Counting and generating terms in the binary lambda calculus extended version. Quantitative aspects of linear and affine closed lambda. The following programs were generated by my learning program in a few seconds. Citeseerx binary lambda calculus and combinatory logic. Counting and generating terms in the binary lambda calculus. Bibtex entry for this abstract preferred format for this abstract see preferences. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. John tromp introduced the socalled binary lambda calculus as a way to encode lambda terms in terms of binary words. Affine terms are terms in which each bound variable occurs at most once, and linear terms are terms in which each bound variable occurs once and. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given. Binary lambda calculus is a new idea introduced by john tromp in 2008 apparently so. Pdf binary lambda calculus and combinatory logic semantic. I am trying to create a very efficient implentation of john tromp s binary lambda calculus so that it can be used as a toy language at golf it is the pure untyped lambda calculus with only 3 types of operations, lambda abstraction, variable argument lookup, and apply.
John tromp provides an answer to this question in his always fun lambda calculus and combinatory logic playground. More specifically, it defines a universal machine, which, from an input stream of bits, parses the binary encoding of a lambda calculus term, applies that to the remainder of input translated to a lazy list of booleans, which have a standard representation in lambda calculus, and translates the evaluated result back into a stream of bits to. We introduce binary representations of both lambda calculus. Heres how to multiply two numbers in lambda calculus. Later, grygiel and lescanne conjectured that the number of. Binary lambda calculus and combinatory logic john tromp. This approach has the advantage of not requiring any other mathematical dependencies, and it should thus provide the simplest possible way of providing your argument. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Calculo lambda binario wikipedia, a enciclopedia livre. Jan 22, 2016 if you find our videos helpful you can support us by buying something from amazon. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of. Along the way we present a simple graphical notation for lambda calculus, a new empty list representation, improved bracket abstraction, and a.
Reduction consists of replacing a part pof eby another expression p0 according to the given rewrite rules. This design of a minimalistic universal computer was motivated by my. With a little syntax sugar, lambda calculus becomes a practical programming language. John tromp defines a version of the lambda calculus that is encoded in binary.
Nov 01, 2019 as an universal model of computation, lambda calculus is important in programming language theory, and especially it is the foundation of functional programming. Calude, world scientific publishing company, october 2008, pages 237260 the last reference, to an initial haskell implementation, is dated 2004 pdf version. The church numerals are a representation of the natural numbers using lambda notation. Lambda calculus is certainly tricky for counting steps, since a single step of betareduction can actually do a lot of work, unlike e. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n. Much of the fun involves tiny selfinterpreters that read binary. Instead, blc requires translating bitstrings into lambda terms, to which the machine itself a lambda term can be readily applied. Pdf in a paper entitled binary lambda calculus and combinatory logic, john tromp presents a simple way of encoding lambda calculus terms as binary.
Binary lambda calculus is explained in detail in my latest paper available in postscript and pdf. Binary lambda calculus and combinatory logic lambda the. Binary lambda calculus and combinatory logic john tromp august 26, 2011 abstract in the. Ive taken some graduate courses in programming languages where ive learned about the lambda calculus and about type theory, and i really appreciate this post since this provides a nice timeline of how programming language theory has evolved. Binary combinatory logic bcl is a formulation of combinatory logic using only the symbols 0 and 1. Most functional international obfuscated c code contest. I am currently reading lambdacalculus and combinators by hindley and seldin. Binary lambda calculus and combinatory logic john tromp may 14, 2014 abstract in the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. A protocol for randomized anonymous twoprocess waitfree testandset with finitestate verification.
Im not an expert, but have always taken an interest in lambda calculus because of involvement with functional programming starting with lisp and sicp, and now with r and haskell. Implementations of untyped lazy lambda calculus lambda. Now for some interesting comparisons to john tromps binary lambda calculus, which is known for its conciseness, but does have the problem of possible syntax errors. Concatenative binary lambda calculuscombinatory logic. In a paper entitled binary lambda calculus and combinatory logic, john tromp presents a simple way of encoding lambda calculus terms as binary sequences. International obfuscated c code contest winners hacker news. I wrote this package to play around with the binary lambda calculus originally defined by john tromp. The data and operators form a mathematical structure which is embedded in the lambda calculus. Contribute to sdiehlbnlc development by creating an account on github.
John tromp has invented binary lambda calculus, which is a much more efficient representation than this. On the number of lambda terms with prescribed size of. The smallest lambda interpreter in javascript m1el. John tromp presents a simple way of encoding lambda calculus terms as binary sequences. How to model the output of the binary lambda calculus. In a paper entitled binary lambda calculus and combinatory logic, john tromp presents a simple way of encoding lambda calculus. In binary lambda calculus and combinatory logic, john tromp states. We propose meta lambda calculus lambda as a basic model of textual.
Along the way we also present new results on list representations, bracket abstraction, and fixpoint. A short introduction to the lambda calculus achim jung. Binary lambda calculus and combinatory logic john tromp march 8, 2006 abstract in the rst part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. Suppose tm is a lambda term that simulates a turing machine consider. References a b john tromp, binary lambda calculus and combinatory logic, in randomness and complexity, from leibniz to chaitin, ed. They concluded that typed selfrepresentation for system f seems to be impossible. Binary lambda calculus and combinatory logic drops. Pictured you can see the 210 bit binary lambda calculus selfinterpreter, and the 272 bit binary combinatory logic selfinterpreter. The most straightforward way is to implement a turing machine in the lambda calculus. Counting terms in the binary lambda calculus nasaads. Binary lambda calculus and combinatory logic john tromp may 14, 2014 abstract in the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary. Along the way we also present new results on list representations, bracket. Algorithmic information theory, using binary lambda calculus html 40 4 tromp. We demonstrate their application to algorithmic information theory with several concrete upper bounds on programsize complexity, including an elegant.
Are lambda calculus and combinatory logic the same. In mathematics, church encoding is a means of representing data and operators in the lambda calculus. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows roughly like 1. I turing complete i higher order functions as data i indeed, in the lambda calculus, all computation happens by. March 18, 2004 abstract the lambda calculus can appear arcane on.
During working on that, i separated out the representation of terms into the package lambdacalculus. We introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. He actually first mentioned the idea as early as 2004. Bcl has applications in the theory of programsize complexity kolmogorov complexity.
For the purpose of making a simple interpreter, a binary tree form is used to represent lambda terms. Epistemology as computation information processing 263 gordana dodigcrnkovic 16. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their. Tromp, john 2007, binary lambda calculus and combinatory logic, randomness and complexity pdf, world sci. In the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. John tromp s reworking of chaitins ideas in lambda calculus and combinatory logic is a fascinating read. Counting terms in the binary lambda calculus aofa 2014. Both are explained in detail in my latest paper available in postscript and pdf.
Theres a 1to1 mapping between binary tree form and binary representation. Selfrepresentation in girards system u proceedings of. Already, our factorial example above is shorter than. Now for some interesting comparisons to john tromp s binary lambda calculus, which is known for its conciseness, but does have the problem of possible syntax errors. Vit anyi, editors, kolmogorov complexity and applications, volume 06051 of dagstuhl seminar proceedings. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size ngrows roughly like 1. It reads from standard input a program description followed by input for the program, and output the result to the standard output. Counting and generating terms in the binary lambda.
If you find our videos helpful you can support us by buying something from amazon. Jan 02, 2014 pdf in a paper entitled binary lambda calculus and combinatory logic, john tromp presents a simple way of encoding lambda calculus terms as binary. Before discovering how to interpret lambda calculus in binary, i figured out how to make a universal machine in binary combinatory logic. Binary lambda calculus and combinatory logic 237 john tromp philosophy 15. In what follows, we study the numbers of binary strings of a given size that represent lambda. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university,halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of ottawa in 2001 and at dalhousie university in 2007 and 20. The former turns out to be a lot more descriptive, i. Introduction to the lambda calculus iowa state university. John s lambda calculus and combinatory logic playground pictured above you can see on the left the 210 bit binary lambda calculus selfinterpreter, and on the right the 272 bit binary combinatory logic selfinterpreter both are explained in detail in my latest paper available in postscript and pdf.
Algorithmic information theory, using binary lambda calculus trompait. The objective of this project is to understand john tromp s 2012 ioccc entry. Spare a thought for students struggling to make turing machines do simple tasks. The objective of this project is to understand john tromps 2012 ioccc entry. O calculo lambda binario foi uma nova ideia introduzida por john tromp em 2004. Binary lambda calculus blc is a minimal, purely functional programming language invented by john tromp in 2004, based on a binary encoding of the untyped lambda calculus in. The lambda calculus the lambda calculus i if our previous language of arithmetic expressions was the simplest nontrivial programming language, then the lambda calculus is the simplest interesting programming language. Lescanne, counting terms in the binary lambda calculus, arxiv preprint arxiv. This is quite easy, because the lambda calculus is practically a high level programming language. Binary lambda calculus and combinatory logic john tromp march 8, 2006 abstract in the rst part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary. John tromp, john s lambda calculus and combinatory logic. The course is entitled lambda calculus and types and it appears in various forms.
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