Programming language theory

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The lowercase Greek letter l (lambda) is an unofficial symbol of the field of programming language theory. This usage derives from the lambda calculus, a model of computation introduced by Alonzo Church in the 1930s and widely used by programming language researchers. It graces the cover of the classic text Structure and Interpretation of Computer Programs, and the title of the so-called Lambda Papers, written by Gerald Jay Sussman and Guy Steele, the developers of the Scheme programming language. Lambda lc.svg
The lowercase Greek letter λ (lambda) is an unofficial symbol of the field of programming language theory. This usage derives from the lambda calculus, a model of computation introduced by Alonzo Church in the 1930s and widely used by programming language researchers. It graces the cover of the classic text Structure and Interpretation of Computer Programs , and the title of the so-called Lambda Papers, written by Gerald Jay Sussman and Guy Steele, the developers of the Scheme programming language.

Programming language theory (PLT) is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features. It falls within the discipline of computer science, both depending on and affecting mathematics, software engineering, linguistics and even cognitive science. It is a well-recognized branch of computer science, and an active research area, with results published in numerous journals dedicated to PLT, as well as in general computer science and engineering publications.

Computer science Study of the theoretical foundations of information and computation

Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.

Programming language Language designed to communicate instructions to a machine

A programming language is a formal language, which comprises a set of instructions that produce various kinds of output. Programming languages are used in computer programming to implement algorithms.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.



In some ways, the history of programming language theory predates even the development of programming languages themselves. The lambda calculus, developed by Alonzo Church and Stephen Cole Kleene in the 1930s, is considered by some to be the world's first programming language, even though it was intended to model computation rather than being a means for programmers to describe algorithms to a computer system. Many modern functional programming languages have been described as providing a "thin veneer" over the lambda calculus, [1] and many are easily described in terms of it.

Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was first introduced by mathematician Alonzo Church in the 1930s as part of his research of the foundations of mathematics.

Alonzo Church American mathematician

Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology, and the Church–Rosser theorem. He also worked on philosophy of language.

Stephen Cole Kleene American mathematician and theoretical computer scientist

Stephen Cole Kleene was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star, Kleene's recursion theorem and the Kleene fixpoint theorem. He also invented regular expressions, and made significant contributions to the foundations of mathematical intuitionism.

The first programming language to be invented was Plankalkül, which was designed by Konrad Zuse in the 1940s, but not publicly known until 1972 (and not implemented until 1998). The first widely known and successful high-level programming language was Fortran, developed from 1954 to 1957 by a team of IBM researchers led by John Backus. The success of FORTRAN led to the formation of a committee of scientists to develop a "universal" computer language; the result of their effort was ALGOL 58. Separately, John McCarthy of MIT developed the Lisp programming language (based on the lambda calculus), the first language with origins in academia to be successful. With the success of these initial efforts, programming languages became an active topic of research in the 1960s and beyond.

Plankalkül is a programming language designed for engineering purposes by Konrad Zuse between 1942 and 1945. It was the first high-level programming language to be designed for a computer.

Konrad Zuse 20th-century German computer scientist and engineer

Konrad Zuse was a German civil engineer, computer scientist, inventor, businessman and computer pioneer. His greatest achievement was the world's first programmable computer; the functional program-controlled Turing-complete Z3 became operational in May 1941. Thanks to this machine and its predecessors, Zuse has often been regarded as the inventor of the modern computer.

In computer science, a high-level programming language is a programming language with strong abstraction from the details of the computer. In contrast to low-level programming languages, it may use natural language elements, be easier to use, or may automate significant areas of computing systems, making the process of developing a program simpler and more understandable than when using a lower-level language. The amount of abstraction provided defines how "high-level" a programming language is.

Some other key events in the history of programming language theory since then:


Noam Chomsky American linguist, philosopher and activist

Avram Noam Chomsky is an American linguist, philosopher, cognitive scientist, historian, social critic, and political activist. Sometimes called "the father of modern linguistics", Chomsky is also a major figure in analytic philosophy and one of the founders of the field of cognitive science. He holds a joint appointment as Institute Professor Emeritus at the Massachusetts Institute of Technology (MIT) and laureate professor at the University of Arizona, and is the author of more than 100 books on topics such as linguistics, war, politics, and mass media. Ideologically, he aligns with anarcho-syndicalism and libertarian socialism.

In the formal languages of computer science and linguistics, the Chomsky hierarchy is a containment hierarchy of classes of formal grammars.

Linguistics is the scientific study of language. It involves analysing language form, language meaning, and language in context. The earliest activities in the documentation and description of language have been attributed to the 6th-century-BC Indian grammarian Pāṇini who wrote a formal description of the Sanskrit language in his Aṣṭādhyāyī.


Simula Early object-oriented programming language

Simula is the name of two simulation programming languages, Simula I and Simula 67, developed in the 1960s at the Norwegian Computing Center in Oslo, by Ole-Johan Dahl and Kristen Nygaard. Syntactically, it is a fairly faithful superset of ALGOL 60, also influenced by the design of Simscript.

Ole-Johan Dahl was a Norwegian computer scientist. Dahl was a professor of computer science at the University of Oslo and is considered to be one of the fathers of Simula and object-oriented programming along with Kristen Nygaard.

Kristen Nygaard Computer scientist, Mathematician

Kristen Nygaard was a Norwegian computer scientist, programming language pioneer, and politician. Internationally Nygaard is acknowledged as the co-inventor of object-oriented programming and the programming language Simula with Ole-Johan Dahl in the 1960s. Nygaard and Dahl received the 2001 A. M. Turing Award for their contribution to computer science.


Dana Scott American mathematician and computer scientist

Dana Stewart Scott is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His research career involved computer science, mathematics, and philosophy. His work on automata theory earned him the ACM Turing Award in 1976, while his collaborative work with Christopher Strachey in the 1970s laid the foundations of modern approaches to the semantics of programming languages. He has worked also on modal logic, topology, and category theory.

In computer science, denotational semantics is an approach of formalizing the meanings of programming languages by constructing mathematical objects that describe the meanings of expressions from the languages. Other approaches provide formal semantics of programming languages including axiomatic semantics and operational semantics.

Logic programming is a type of programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic programming language families include Prolog, answer set programming (ASP) and Datalog. In all of these languages, rules are written in the form of clauses:



There are several fields of study which either lie within programming language theory, or which have a profound influence on it; many of these have considerable overlap. In addition, PLT makes use of many other branches of mathematics, including computability theory, category theory, and set theory.

Formal semantics

Formal semantics is the formal specification of the behaviour of computer programs and programming languages. Three common approaches to describe the semantics or "meaning" of a computer program are denotational semantics, operational semantics and axiomatic semantics.

Type theory

Type theory is the study of type systems; which are "a tractable syntactic method for proving the absence of certain program behaviors by classifying phrases according to the kinds of values they compute". [3] Many programming languages are distinguished by the characteristics of their type systems.

Program analysis and transformation

Program analysis is the general problem of examining a program and determining key characteristics (such as the absence of classes of program errors). Program transformation is the process of transforming a program in one form (language) to another form.

Comparative programming language analysis

Comparative programming language analysis seeks to classify programming languages into different types based on their characteristics; broad categories of programming languages are often known as programming paradigms.

Generic and metaprogramming

Metaprogramming is the generation of higher-order programs which, when executed, produce programs (possibly in a different language, or in a subset of the original language) as a result.

Domain-specific languages

Domain-specific languages are languages constructed to efficiently solve problems of a particular part of domain.

Compiler construction

Compiler theory is the theory of writing compilers (or more generally, translators); programs which translate a program written in one language into another form. The actions of a compiler are traditionally broken up into syntax analysis (scanning and parsing), semantic analysis (determining what a program should do), optimization (improving the performance of a program as indicated by some metric; typically execution speed) and code generation (generation and output of an equivalent program in some target language; often the instruction set of a CPU).

Run-time systems

Runtime systems refers to the development of programming language runtime environments and their components, including virtual machines, garbage collection, and foreign function interfaces.

Journals, publications, and conferences

Conferences are the primary venue for presenting research in programming languages. The most well known conferences include the Symposium on Principles of Programming Languages (POPL), Programming Language Design and Implementation (PLDI), the International Conference on Functional Programming (ICFP), the International Conference on Object Oriented Programming, Systems, Languages and Applications (OOPSLA) and the International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS).

Notable journals that publish PLT research include the ACM Transactions on Programming Languages and Systems (TOPLAS), Journal of Functional Programming (JFP), Journal of Functional and Logic Programming , and Higher-Order and Symbolic Computation .

See also

Related Research Articles

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type.

Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions - and to remove any mention of variables - particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.

ISWIM is an abstract computer programming language devised by Peter J. Landin and first described in his article The Next 700 Programming Languages, published in the Communications of the ACM in 1966. The acronym stands for "If you See What I Mean".

In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.

A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered. From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type.

In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages. It does so by evaluating the meaning of syntactically valid strings defined by a specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically invalid strings, the result would be non-computation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will be executed on a certain platform, hence creating a model of computation.

Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.

The actor model in computer science is a mathematical model of concurrent computation that treats "actors" as the universal primitives of concurrent computation. In response to a message that it receives, an actor can: make local decisions, create more actors, send more messages, and determine how to respond to the next message received. Actors may modify their own private state, but can only affect each other indirectly through messaging.

Quantum programming is the process of assembling sequences of instructions, called quantum programs, that are capable of running on a quantum computer. Quantum programming languages help express quantum algorithms using high-level constructs.

In computer science, the Actor model and process calculi are two closely related approaches to the modelling of concurrent digital computation. See Actor model and process calculi history.

In computer science, Programming Computable Functions, or PCF, is a typed functional language introduced by Gordon Plotkin in 1977, based on previous unpublished material by Dana Scott. It can be considered to be an extended version of the typed lambda calculus or a simplified version of modern typed functional languages such as ML or Haskell.

In programming languages and type theory, parametric polymorphism is a way to make a language more expressive, while still maintaining full static type-safety. Using parametric polymorphism, a function or a data type can be written generically so that it can handle values identically without depending on their type. Such functions and data types are called generic functions and generic datatypes respectively and form the basis of generic programming.

The Actor model and process calculi share an interesting history and co-evolution.

Harry George Mairson is a theoretical computer scientist and Professor of Computer Science in the Volen National Center for Complex Systems at Brandeis University in Waltham, Massachusetts. His research is in the fields of logic in computer science, lambda calculus and functional programming, type theory and constructive mathematics, computational complexity theory, and algorithmics.

The categorical abstract machine (CAM) is a model of computation for programs that preserves the abilities of applicative, functional, or compositional style. It is based on the techniques of applicative computing.

Applicative 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 in these systems is the representation of object. In combinatory logic the only metaoperator is application in a sense of applying one object to other. In lambda calculus two metaoperators are used: application – the same as in combinatory logic, and functional abstraction which binds the only variable in one object.

Carl Hewitt American mathematician and computer scientist

Carl Eddie Hewitt is an American computer scientist who designed the Planner programming language for automated planning and the actor model of concurrent computation, which have been influential in the development of logic, functional and object-oriented programming. Planner was the first programming language based on procedural plans invoked using pattern-directed invocation from assertions and goals. The actor model influenced the development of the Scheme programming language, the π-calculus, and served as an inspiration for several other programming languages.


  2. C. Böhm and W. Gross (1996). Introduction to the CUCH. In E. R. Caianiello (ed.), Automata Theory, p. 35-64/
  3. Benjamin C. Pierce. 2002. Types and Programming Languages. MIT Press, Cambridge, Massachusetts, USA.

Further reading