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Lecture

Church Numerals and Conditionals

In course

CS-210: Functional programming

Understanding of the principles and applications of functional programming, the fundamental models of program execution, application of fundamental methods of program composition, meta-programming through the construction of interpreters and advanced progr ...

Description

This lecture introduces Church numerals as a way to represent numbers using lambda calculus. It explores the concept of encoding conditionals in lambda calculus, demonstrating how to check if a numeral is nonzero and automating the encoding of conditionals.

Instructors (2)

Martin Odersky

Martin Odersky heads the programming research group at EPFL. His research interests cover fundamental as well as applied aspects of programming languages. They include semantics, type systems, programming language design, and compiler construction. The main focus if his work lies in the integration of object-oriented and functional programming. His research thesis is that the two paradigms are just two sides of the same coin and should be unified as much as possible. To prove this he has experimented a number of language designs, from Pizza to GJ to Functional Nets. He has also influenced the development of Java as a co-designer of Java generics and as the original author of the current javac reference compiler. His current work concentrates on the Scala programming language, which unifies FP and OOP, while staying completely interoperable with Java and .NET. Martin Odersky got his doctorate from ETHZ, in 1989. He held research positions at the IBM T.J. Watson Research Center from 1989 and at Yale University from 1991. He was then a professor at the University of Karlsruhe from 1993 and at the University of South Australia from 1997. He joined EPFL as full professor in 1999. He is associate editor of the Journal of Functional Programming and member of IFIP WG 2.8. He was conference chair for ICFP 2000, and program chair for ECOOP 2004 as well as ETAPS/CC 2007.

Viktor Kuncak

Viktor Kunčak joined EPFL in 2007, after receiving a PhD degree from MIT. Since then has been leading the Laboratory for Automated Reasoning and Analysis and supervised at least 12 completed PhD theses. His works on languages, algorithms and systems for verification and automated reasoning. He served as an initiator and one of the coordinators of a European network (COST action) in the area of automated reasoning, verification, and synthesis. In 2012 he received a 5-year single-investigator European Research Council (ERC) grant of 1.5M EUR. His invited talks include those at Lambda Days, Scala Days, NFM, LOPSTR, SYNT, ICALP, CSL, RV, VMCAI, and SMT. A paper on test generation he co-authored received an ACM SIGSOFT distinguished paper award at ICSE. A PLDI paper he co-authored was published in the Communications of the ACM as a Research Highlight article. His Google Scholar profile reports an over-approximate H-index of 38. He was an associate editor of ACM Transactions on Programming Languages and Systems (TOPLAS) and served as a co-chair of conferences on Computer-Aided Verification (CAV), Formal Methods in Computer Aided Design (FMCAD), Workshop on Synthesis (SYNT), and Verification, Model Checking, and Abstract Interpretation (VMCAI). At EPFL he teaches courses on functional and parallel programming, compilers, and verification. He has co-taught the MOOC "Parallel Programming" that was visited by over 100'000 learners and completed by thousands of students from all over the world.

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Related concepts (32)

Simply typed lambda calculus

The simply typed lambda calculus (), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor () that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus. The term simple type is also used to refer extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF).

Typed lambda calculus

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 (see kinds below). 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.

Lambda calculus

Lambda calculus (also written as λ-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 introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing lambda terms and performing reduction operations on them.

Lambda cube

In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to: x-axis (): types that can bind terms, corresponding to dependent types.

Counterfactual conditional

Counterfactual conditionals (also subjunctive or X-marked) are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood.

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