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Representation of natural numbers and other data types in lambda calculus
In mathematics, Church encoding is a way of representing various data types in the lambda calculus. In the untyped lambda calculus the only primitive
Church_encoding
Way to represent data types in the lambda calculus
Mogensen–Scott encoding extends and slightly modifies Scott encoding by applying the encoding to Metaprogramming[citation needed]. This encoding allows the
Mogensen–Scott_encoding
American mathematician and computer scientist (1903–1995)
functional programming languages in general. The Church encoding is named in his honor. In his honor the Alonzo Church Award for Outstanding Contributions to Logic
Alonzo_Church
Natural number
computability theory, natural numbers are represented by Church encoding as functions, where the Church numeral for 1 is represented by the function f {\displaystyle
1
Function in mathematical logic
structure of sets. Gödel sets can also be used to encode formulas in infinitary languages. Church encoding Description number Gödel numbering for sequences
Gödel_numbering
Higher-order function Y for which Y f = f (Y f)
apply the fixed-point combinator to may be expressed using an encoding, like Church encoding. In this case particular lambda terms (which define functions)
Fixed-point_combinator
Function and primitive data structure in Lisp and other functional programming languages
it may even turn out to be more efficient than other kinds of encoding. This encoding also has the advantage of being implementable in a statically typed
Cons
Logical formalism using combinators instead of variables
input syntactic representations of terms under a suitable encoding (e.g., Church encoding). One may also consider a toy trivial computation model where
Combinatory_logic
Paradox in set theory
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Russell's_paradox
Theorem for proving more complex theorems
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Lemma_(mathematics)
Thesis on the nature of computability
In 1932–33, Alonzo Church created a method for defining functions called the λ-calculus. Within λ-calculus, he defined an encoding of the natural numbers
Church–Turing_thesis
Method of deriving conclusions
the limitations pointed out by Kurt Gödel regarding formal systems that encode arithmetic. This example assumes that a {\displaystyle a} refers to an individual
Rule_of_inference
Programming language
1 The following code defines a Church encoding of natural numbers, with successor (succ) and addition (add). A Church numeral n is a higher-order function
OCaml
Impossible task in computing
every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic
Entscheidungsproblem
Set whose elements all belong to another set
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Subset
Input to a mathematical function
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Argument_of_a_function
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Mathematical_object
Statement that is taken to be true
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Axiom
Mathematical use of "there exists"
in the article on quantification (logic). The existential quantifier is encoded as U+2203 ∃ THERE EXISTS in Unicode, and as \exists in LaTeX and related
Existential_quantification
Type theory created by Thierry Coquand
{\displaystyle A\vee B} Booleans and Naturals are defined in the same way as in Church encoding. However, added problems arise from propositional extensionality and
Calculus_of_constructions
Infinite cardinal number
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Aleph_number
Symbol representing a property or relation in logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Predicate_(logic)
Any one of the distinct objects that make up a set in set theory
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Element_of_a_set
Mathematical operation with two operands
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Binary_operation
Set of all things that may be the input of a mathematical function
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Domain_of_a_function
Function, homomorphism, or morphism
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Map_(mathematics)
Process of repeating items in a self-similar way
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Recursion
Mathematical theory of data types
concept of equality. Set theory encodes numbers as sets. Type theory can encode numbers as functions using Church encoding, or more naturally as inductive
Type_theory
Collection of mathematical objects
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Set_(mathematics)
Class of formal logics
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Classical_logic
Base-1 numeral system
arithmetic within mathematical logic. A form of unary notation called Church encoding is used to represent numbers within lambda calculus. Some email spam
Unary_numeral_system
Additional mathematical object
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Mathematical_structure
Set of the elements not in a given subset
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Complement_(set_theory)
Logical incompatibility between two or more propositions
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Contradiction
Value indicating the relation of a proposition to truth
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Truth_value
Problem in computer science
fractions are uncomputable and also highly dependent on the choice of program encoding used to determine "size". For example, consider classifying programs by
Halting_problem
Mathematical set containing no elements
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Empty_set
Logical connective AND
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Logical_conjunction
Characteristic of some logical systems
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Completeness_(logic)
Set of elements common to all of some sets
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Intersection_(set_theory)
Subset of a function's codomain
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Range_of_a_function
Set that is not a finite set
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Infinite_set
Mathematical-logic system based on functions
A version of lambda calculus with binary input/output (I/O), a binary encoding of terms, and a designated universal machine. Lambda-mu calculus – An extension
Lambda_calculus
In logic, a statement which is always true
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Tautology_(logic)
Concept in logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Logical_equivalence
Mathematical set of all subsets of a set
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Power_set
Theory of truth in the philosophy of language
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Semantic_theory_of_truth
Function that preserves distinctness
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Injective_function
Formal system in mathematical logic
except for full recursion, are still considered simple because the Church encodings of such structures can be done using only → {\displaystyle \to } and
Simply_typed_lambda_calculus
Fundamental theorem in mathematical logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Gödel's_completeness_theorem
Target set of a mathematical function
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Codomain
Mathematical function such that every output has at least one input
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Surjective_function
Mathematical use of "for all"
in the article on quantification (logic). The universal quantifier is encoded as U+2200 ∀ FOR ALL in Unicode, and as \forall in LaTeX and related formula
Universal_quantification
Set of elements in any of some sets
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Union_(set_theory)
arithmetic within mathematical logic. A form of unary notation called Church encoding is used to represent numbers within lambda calculus. Some email spam
List_of_numeral_systems
Proof in set theory
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Cantor's_diagonal_argument
Undecidability of equality of real numbers
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Richardson's_theorem
Mathematical concept
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Transfinite_induction
Branch of mathematics that studies sets
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Set_theory
Type of logical argument that applies deductive reasoning
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Syllogism
Mathematical model for deduction or proof systems
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Formal_system
Number of arguments required by a function
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Arity
Token in a mathematical or logical formula
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Symbol_(formal)
Index of articles associated with the same name
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Stratification_(mathematics)
Computation model defining an abstract machine
"universal" nature was introduced by Alonzo Church. Church's work intertwined with Turing's to form the basis for the Church–Turing thesis. This thesis states that
Turing_machine
Standard system of axiomatic set theory
connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as functional completeness. This
Zermelo–Fraenkel_set_theory
Logical quantifier
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Uniqueness_quantification
Term in mathematical logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Independence (mathematical logic)
Independence_(mathematical_logic)
Mathematical set formed from two given sets
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Cartesian_product
Software design pattern
without making a type differentiation within the traversed objects Church encoding – a related concept from functional programming, in which tagged union/sum
Visitor_pattern
Establishment of a theorem using inference from the axioms
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Formal_proof
Limitative results in mathematical logic
Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Properties linking logical conjunction and disjunction
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Conjunction/disjunction duality
Conjunction/disjunction_duality
Diagram that shows all possible logical relations between a collection of sets
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Venn_diagram
Logical principle
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Law_of_excluded_middle
Infinite set that is not countable
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Uncountable_set
Set with algorithmic membership test
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Computable_set
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
List_of_mathematical_proofs
Mathematical logic concept
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Atomic_formula
Complexity class used to classify decision problems
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
NP_(complexity)
Argument whose conclusion must be true if its premises are
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Validity_(logic)
Writing system
Kurdish, and Moksha. Other character encoding systems for Cyrillic: CP866 – 8-bit Cyrillic character encoding established by Microsoft for use in MS-DOS
Cyrillic_script
Garbled text as a result of incorrect character encodings
one encoding, when the same binary code constitutes one symbol in the other encoding. This is either because of differing constant length encoding (as
Mojibake
Mathematical proof expressed visually
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Proof_without_words
Term in logic and deductive reasoning
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Soundness
Logic theorem
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Law_of_noncontradiction
Ordered listing of items in collection
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Enumeration
Formal system of logic
logic", Report No. 15 (2009/2010) of the Mittag-Leffler Institute. Alonzo Church, A formulation of the simple theory of types, The Journal of Symbolic Logic
Higher-order_logic
Yes/no problem in computer science
\{0,1\}^{*}} . For another example, using an encoding such as Gödel numbering, any string can be encoded as a natural number, via which a decision problem
Decision_problem
Proposition in mathematical logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Continuum_hypothesis
Formal system in mathematical logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Abstract_logic
Description of non-logical symbols
For many-sorted logic and for many-sorted structures, signatures must encode information about the sorts. The most straightforward way of doing this
Signature_(logic)
Relationship where one statement follows from another
New York: Raven Press, ISBN 9780486432281. Papers include those by Gödel, Church, Rosser, Kleene, and Post. Dummett, Michael (1991), The Logical Basis of
Logical_consequence
Yes-or-no question that cannot ever be solved by a computer
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Undecidable_problem
Symbol representing a mathematical object
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Variable_(mathematics)
Axiom in the mathematical field of set theory
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Martin's_axiom
Concept in mathematical logic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Hereditary_set
Consistency of the axioms of arithmetic
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Hilbert's_second_problem
Axioms for the natural numbers
proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing
Peano_axioms
In mathematical logic, a well-formed formula with no free variables
predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable
Sentence_(mathematical_logic)
CHURCH ENCODING
CHURCH ENCODING
Boy/Male
English Scandinavian American Scottish Norse Teutonic
Church.
Boy/Male
Scandinavian Scottish
Church.
Boy/Male
American, British, Chinese, Christian, English, French, German, Indian, Norse, Scandinavian, Scottish, Teutonic
Dweller by the Church; Church
Surname or Lastname
English
English : topographic surname for someone who lived near a church. The word comes from Old English cyrice, ultimately from medieval Greek kyrikon, for earlier kyriakÅn (dÅma) ‘(house) of the Lord’, from kyrios ‘lord’.Translation of German Kirch.
Boy/Male
English American
A from the Old English 'ceorl' meaning man. Famous bearer: American singer Chuck Berry.
Surname or Lastname
English
English : topographic name for someone who lived by a churchyard, Middle English chircheheye literally ‘church enclosure’.
Boy/Male
Christian & English(British/American/Australian)
Church
Boy/Male
Scandinavian Scottish
Church.
Girl/Female
Norse
Church farm. Church village.
Boy/Male
Scandinavian Scottish
Church.
Male
English
Pet form of English Charles, CHUCK means "man."
Boy/Male
Scandinavian Scottish
Church.
Surname or Lastname
English
English : probably an occupational name for someone who worked at a ‘church house’ (Middle English chirche + h(o)us), a building, usually adjoining the church, which served as a parish room.
Surname or Lastname
English
English : habitational name from Upchurch, a place in Kent, named from Old English upp ‘up’ + cirice ‘church’, i.e. ‘church standing high up’.
Boy/Male
British, English
From the Upper Church
Male
Spanish
Pet form of Spanish Jesús, CHUCHO means "God is salvation."
Surname or Lastname
English (Devon)
English (Devon) : nickname for a dwarf, Middle English murch(e).French (Lorraine) : nickname for a lethargic, feeble man, from a Middle High German loanword, mursch, murz.
Boy/Male
Scandinavian Scottish
Church.
Boy/Male
American, British, English, Scandinavian, Scottish
Dweller by the Church; Church
Boy/Male
English Scandinavian Scottish
Church.
CHURCH ENCODING
CHURCH ENCODING
Girl/Female
American, Australian, French, German, Latin
Farmer; Female Version of George
Boy/Male
British, English
From the Mill Stream
Girl/Female
Arabic
Pure; Holy
Girl/Female
Muslim/Islamic
Praising Allah
Boy/Male
Muslim
One who gives protection, The giver of might and glory
Male
Greek
(ÎάÏκισσος) Greek name possibly derived from the word narke, NARKISSOS means "numbness; sleep." In mythology, this is the name of a vain youth who fell in love with his own reflection and eventually was turned into a kind of lily or daffodil flower known as the narkissos.Â
Boy/Male
Hebrew
Father of light.
Girl/Female
Indian
Beautiful Angel, Night
Girl/Female
French
Great happiness.
Boy/Male
Indian
Servant of the everlasting, Slave of the eternal
CHURCH ENCODING
CHURCH ENCODING
CHURCH ENCODING
CHURCH ENCODING
CHURCH ENCODING
v. t.
To expel, or cause to separate, from a church; to excommunicate.
n.
A church or the church, in the various senses of the word; esp., the Church of Scotland as distinguished from other reformed churches, or from the Roman Catholic Church.
v. t.
To deprive of the character, privileges, and authority of a church.
v. t.
To bless according to a prescribed form, or to unite with in publicly returning thanks in church, as after deliverance from the dangers of childbirth; as, the churching of women.
n.
Any body of worshipers; as, the Jewish church; the church of Brahm.
n.
See Courche.
n.
A body of Christian believers, holding the same creed, observing the same rites, and acknowledging the same ecclesiastical authority; a denomination; as, the Roman Catholic church; the Presbyterian church.
a.
Of or pertaining to, or favoring, the party called the High Church, or their doctrines or policy. See High Church, under High, a.
a.
Not placing a high estimate on ecclesiastical organizations or forms; -- applied especially to Episcopalians, and opposed to high-church. See High Church, under High.
a.
Relating to a church; unduly fond of church forms.
a.
Pertaining to, or suitable for, the church; ecclesiastical.
n.
The chuck or call of a hen.
n.
The aggregate of religious influences in a community; ecclesiastical influence, authority, etc.; as, to array the power of the church against some moral evil.
n.
A hunch.
v. t.
To leave in the lurch; to cheat.
n.
A seat in the porch of a church.
imp. & p. p.
of Church
n.
A church or parish festival (as in commemoration of the dedication of a church), at which much ale was used.