Packages

  • package root
    Definition Classes
    root
  • package provingground

    This is work towards automated theorem proving based on learning, using homotopy type theory (HoTT) as foundations and natural language processing.

    This is work towards automated theorem proving based on learning, using homotopy type theory (HoTT) as foundations and natural language processing.

    The implementation of homotopy type theory is split into:

    • the object HoTT with terms, types, functions and dependent functions, pairs etc
    • the package induction with general inductive types and recursion/induction on these.

    The learning package has the code for learning.

    Scala code, including the spire library, is integrated with homotopy type theory in the scalahott package

    We have implemented a functor based approach to translation in the translation package, used for nlp as well as serialization and parsing.

    The library package is contains basic structures implemented in HoTT.

    Definition Classes
    root
  • package andrewscurtis
    Definition Classes
    provingground
  • ACElem
  • ACPortableResults
  • ACStates
  • ACThm
  • ACevolution
  • ACresults
  • AndrewsCurtis
  • AndrewsCurtisInterface
  • AtomicMove
  • Conj
  • DiffStructure
  • FreeGroups
  • Id
  • Inv
  • LftMult
  • LftMultInv
  • MoveGenerator
  • Moves
  • Param
  • RtMult
  • RtMultInv
  • SimpleAcEvolution
  • Transpose
  • package examples
    Definition Classes
    provingground
  • package induction

    Much of the richness of HoTT is in the definitions of Inductive types (and their indexed counterparts) and of (dependent) functions on these by recursion and induction These are implemented using several layers of recursive definitions and diagonals (i.e., fixed points).

    Much of the richness of HoTT is in the definitions of Inductive types (and their indexed counterparts) and of (dependent) functions on these by recursion and induction These are implemented using several layers of recursive definitions and diagonals (i.e., fixed points). In HoTT, recursion and induction are applications of (dependent) functions rec_W,X and ind_W, Xs to the definition data.

    It is useful to capture information regarding inductive types and the recursion and induction functions in scala types. Our implementation is designed to do this.

    Inductive Type Definitions

    Inductive types are specified by introduction rules. Each introduction rule is specified in ConstructorShape (without specifying the type) and ConstructorTL including the specific type. The full definition is in ConstructorSeqTL.

    Recursion and Induction functions

    These are defined recursively, first for each introduction rule and then for the inductive type as a whole. A subtlety is that the scala type of the rec_W,X and induc_W, Xs functions depends on the scala type of the codomain X (or family Xs). To make these types visible, some type level calculations using implicits are done, giving traits ConstructorPatternMap and ConstructorSeqMap that have recursive definition of the recursion and induction functions, the former for the case of a single introduction rule. Traits ConstructorSeqMapper and ConstructorPatternMapper provide the lifts.

    Indexed Versions

    There are indexed versions of all these definitions, to work with indexed inductive type families.

    Definition Classes
    provingground
  • package interface
    Definition Classes
    provingground
  • package learning
    Definition Classes
    provingground
  • package library
    Definition Classes
    provingground
  • package scalahott
    Definition Classes
    provingground
  • package scratch
    Definition Classes
    provingground
  • package translation

    Translation primarily using a functorial framework - see Translator$, for natural language processing as well as serialization, formatted output, parsing, interface with formal languages etc.

    Translation primarily using a functorial framework - see Translator$, for natural language processing as well as serialization, formatted output, parsing, interface with formal languages etc.

    Besides the Translator framework and helper typeclasses is Functors, several structures for concrete languages including our implementation of HoTT are in this package.

    Definition Classes
    provingground
p

provingground

andrewscurtis

package andrewscurtis

Content Hierarchy
Ordering
  1. Alphabetic
Visibility
  1. Public
  2. Protected

Type Members

  1. case class ACElem(name: String, moves: Moves, rank: Int, pres: Presentation, weight: Double, loops: Int) extends Product with Serializable

    An element in the Andrews-Curtis evolution, namely: * actor name, * number of loops, * rank * associated presentation * weight in distribution

  2. case class ACPortableResults(paths: Map[String, LazyList[(FiniteDistribution[AtomicMove], FiniteDistribution[Moves])]]) extends ACresults with Product with Serializable
  3. trait ACStates extends AnyRef
  4. case class ACThm(name: String, pres: Presentation, weight: Double, loops: Int) extends Product with Serializable

    An presentation (theorem) in the Andrews-Curtis evolution, namely: * actor name, * number of loops, * presentation * weight in distribution on presentations

  5. class ACresults extends ACStates
  6. sealed trait AtomicMove extends (Moves) => Option[Moves]
  7. case class Conj(k: Int, l: Int) extends AtomicMove with Product with Serializable
  8. case class Inv(k: Int) extends AtomicMove with Product with Serializable
  9. case class LftMult(k: Int, l: Int) extends AtomicMove with Product with Serializable
  10. case class LftMultInv(k: Int, l: Int) extends AtomicMove with Product with Serializable
  11. final case class Moves(moves: List[AtomicMove]) extends AnyVal with Product with Serializable
  12. case class Param(rank: Int = 2, size: Int = 1000, wrdCntn: Double = 0.1) extends Product with Serializable
  13. case class RtMult(k: Int, l: Int) extends AtomicMove with Product with Serializable
  14. case class RtMultInv(k: Int, l: Int) extends AtomicMove with Product with Serializable
  15. case class Transpose(k: Int, l: Int) extends AtomicMove with Product with Serializable

Value Members

  1. object ACElem extends Serializable
  2. object ACThm extends Serializable
  3. object ACevolution

  4. object ACresults
  5. object AndrewsCurtis
  6. object AndrewsCurtisInterface
  7. object AtomicMove
  8. object Conj extends Serializable
  9. object DiffStructure
  10. object FreeGroups
  11. case object Id extends AtomicMove with Product with Serializable
  12. object Inv extends Serializable
  13. object LftMult extends Serializable
  14. object LftMultInv extends Serializable
  15. object MoveGenerator
  16. object Moves extends Serializable
  17. object Param extends Serializable
  18. object RtMult extends Serializable
  19. object RtMultInv extends Serializable
  20. object SimpleAcEvolution

  21. object Transpose extends Serializable

Ungrouped