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Logical formalism using combinators instead of variables
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell
Combinatory_logic
Computer programming language
Binary combinatory logic (BCL) is a computer programming language that uses binary terms 0 and 1 to create a complete formulation of combinatory logic using
Binary_combinatory_logic
Relationship between programs and proofs
with the typed fragment of a standard model of computation known as combinatory logic. In 1969 Howard observes that another, more "high-level" proof system
Curry–Howard_correspondence
American mathematician (1900-1982)
mathematician and computer scientist. Curry is best known for his work in combinatory logic. Although its initial concept was based on a paper by Moses Schönfinkel
Haskell_Curry
Russian logician and mathematician
1942 (1943)) was a logician and mathematician, known for the invention of combinatory logic. Moses Schönfinkel was born on (1888-09-29)29 September 1888 in Ekaterinoslav
Moses_Schönfinkel
Book by Raymond Smullyan
To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic (1985, ISBN 0-19-280142-2) is a book by the mathematician
To_Mock_a_Mockingbird
Mathematical paradox
logics, including certain forms of set theory, lambda calculus, and combinatory logic. The paradox is named after the logician Haskell Curry, who wrote
Curry's_paradox
Mathematical-logic system based on functions
formal systems are related to lambda calculus: Combinatory logic – A notation for mathematical logic without variables SKI combinator calculus – A computational
Lambda_calculus
Algebraization of first-order logic
combinator. Hence combinatory logic goes beyond first-order logic by having the expressive power of set theory, which makes combinatory logic vulnerable to
Predicate_functor_logic
Abstract grammatical formalism
structure grammar (as opposed to a dependency grammar). CCG relies on combinatory logic, which has the same expressive power as the lambda calculus, but builds
Combinatory categorial grammar
Combinatory_categorial_grammar
Simple Turing complete logic
The SKI combinator calculus is a combinatory logic system and a computational system. It can be thought of as a computer programming language, though
SKI_combinator_calculus
Last letter of the Greek alphabet
(codomain of the) subobject classifier of an elementary topos. In combinatory logic, the looping combinator, (S I I (S I I)). In group theory, the omega
Omega
Programming paradigm based on applying and composing functions
functional programming languages. An equivalent theoretical formulation, combinatory logic, was developed by Moses Schönfinkel and Haskell Curry in the 1920s
Functional_programming
theoretical computer science and mathematical logic, specifically in realizability, a partial combinatory algebra (pca) is an algebraic structure which
Partial_combinatory_algebra
Software programming optimization technique
memoization on a class webpage. Memoization in Combinatory Logic – A web service to reduce Combinatory Logic while memoizing every step in a database. MbCache
Memoization
Combinatory logic system
The B, C, K, W system is a variant of combinatory logic that takes as primitive the combinators B, C, K, and W. This system was discovered by Haskell
B,_C,_K,_W_system
Formal system in mathematical logic
typed lambda calculus representations of the basic combinators of combinatory logic. Each type τ {\displaystyle \tau } is assigned an order, a number
Simply_typed_lambda_calculus
Academic subfield of computer science
combinator Y has normal form in combinatory logic but not in λ {\displaystyle \lambda } -calculus). Combinatory logic was developed with great ambitions:
Theory_of_computation
Branch of logic
function Categorical logic Combinational logic Combinatory logic Conceptual graph Disjunctive syllogism Entitative graph Equational logic Existential graph
Propositional_logic
Intuitionistic type theory BHK interpretation Curry–Howard correspondence Linear logic Game semantics Typed lambda calculus Typed and untyped languages Type signature
List of functional programming topics
List_of_functional_programming_topics
Higher-order function Y for which Y f = f (Y f)
In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function (i.e., a function that takes a
Fixed-point_combinator
Ability to solve a problem by an effective procedure
combinator Y has normal form in combinatory logic but not in λ {\displaystyle \lambda } -calculus). Combinatory logic was developed with great ambitions:
Computability
About mathematical functions
building blocks of mathematical logic". ibid. pp. 355–366. With commentary by Willard Quine. The start of combinatory logic. ——; von Neumann, John (1967)
History of the function concept
History_of_the_function_concept
Branch of computer science
Object in Combinatory Logic, later generalized into the Hindley–Milner type inference algorithm. In 1969, Tony Hoare introduces the Hoare logic, a form
Programming_language_theory
Constructed human language based on predicate logic
translated in some of its parts into predicate logic. There are also analogies between Lojban and combinatory logic. There have been proposals to use Lojban
Lojban
Topics referred to by the same term
mathematics to denote integrals and antiderivatives S combinator in combinatory logic Sphere S (set theory), an axiomatic set theory s(n), the aliquot sum
S_(disambiguation)
American mathematician and logician (1919–2017)
Puzzles Including An Amazing Adventure in Combinatory Logic. ISBN 0192801422. puzzles based on combinatory logic — (1987). Forever Undecided. ISBN 0192801414
Raymond_Smullyan
American logician (1908–1987)
in context of the knowability paradox. Fitch worked primarily in combinatory logic, authoring an undergraduate-level textbook on the subject (1974),
Frederic_Fitch
Subfield of mathematics
proof theory, especially intuitionistic logic. Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming
Mathematical_logic
structure, whether that structure is definable or not. combinatory logic A branch of mathematical logic that seeks to eliminate the need for variables in mathematical
Glossary_of_logic
Functional programming language
functional programming language invented by David Madore. It is based on combinatory logic, an expression system without the lambda operator or free variables
Unlambda
Reasoning about equations with free variables
respects include: Combinatory logic, having the expressive power of set theory; Relation algebra, arguably the paradigmatic algebraic logic, can express Peano
Algebraic_logic
System of formal deduction in logic
B.; Robert Feys (1958). Combinatory Logic Vol. I. Vol. 1. Amsterdam: North Holland. Monk, J. Donald (1976). Mathematical Logic. Graduate Texts in Mathematics
Hilbert_system
Programming paradigm
equivalent point-free definition would be: def example: foo | bar | baz; Combinatory logic Concatenative programming language Function-level programming Joy
Tacit_programming
Element mapped to itself by a mathematical function
greatest fixed point coincides with its greatest postfixpoint). In combinatory logic for computer science, a fixed-point combinator is a higher-order function
Fixed_point_(mathematics)
Esoteric programming languages
continuing through the continuation w. Lambda calculus Combinatory logic Binary combinatory logic SKI combinator calculus Barker, Chris. "Zot". The Esoteric
Iota_and_Jot
computing systems, or ACS are the systems of object calculi founded on combinatory logic and lambda calculus. The only essential notion which is under consideration
Applicative_computing_systems
programmers, represented by Cartesian closed category and embedded into the combinatory logic. CAM is a transparent and sound mathematical representation for the
Categorical_abstract_machine
Representation of data types in lambda calculus
ISBN 978-0-262-16209-8. Tromp, John (2007). "14. Binary Lambda Calculus and Combinatory Logic". In Calude, Cristian S (ed.). Randomness And Complexity, From Leibniz
Church_encoding
Mathematical model describing how an output of a function is computed given an input
memory model Functional models include: Abstract rewriting systems Combinatory logic General recursive functions Lambda calculus Concurrent models include:
Model_of_computation
Mathematical theory of data types
theory UTT (Luo's unified theory of dependent types) some forms of combinatory logic others defined in the lambda cube (also known as pure type systems)
Type_theory
Transforming a function in such a way that it only takes a single argument
Aspects of Combinatory Logic". The Kleene Symposium: Proceedings of the Symposium Held June 18-24, 1978 at Madison, Wisconsin, U.S.A. (Studies in Logic and the
Currying
Dutch computer scientist
"the logical rules of Go". He is also known for his work in binary combinatory logic (binary lambda calculus) [citation needed] and lambda diagrams that
John_Tromp
Operation on mathematical functions
\circ. Cobweb plot – a graphical technique for functional composition Combinatory logic Composition ring, a formal axiomatization of the composition operation
Function_composition
Paradox that shows that certain systems of formal logic are inconsistent
that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's
Kleene–Rosser_paradox
Register that stores where in a program a processor is executing
desired function and the low-level programmer might specify this using combinatory logic. This research also led to ways to making conventional, PC-based,
Program_counter
Approach in computer programming
result); return 0; } The program output is: 0 1 Backus–Naur form Combinatory logic in computing Design Patterns Domain-specific language Interpreter
Interpreter_pattern
Italian computer scientist (1923–2017)
to the theory of structured programming, constructive mathematics, combinatory logic, lambda calculus, and the semantics and implementation of functional
Corrado_Böhm
algorithm Lambda calculus Church–Rosser theorem Calculus of constructions Combinatory logic Post correspondence problem Kleene's recursion theorem Recursively
List of mathematical logic topics
List_of_mathematical_logic_topics
Variable used for specification
programming and its foundational disciplines, lambda calculus and combinatory logic. Terminology varies between languages; some computer languages such
Parameter
Type whose definition depends on a value
in its combinatory logic counterpart, followed the same pattern as axioms in propositional logic. Going further, for every proof in the logic, there was
Dependent_type
Topics referred to by the same term
close front rounded vowel Y combinator, a fixed-point combinator in combinatory logic Y, the Bessel function Y, dependent variable Y or y, vertical axis
Y_(disambiguation)
Symbolic description of a mathematical object
history, see Cardone and Hindley's "History of Lambda-calculus and Combinatory Logic" (2006). Oxford English Dictionary, s.v. “Expression (n.), sense II
Expression_(mathematics)
Software optimization technique
sum = prev + next; prev = next; next = sum; yield return sum; } } Combinatory logic Currying Dataflow Eager evaluation Functional programming Futures
Lazy_evaluation
Aspect of mathematical logic
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski
Abstract_algebraic_logic
Overview of and topical guide to logic
Classical logic Computability logic Deontic logic Dependence logic Description logic Deviant logic Doxastic logic Epistemic logic First-order logic Formal
Outline_of_logic
Layer of hardware-level instructions or data structures
encoded, that is, the bit fields generally pass through intermediate combinatory logic that, in turn, generates the control and sequencing signals for internal
Microcode
Type system used in computer programming and mathematics
Hindley, J. Roger (1969). "The Principal Type-Scheme of an Object in Combinatory Logic". Transactions of the American Mathematical Society. 146: 29–60. doi:10
Hindley–Milner_type_system
American logician (born 1932)
to programming languages as well as to construct models of Curry's combinatory logic and Church's calculus of lambda conversion; and The 2001 Bolzano Prize
Dana_Scott
Thesis on the nature of computability
cousin to the modern notion of the computer. Other models include combinatory logic and Markov algorithms. Gurevich adds the pointer machine model of
Church–Turing_thesis
Measure of algorithmic complexity
"Review of Li Vitányi 1997". Tromp, John. "John's Lambda Calculus and Combinatory Logic Playground". Tromp's lambda calculus computer model offers a concrete
Kolmogorov_complexity
Italian mathematician
fundamental functional programming languages including Lambda Calculus and Combinatory Logic. He integrated his research findings into a broader framework that
Giuseppe_Longo
Topics referred to by the same term
used by approximately one third of the global population. Binary combinatory logic Bunga Citra Lestari, Indonesian actress and singer Members of the
BCL
Explicit substitutions were sketched in the preface of Curry's book on Combinatory logic and grew out of an ‘implementation trick’ used, for example, by AUTOMATH
Explicit_substitution
Family of formalisms in natural language syntax
recent years is Steedman and Szabolcsi's combinatory categorial grammar, which builds on combinatory logic invented by Moses Schönfinkel and Haskell
Categorial_grammar
Mathematical notation in lambda calculus
and are free in the conclusion The de Bruijn notation for λ-terms. Combinatory logic, a more essential way to eliminate variable names. de Bruijn, Nicolaas
De_Bruijn_index
Belgian logician and philosopher (1889-1961)
simply typed lambda calculus (Combinatory Logic). Haskell B. Curry, Robert Feys, and William Craig. Combinatory Logic. Amsterdam, North-Holland Pub.
Robert_Feys
Symbols for constants, special functions
transport Ψ {\displaystyle \Psi } represents: a quaternary combinator in combinatory logic a symbol for psychology the wave function in the Schrödinger equation
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Spanish philosopher and lawyer
logic. TL is thus a logical blend or rather a crossbreed. Peña's plan to investigate the grounds of his logical system as a nonclassical combinatory logic
Lorenzo_Peña
Metatheorem in mathematical logic
to the conversion process from lambda calculus terms to terms of combinatory logic, where axiom 1 corresponds to the K combinator, and axiom 2 corresponds
Deduction_theorem
Function that takes one or more functions as an input or that outputs a function
in this example has the signature auto apply. First-class function Combinatory logic Function-level programming Functional programming Kappa calculus -
Higher-order_function
uncountability proof uncountability of the real numbers Combinatorics Combinatory logic Co-NP Coset Countable countability of a subset of a countable set
List_of_mathematical_proofs
Family of type systems based on substructural logic
diagonal functor which could duplicate states; similarly, from the combinatory logic point of view, there is no K-combinator which can destroy states.
Substructural_type_system
articles that criticised socialism Moses Schönfinkel, inventor of combinatory logic Sara Shakulova, first female mathematician of Tatar descent Yakov
List of Russian mathematicians
List_of_Russian_mathematicians
Decision problem pertaining to equivalence of expressions
of the earliest proofs that a word problem is undecidable was for combinatory logic: when are two strings of combinators equivalent? Because combinators
Word_problem_(mathematics)
Basis of generic programming
Hindley, J. Roger (1969), "The principal type scheme of an object in combinatory logic", Transactions of the American Mathematical Society, 146: 29–60, doi:10
Parametric_polymorphism
Mathematical formalism
expressions are also known as combinators and are equivalent to terms in combinatory logic. It is common to restrict discussion to only closed terms, and some
Lambda_calculus_definition
Algebraization of first-order logic with equality
algebra to the one variable case. Abstract algebraic logic Lambda calculus and Combinatory logic—other approaches to modelling quantification and eliminating
Cylindric_algebra
Concept in mathematics or computer science
science) Combinatory logic Lambda lifting Name binding Scope (programming) Scope (logic) Quine, Willard Van Orman (1982). Mathematical Logic (Revised ed
Free variables and bound variables
Free_variables_and_bound_variables
Logician
Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi (with J. M. Dunn, CSLI Publications, 2008) Combinatory Logic: Pure, Applied
Katalin_Bimbó
Extension of linear logic
proposed the first non-commutative logic in his 1958 paper Mathematics of Sentence Structure to model the combinatory possibilities of the syntax of natural
Noncommutative_logic
British computer scientist (1946–2023)
Programming Languages: The Compilation of an Applicative Language to Combinatory Logic", supervised by Christopher Strachey and Dana Scott. He held professorships
David Turner (computer scientist)
David_Turner_(computer_scientist)
Sequence of operations for a task
Infinite machines. Post, Emil (1936). "Finite Combinatory Processes, Formulation I". The Journal of Symbolic Logic. 1 (3): 103–105. doi:10.2307/2269031. JSTOR 2269031
Algorithm
Relation specifying a rewrite for each object, compatible with a reduction relation
ISBN 978-3-540-20861-7. Curry, Haskell B.; Feys, Robert (1958). Combinatory Logic. Vol. I. Amsterdam: North Holland. pp. 139–142. ISBN 0-7204-2208-6
Reduction_strategy
British logician
Hindley, J. Roger (1969), "The principal type-scheme of an object in combinatory logic", Transactions of the American Mathematical Society, 146: 29–60, doi:10
J._Roger_Hindley
In Hindley J. R. and Seldin J. P. (ed.). To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press. pp. 29–61. ISBN 978-0-12-349050-6
De_Bruijn_notation
and Schnirelmann density of numbers Moses Schönfinkel, inventor of combinatory logic Yakov Sinai, developed the Kolmogorov–Sinai entropy and Sinai billiard
List_of_Russian_scientists
Topics referred to by the same term
Skikda Airport, Algeria, IATA code SKI SKI combinator calculus, a combinatory logic system and a computational system SKI protein, a nuclear proto-oncogene
Ski_(disambiguation)
Schoenflies (1853–1928), mathematician Moses Schönfinkel (1889–1942), combinatory logic Oded Schramm (1961–2008), conformal field theory and probability theory;
List_of_Jewish_mathematicians
K), K) This is the well-known encoding of the identity function in combinatory logic. Normalising it at an identity type produces: - nbe (Arrow (Basic
Normalisation_by_evaluation
Topics referred to by the same term
created by the Wrigley Company Binary lambda calculus (or Binary combinatory logic), a concept in information theory Bloodline Champions, a video game
BLC
Test to determine whether a user is human
JavaScript. Since it is hard for most bots to parse and execute JavaScript, a combinatory method which fills the CAPTCHA fields and hides both the image and the
CAPTCHA
POPL 1983 R. Hindley, "The Principal Type-Scheme of an Object in Combinatory Logic", Transactions of the American Mathematical Society, Vol. 146 (Dec
Principal_type
Overview of and topical guide to software engineering
Sequences Trees Graph (data structure) Logic Deduction First-order logic Higher-order logic Combinatory logic Induction Combinatorics Other Domain knowledge
Outline of software engineering
Outline_of_software_engineering
and Schnirelmann density of numbers Moses Schönfinkel, inventor of combinatory logic Yakov Sinai, developed the Kolmogorov–Sinai entropy and Sinai billiard
List_of_Russian_people
otherwise. The Scott–Curry theorem applies equally to sets of terms in combinatory logic with weak equality. It has parallels to Rice's theorem in computability
Scott–Curry_theorem
Highest academic rank at Yale University
1929 Frederic Brenton Fitch Philosophy 1974 Logician; symbolic and combinatory logic; Fitch-style calculus John Farquhar Fulton Physiology and History
Sterling_Professor
Computation model defining an abstract machine
1145/321356.321362. Post, Emil (1936). "Finite Combinatory Processes—Formulation 1". Journal of Symbolic Logic. 1: 103–105. doi:10.2307/2269031. JSTOR 2269031
Turing_machine
Type of fiction writing
Combinatory literature is a type of fiction writing in which the author relies and draws on concepts outside of general writing practices and applies
Combinatory_literature
Theory on interaction between feeling and thinking
Affect-logics or Affect logic is a biopsychosocial notion, introduced in 1988 by Swiss psychiatrist Luc Ciompi, relating initially to schizophrenia and
Affect-logics
COMBINATORY LOGIC
COMBINATORY LOGIC
Boy/Male
Hindu
Love and kindness, Analytical, Logical
Girl/Female
Tamil
Viviktha | விவீகà¯à®¤à®¾Â
Distinguished, Pure, Deep, Logically intelligent
Viviktha | விவீகà¯à®¤à®¾Â
Girl/Female
Hindu
Distinguished, Pure, Deep, Logically intelligent
Girl/Female
African, Arabic, French, Indian, Muslim, Swahili, Tamil
Intelligent; Logical; Intelligent One who Reasons; Wise
Girl/Female
Indian
Successful; Logical Thinkers
Girl/Female
Danish, Hindu, Indian, Japanese
Ray of Light; Logical
Boy/Male
Tamil
Intelligent, Logical
Girl/Female
Tamil
Vivikta | விவிகதா
Distinguished, Pure, Deep, Logically intelligent
Vivikta | விவிகதா
Boy/Male
Hindu
Full of feathers, Full of logic, Name of sage, Vatsyayan
Boy/Male
Tamil
Love and kindness, Analytical, Logical
Girl/Female
Tamil
Trick, Power, Strategy, Solution by logic, By reasoning
Girl/Female
Hindu
Distinguished, Pure, Deep, Logically intelligent
Girl/Female
Bengali, Hindu, Indian, Tamil, Telugu
Logically Intelligent; Who Stands Alone
Boy/Male
Indian
Intelligent, Logical
Girl/Female
Tamil
Trick, Power, Strategy, Solution by logic, By reasoning
Girl/Female
Arabic, Muslim, Pashtun
Logic; Reason
Girl/Female
Hindu
Trick, Power, Strategy, Solution by logic, By reasoning
Girl/Female
Gujarati, Hindu, Indian, Kannada, Tamil
Trick; Power; Strategy; Solution by Logic; By Reasoning
Boy/Male
Tamil
Full of feathers, Full of logic, Name of sage, Vatsyayan
Boy/Male
Hindu, Indian
Logical
COMBINATORY LOGIC
COMBINATORY LOGIC
Girl/Female
Tamil
Night
Boy/Male
English
From Wine's forest.
Boy/Male
Greek
Fertile.
Girl/Female
Hindu
(Maid servant of Keikeyi who convinced her for Bharat's thorn and exile of Rama)
Girl/Female
Bengali, Gujarati, Hindu, Indian
Always Smiling
Boy/Male
Hindu, Indian
Peace of God and Divine Light
Boy/Male
Indian
The most high
Boy/Male
Tamil
Girl/Female
Hindu
Look, Blessed with beauty, Shape, Beauty
Girl/Female
American, British, English, French, Jamaican, Latin
Domain of Curtis; Short Nose; From Courtenay
COMBINATORY LOGIC
COMBINATORY LOGIC
COMBINATORY LOGIC
COMBINATORY LOGIC
COMBINATORY LOGIC
a.
Not skillful; inexperienced; awkward; bungling; as, an unskillful surgeon or mechanic; an unskillful logician.
n.
A person skilled in logic.
n.
A treatise on logic; as, Mill's Logic.
n.
The three " liberal" arts, grammar, logic, and rhetoric; -- being a triple way, as it were, to eloquence.
a.
Skilled in logic; versed in the art of thinking and reasoning; as, he is a logical thinker.
a.
Threatening or denouncing punishment; as, comminatory terms.
n.
Of or pertaining to a place; limited; logical application; as, a topical remedy; a topical claim or privilege.
a.
Having or observing logical sequence; logically consistent and rigorous; consecutive in development or transition of thought.
adv.
In a logical manner; as, to argue logically.
v. i.
Not possessing or manifesting intellectual, logical, moral, or political strength, vigor, etc.
n.
The art of reasoning; logic.
n.
That which follows as the logical result of reasoning; inference; conclusion; suggestion.
n.
Logicalness.
n.
A European aquatic toad (Bombinator igneus). Its back is dark; its belly is marked with crimson. Called also feuerkrote.
a.
Of or pertaining to logic; used in logic; as, logical subtilties.
a.
Half logical; partly logical; said of fallacies.
n.
See Logic.
a.
According to the rules of logic; as, a logical argument or inference; the reasoning is logical.
n.
The quality of being logical.
n.
One of the seminaries for teaching logic, metaphysics, and theology, which were formed in the Middle Ages, and which were characterized by academical disputations and subtilties of reasoning.