{"id":348,"date":"2007-03-16T15:39:33","date_gmt":"2007-03-16T15:39:33","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/03\/16\/simple-programming-in-binary-binary-combinatory-logic\/"},"modified":"2007-03-16T15:39:33","modified_gmt":"2007-03-16T15:39:33","slug":"simple-programming-in-binary-binary-combinatory-logic","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/03\/16\/simple-programming-in-binary-binary-combinatory-logic\/","title":{"rendered":"Simple Programming in Binary: Binary Combinatory Logic"},"content":{"rendered":"

For reasons that I’ll explain in another post, I don’t have a lot of time for writing a long pathological programming post, so I’m going to hit you with something short, sweet, and beautiful: binary combinatory logic.<\/a><\/p>\n

I’ve written in the past about lambda calculus, and it’s equivalent variable-free form, the SKI combinator calculus. I’ve ever written about other combinator calculus based languages, like Unlambda<\/a> and Iota<\/a>.<\/p>\n

Binary combinatory logic, aka BCL, is a language based on SKI calculus – except that it encodes the entire thing into binary. Two characters, plus two rewrite rules, and that’s it – a complete
\ncombinator calculus based programming language.<\/p>\n

SKI combinator calculus is a simple variable-free calculus with three constructs: S, K, and I; and I isn’t really primitive, but can be defined in terms of S and K.<\/p>\n

    \n
  1. S=λx y z.x z (y z)<\/li>\n
  2. K=λx.(λy.x)<\/li>\n
  3. I=λx.x=SKK<\/li>\n<\/ol>\n

    So, in BCL, S is written “01”; K is written “00”. And there are two rewrite rules, which basically define “1” without a zero prefix as a a paren-like grouping construct:<\/p>\n

      \n
    1. “1100xy”, where “x” and “y” are valid BCL terms (that is, complete syntactic units),
      \ngets rewritten to be “x”. If you follow that through, that means that it reduces to ((Kx)y).<\/li>\n
    2. “11101xzy” gets rewritten to “11xz1yz”. Again, following it through, and that
      \nreduces out to “(((Sx)y)z)”.<\/li>\n<\/ol>\n

      So, following on unlambda’s method of handling IO, “hello world” in BCL is:<\/p>\n

      \n010001101000010000010110000000000101101111\n000010110111110011111111011110000010011010\n<\/pre>\n
      \"bcl.gif\"<\/p>\n
       And here’s the really neat thing. Write an interpreter for BCL in BCL. Take the bit string that results, and convert it to a bitmap. That’s what’s over the right here. So, for example, the first line is “1111100000111001”; keep going, and you’ll find the entire BCL interpreter.<\/p>\n","protected":false},"excerpt":{"rendered":"
      For reasons that I’ll explain in another post, I don’t have a lot of time for writing a long pathological programming post, so I’m going to hit you with something short, sweet, and beautiful: binary combinatory logic. I’ve written in the past about lambda calculus, and it’s equivalent variable-free form, the SKI combinator calculus. I’ve […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[92],"tags":[],"class_list":["post-348","post","type-post","status-publish","format-standard","hentry","category-pathological-programming"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-5C","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/348","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=348"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/348\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=348"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=348"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}