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
  • package examples
  • 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.

  • package interface
  • package learning
  • package library
  • package scalahott
  • package scratch
  • 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.

  • Base
  • Context
  • FiniteDistribution
  • Frankl
  • HoTT
  • JvmUtils
  • LinearStructure
  • MereProposition
  • PickledWeighted
  • ProbabilityDistribution
  • Subst
  • SubstImplicits
  • TermList
  • TermListImplicits
  • Utils
  • Weighted
p

provingground

package provingground

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.

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Package Members

  1. package andrewscurtis
  2. package examples
  3. 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.

  4. package interface
  5. package learning
  6. package library
  7. package scalahott
  8. package scratch
  9. 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.

Type Members

  1. sealed trait Context extends AnyRef
  2. final case class FiniteDistribution[T](pmf: Vector[Weighted[T]]) extends AnyVal with ProbabilityDistribution[T] with Product with Serializable

    Finite distributions, often supposed to be probability distributions, but may also be tangents to this or intermediates.

    Finite distributions, often supposed to be probability distributions, but may also be tangents to this or intermediates.

    pmf

    probability mass function, may have same object split.

  3. case class LinearStructure[A](zero: A, sum: (A, A) => A, mult: (Double, A) => A) extends Product with Serializable
  4. case class PickledWeighted(elem: String, weight: Double) extends Product with Serializable
  5. trait ProbabilityDistribution[A] extends Any

    A probability distribution, from which we can pick values at random (the only abstract method).

    A probability distribution, from which we can pick values at random (the only abstract method). We can obtain a random variable from this, which is an iterator.

  6. trait Subst[A] extends AnyRef

    allows substitution of a Term by another.

  7. trait SubstImplicits extends AnyRef
  8. sealed trait TermList[A] extends Subst[A]

    allows substitution of a Term by another, as well as mapping to a vector of terms chiefly subtypes of Term and HLists of these;

  9. trait TermListImplicits extends SubstImplicits
  10. case class Weighted[T](elem: T, weight: Double) extends Product with Serializable

Value Members

  1. object Base
  2. object Context
  3. object FiniteDistribution extends Serializable
  4. object Frankl
  5. object HoTT

    Core of Homotopy Type Theory (HoTT) implementation.

    Core of Homotopy Type Theory (HoTT) implementation. Includes: - terms : Term, - types : Typ - universes - functions and dependent functions (see [FuncLike], [Func]) - function types FuncTyp and pi-types PiDefn, - lambda definitions LambdaLike, - pairs PairTerm and dependent pairs DepPair - product types ProdTyp and sigma types SigmaTyp - Coproduct types PlusTyp, the Unit type Unit and the empty type Zero - recursion and induction functions for products, coproducts

    General inductive types are not implemented here, but in the induction package.

  6. object JvmUtils
  7. object LinearStructure extends Serializable
  8. object MereProposition
  9. object PickledWeighted extends Serializable
  10. object ProbabilityDistribution
  11. object Subst
  12. object TermList extends TermListImplicits
  13. object Utils
  14. object Weighted extends Serializable

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