In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that map values to other values, rather than a sequence of imperative statements which update the running state of the program.
In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or datatypes, are defined in terms of each other. Mutual recursion is very common in functional programming and in some problem domains, such as recursive descent parsers, where the datatypes are naturally mutually recursive.
ML is a general-purpose, high-level, functional programming language. It is known for its use of the polymorphic Hindley–Milner type system, which automatically assigns the data types of most expressions without requiring explicit type annotations, and ensures type safety; there is a formal proof that a well-typed ML program does not cause runtime type errors. ML provides pattern matching for function arguments, garbage collection, imperative programming, call-by-value and currying. While a general-purpose programming language, ML is used heavily in programming language research and is one of the few languages to be completely specified and verified using formal semantics. Its types and pattern matching make it well-suited and commonly used to operate on other formal languages, such as in compiler writing, automated theorem proving, and formal verification.
In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type to every term. Usually the terms are various language constructs of a computer program, such as variables, expressions, functions, or modules. A type system dictates the operations that can be performed on a term. For variables, the type system determines the allowed values of that term.
Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics (procedures) and various decision procedures.
A data manipulation language (DML) is a computer programming language used for adding (inserting), deleting, and modifying (updating) data in a database. A DML is often a sublanguage of a broader database language such as SQL, with the DML comprising some of the operators in the language. Read-only selecting of data is sometimes distinguished as being part of a separate data query language (DQL), but it is closely related and sometimes also considered a component of a DML; some operators may perform both selecting (reading) and writing.
Lennart Augustsson is a Swedish computer scientist. He was formerly a lecturer at the Computing Science Department at Chalmers University of Technology. His research field is functional programming and implementations of functional programming languages.
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, Idris, and Lean, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.
The ACM–IEEE Symposium on Logic in Computer Science (LICS) is an annual academic conference on the theory and practice of computer science in relation to mathematical logic. Extended versions of selected papers of each year's conference appear in renowned international journals such as Logical Methods in Computer Science and ACM Transactions on Computational Logic.
In functional programming, a generalized algebraic data type is a generalization of a parametric algebraic data type (ADT).
In computer science, termination analysis is program analysis which attempts to determine whether the evaluation of a given program halts for each input. This means to determine whether the input program computes a total function.
In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its primitive data types. The expression of an instance of a product type will be a tuple, and is called a "tuple type" of expression. A product of types is a direct product of two or more types.
In computing, ATS is a multi-paradigm, general-purpose, high-level, functional programming language. It is a dialect of the programming language ML, designed by Hongwei Xi to unify computer programming with formal specification. ATS has support for combining theorem proving with practical programming through the use of advanced type systems. A past version of The Computer Language Benchmarks Game has demonstrated that the performance of ATS is comparable to that of the languages C and C++. By using theorem proving and strict type checking, the compiler can detect and prove that its implemented functions are not susceptible to bugs such as division by zero, memory leaks, buffer overflow, and other forms of memory corruption by verifying pointer arithmetic and reference counting before the program compiles. Also, by using the integrated theorem-proving system of ATS (ATS/LF), the programmer may make use of static constructs that are intertwined with the operative code to prove that a function conforms to its specification.
In type theory, a refinement type is a type endowed with a predicate which is assumed to hold for any element of the refined type. Refinement types can express preconditions when used as function arguments or postconditions when used as return types: for instance, the type of a function which accepts natural numbers and returns natural numbers greater than 5 may be written as . Refinement types are thus related to behavioral subtyping.
Haskell is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell has pioneered several programming language features such as type classes, which enable type-safe operator overloading, and monadic input/output (IO). It is named after logician Haskell Curry. Haskell's main implementation is the Glasgow Haskell Compiler (GHC).
Conor McBride is a Reader in the department of Computer and Information Sciences at the University of Strathclyde. In 1999, they completed a Doctor of Philosophy (Ph.D.) in Dependently Typed Functional Programs and their Proofs at the University of Edinburgh for their work in type theory. They formerly worked at Durham University and briefly at Royal Holloway, University of London before joining the academic staff at the University of Strathclyde.
F* is a high-level, multi-paradigm, functional and object-oriented programming language inspired by the languages ML, Caml, and OCaml, and intended for program verification. It is a joint project of Microsoft Research, and the French Institute for Research in Computer Science and Automation (Inria). Its type system includes dependent types, monadic effects, and refinement types. This allows expressing precise specifications for programs, including functional correctness and security properties. The F* type-checker aims to prove that programs meet their specifications using a combination of satisfiability modulo theories (SMT) solving and manual proofs. For execution, programs written in F* can be translated to OCaml, F#, C, WebAssembly, or assembly language. Prior F* versions could also be translated to JavaScript.
Ur also called Ur/Web is a multi-paradigm, high-level, pure, strict, functional programming language. It is a dialect of the language ML, designed for web development, created by Adam Chlipala at the Massachusetts Institute of Technology that one program can emit code for a server, web browser client, and SQL specific to a given database backend. The full implementation is free and open-source software released under an MIT License.
Apache SystemDS is an open source ML system for the end-to-end data science lifecycle.
Futhark is a multi-paradigm, high-level, functional, data parallel, array programming language. It is a dialect of the language ML, originally developed at UCPH Department of Computer Science (DIKU) as part of the HIPERFIT project. It focuses on enabling data parallel programs written in a functional style to be executed with high performance on massively parallel hardware, especially graphics processing units (GPUs). Futhark is strongly inspired by NESL, and its implementation uses a variant of the flattening transformation, but imposes constraints on how parallelism can be expressed in order to enable more aggressive compiler optimisations. In particular, irregular nested data parallelism is not supported. It is free and open-source software released under an ISC license.
