Logical consequenceLogical consequence (also entailment) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises.
CoinductionIn computer science, coinduction is a technique for defining and proving properties of systems of concurrent interacting objects. Coinduction is the mathematical to structural induction. Coinductively defined types are known as codata and are typically infinite data structures, such as streams. As a definition or specification, coinduction describes how an object may be "observed", "broken down" or "destructed" into simpler objects. As a proof technique, it may be used to show that an equation is satisfied by all possible implementations of such a specification.
Agda (programming language)Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally known as Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the propositions-as-types paradigm, but unlike Coq, has no separate tactics language, and proofs are written in a functional programming style.
Bounded quantificationIn type theory, bounded quantification (also bounded polymorphism or constrained genericity) refers to universal or existential quantifiers which are restricted ("bounded") to range only over the subtypes of a particular type. Bounded quantification is an interaction of parametric polymorphism with subtyping. Bounded quantification has traditionally been studied in the functional setting of System F
Gödel's completeness theoremGödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable".
Kotlin (programming language)Kotlin (ˈkɒtlɪn) is a cross-platform, statically typed, general-purpose high-level programming language with type inference. Kotlin is designed to interoperate fully with Java, and the JVM version of Kotlin's standard library depends on the Java Class Library, but type inference allows its syntax to be more concise. Kotlin mainly targets the JVM, but also compiles to JavaScript (e.g., for frontend web applications using React) or native code via LLVM (e.g., for native iOS apps sharing business logic with Android apps).
OCamlOCaml (oʊˈkæməl , formerly Objective Caml) is a general-purpose, high-level multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy, Ascánder Suárez, and others. The OCaml toolchain includes an interactive top-level interpreter, a bytecode compiler, an optimizing native code compiler, a reversible debugger, and a package manager (OPAM).
Multiple inheritanceMultiple inheritance is a feature of some object-oriented computer programming languages in which an object or class can inherit features from more than one parent object or parent class. It is distinct from single inheritance, where an object or class may only inherit from one particular object or class. Multiple inheritance has been a controversial issue for many years, with opponents pointing to its increased complexity and ambiguity in situations such as the "diamond problem", where it may be ambiguous as to which parent class a particular feature is inherited from if more than one parent class implements said feature.
Epigram (programming language)Epigram is a functional programming language with dependent types, and the integrated development environment (IDE) usually packaged with the language. Epigram's type system is strong enough to express program specifications. The goal is to support a smooth transition from ordinary programming to integrated programs and proofs whose correctness can be checked and certified by the compiler. Epigram exploits the Curry–Howard correspondence, also termed the propositions as types principle, and is based on intuitionistic type theory.
Quantifier eliminationQuantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement " such that " can be viewed as a question "When is there an such that ?", and the statement without quantifiers can be viewed as the answer to that question. One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest.
