAlgebrascapturegeneralstructuredcorecursionandcoinduction • Freecorecursivealgebrasexistandcaniterativelybeconstructionasthealgebracoproduct • Bloom algebrasaretheEilenberg-Moore algebrasforthemonadoffreecorecursivealgebras • The monadoffreecorecursivealgebrasisthefreecorecursivemonad. • Future work • Finitarycorecursivealgebras, i.e., finitecoalgebrasadmit a uniquecoalgebra-to-algebrahomomorphism • The analog ofIteration Theoriesof Bloom andÉsik

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