/*jslint bitwise: true,
- eqeqeq: true,
- immed: true,
- newcap: true,
- nomen: true,
- onevar: true,
- plusplus: true,
- regexp: true,
- rhino: true,
- browser: false,
- undef: true,
- white: true,
-
- evil: false,
- regexp: false,
- sub: true,
- laxbreak: true,
- onevar: false,
- debug: true */
+ eqeqeq: true,
+ immed: true,
+ newcap: true,
+ nomen: true,
+ onevar: true,
+ plusplus: true,
+ regexp: true,
+ rhino: true,
+ browser: false,
+ undef: true,
+ white: true,
+
+ evil: false,
+ regexp: false,
+ sub: true,
+ laxbreak: true,
+ onevar: false,
+ debug: true */
// DeBruijn terms
// substitution and translation algorithms from Chris Hankin, An Introduction to Lambda Calculi for Comptuer Scientists
//
function Db_free(variable) {
- this.variable = variable;
- this.subst = function (m, new_term) {
- return this;
- };
- this.renumber = function (m, i) {
- return this;
- };
- // we assume that other will have variable iff it's a Db_free
- this.equal = function (other) {
- return other.variable && this.variable.equal(other.variable);
- };
- this.contains = this.equal;
+ this.variable = variable;
+ this.subst = function (m, new_term) {
+ return this;
+ };
+ this.renumber = function (m, i) {
+ return this;
+ };
+ // we assume that other will have variable iff it's a Db_free
+ this.equal = function (other) {
+ return other.variable && this.variable.equal(other.variable);
+ };
+ this.contains = this.equal;
}
function Db_index(i) {
- this.index = i;
- this.subst = function (m, new_term) {
- if (this.index < m) {
- return this;
- } else if (this.index > m) {
- return new Db_index(this.index - 1);
- } else {
- return new_term.renumber(this.index, 1);
- }
- };
- this.renumber = function (m, i) {
- if (this.index < i) {
- return this;
- } else {
- return new Db_index(this.index + m - 1);
- }
- };
- // we assume that other will have index iff it's a Db_index
- this.equal = function (other) {
- return this.index === other.index;
- };
- this.contains = this.equal;
+ this.index = i;
+ this.subst = function (m, new_term) {
+ if (this.index < m) {
+ return this;
+ } else if (this.index > m) {
+ return new Db_index(this.index - 1);
+ } else {
+ return new_term.renumber(this.index, 1);
+ }
+ };
+ this.renumber = function (m, i) {
+ if (this.index < i) {
+ return this;
+ } else {
+ return new Db_index(this.index + m - 1);
+ }
+ };
+ // we assume that other will have index iff it's a Db_index
+ this.equal = function (other) {
+ return this.index === other.index;
+ };
+ this.contains = this.equal;
}
function Db_app(left, right) {
- this.left = left;
- this.right = right;
- this.subst = function (m, new_term) {
- return new Db_app(this.left.subst(m, new_term), this.right.subst(m, new_term));
- };
- this.renumber = function (m, i) {
- return new Db_app(this.left.renumber(m, i), this.right.renumber(m, i));
- };
- // we assume that other will have left iff it's a Db_app
- this.equal = function (other) {
- return other.left && this.left.equal(other.left) && this.right.equal(other.right);
- };
- this.contains = function (other) {
- if (other.left && this.left.equal(other.left) && this.right.equal(other.right)) {
- return true;
- } else {
- return this.left.contains(other) || this.right.contains(other);
- }
- };
+ this.left = left;
+ this.right = right;
+ this.subst = function (m, new_term) {
+ return new Db_app(this.left.subst(m, new_term), this.right.subst(m, new_term));
+ };
+ this.renumber = function (m, i) {
+ return new Db_app(this.left.renumber(m, i), this.right.renumber(m, i));
+ };
+ // we assume that other will have left iff it's a Db_app
+ this.equal = function (other) {
+ return other.left && this.left.equal(other.left) && this.right.equal(other.right);
+ };
+ this.contains = function (other) {
+ if (other.left && this.left.equal(other.left) && this.right.equal(other.right)) {
+ return true;
+ } else {
+ return this.left.contains(other) || this.right.contains(other);
+ }
+ };
}
function Db_lam(body) {
- this.body = body;
- this.subst = function (m, new_term) {
- return new Db_lam(this.body.subst(m + 1, new_term));
- };
- this.renumber = function (m, i) {
- return new Db_lam(this.body.renumber(m, i + 1));
- };
- // we assume that other will have body iff it's a Db_lam
- this.equal = function (other) {
- return other.body && this.body.equal(other.body);
- };
- this.contains = function (other) {
- if (other.body && this.body.equal(other.body)) {
- return true;
- } else {
- return this.body.contains(other);
- }
- };
+ this.body = body;
+ this.subst = function (m, new_term) {
+ return new Db_lam(this.body.subst(m + 1, new_term));
+ };
+ this.renumber = function (m, i) {
+ return new Db_lam(this.body.renumber(m, i + 1));
+ };
+ // we assume that other will have body iff it's a Db_lam
+ this.equal = function (other) {
+ return other.body && this.body.equal(other.body);
+ };
+ this.contains = function (other) {
+ if (other.body && this.body.equal(other.body)) {
+ return true;
+ } else {
+ return this.body.contains(other);
+ }
+ };
}
// http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell
//
function Variable(name, tag) {
- this.name = name;
- this.tag = tag || 0;
- this.to_string = function () {
- // append count copies of str to accum
- function markup(accum, count) {
- if (count === 0) {
- return accum;
- } else {
- return markup(accum + "'", count - 1);
- }
- }
- return markup(this.name, this.tag);
- };
- this.equal = function (other) {
- return (this.tag === other.tag) && (this.name === other.name);
- };
- // position of this in seq
- this.position = function (seq) {
- for (var i = 0; i < seq.length; i += 1) {
- if (this.equal(seq[i])) {
- return new Db_index(i + 1);
- }
- }
- return new Db_free(this);
- };
+ this.name = name;
+ this.tag = tag || 0;
+ this.to_string = function () {
+ // append count copies of str to accum
+ function markup(accum, count) {
+ if (count === 0) {
+ return accum;
+ } else {
+ return markup(accum + "'", count - 1);
+ }
+ }
+ return markup(this.name, this.tag);
+ };
+ this.equal = function (other) {
+ return (this.tag === other.tag) && (this.name === other.name);
+ };
+ // position of this in seq
+ this.position = function (seq) {
+ for (var i = 0; i < seq.length; i += 1) {
+ if (this.equal(seq[i])) {
+ return new Db_index(i + 1);
+ }
+ }
+ return new Db_free(this);
+ };
}
// if v occurs free_in term, returns Some v' where v' is the highest-tagged
// variable with the same name as v occurring (free or bound) in term
//
function free_in(v, term) {
- var res = term.has_free(v);
- return res[0] && res[1];
+ var res = term.has_free(v);
+ return res[0] && res[1];
}
function subst(v, new_term, expr) {
- if (new_term.variable && new_term.variable.equal(v)) {
- return expr;
- } else {
- return expr.subst(v, new_term);
- }
+ if (new_term.variable && new_term.variable.equal(v)) {
+ return expr;
+ } else {
+ return expr.subst(v, new_term);
+ }
}
function equal(expr1, expr2) {
- return expr1.debruijn([]).equal(expr2.debruijn([]));
+ return expr1.debruijn([]).equal(expr2.debruijn([]));
}
function contains(expr1, expr2) {
- return expr1.debruijn([]).contains(expr2.debruijn([]));
+ return expr1.debruijn([]).contains(expr2.debruijn([]));
}
function Lambda_var(variable) {
- this.variable = variable;
- this.debruijn = function (seq) {
- return this.variable.position(seq);
- };
- this.to_string = function (as_atom) {
- return this.variable.to_string();
- };
- this.has_free = function (v) {
- if (v.name !== this.variable.name) {
- return [false, v];
- } else if (v.tag === this.variable.tag) {
- return [true, v];
- } else {
- return [false, this.variable];
- }
- };
- this.subst = function (v, new_term) {
- if (this.variable.equal(v)) {
- return new_term;
- } else {
- return this;
- }
- };
- this.check_eta = function () {
- return this;
- };
- this.eval_loop = function (stack, eta) {
- function unwind(left, stack) {
- if (stack.length === 0) {
- return left;
- } else {
- var x = stack[0];
- var xs = stack.slice(1);
- return unwind(new Lambda_app(left, x.eval_loop([], eta)), xs);
- }
- }
- return unwind(this, stack);
- };
- this.eval_cbv = function (aggressive) {
- return this;
- };
+ this.variable = variable;
+ this.debruijn = function (seq) {
+ return this.variable.position(seq);
+ };
+ this.to_string = function (as_atom) {
+ return this.variable.to_string();
+ };
+ this.has_free = function (v) {
+ if (v.name !== this.variable.name) {
+ return [false, v];
+ } else if (v.tag === this.variable.tag) {
+ return [true, v];
+ } else {
+ return [false, this.variable];
+ }
+ };
+ this.subst = function (v, new_term) {
+ if (this.variable.equal(v)) {
+ return new_term;
+ } else {
+ return this;
+ }
+ };
+ this.check_eta = function () {
+ return this;
+ };
+ this.eval_loop = function (stack, eta) {
+ function unwind(left, stack) {
+ if (stack.length === 0) {
+ return left;
+ } else {
+ var x = stack[0];
+ var xs = stack.slice(1);
+ return unwind(new Lambda_app(left, x.eval_loop([], eta)), xs);
+ }
+ }
+ return unwind(this, stack);
+ };
+ this.eval_cbv = function (aggressive) {
+ return this;
+ };
}
function Lambda_app(left, right) {
- this.left = left;
- this.right = right;
- this.debruijn = function (seq) {
- return new Db_app(this.left.debruijn(seq), this.right.debruijn(seq));
- };
- this.to_string = function (as_atom) {
- var base;
- if (this.left.left) {
- base = this.left.to_string() + " " + this.right.to_string(true);
- } else {
- base = this.left.to_string(true) + " " + this.right.to_string(true);
- }
- if (as_atom) {
- return "(" + base + ")";
- } else {
- return base;
- }
- };
- this.has_free = function (v) {
- var left_res = this.left.has_free(v);
- var right_res = this.right.has_free(v);
- var left_bool = left_res[0];
- var right_bool = right_res[0];
- var left_tag = left_res[1].tag;
- var right_tag = right_res[1].tag;
- var res;
- if (left_tag > right_tag) {
- res = left_res[1];
- } else {
- res = right_res[1];
- }
- return [left_bool || right_bool, res];
- };
- this.subst = function (v, new_term) {
- return new Lambda_app(subst(v, new_term, this.left), subst(v, new_term, this.right));
- };
- this.check_eta = function () {
- return this;
- };
- this.eval_loop = function (stack, eta) {
- var new_stack = stack.slice(0);
- new_stack.unshift(this.right);
- return this.left.eval_loop(new_stack, eta);
- };
- this.eval_cbv = function (aggressive) {
- var left = this.left.eval_cbv(aggressive);
- var right = this.right.eval_cbv(aggressive);
- if (left.body) {
- return subst(left.bound, right, left.body).eval_cbv(aggressive);
- } else {
- return new Lambda_app(left, right);
- }
- };
+ this.left = left;
+ this.right = right;
+ this.debruijn = function (seq) {
+ return new Db_app(this.left.debruijn(seq), this.right.debruijn(seq));
+ };
+ this.to_string = function (as_atom) {
+ var base;
+ if (this.left.left) {
+ base = this.left.to_string() + " " + this.right.to_string(true);
+ } else {
+ base = this.left.to_string(true) + " " + this.right.to_string(true);
+ }
+ if (as_atom) {
+ return "(" + base + ")";
+ } else {
+ return base;
+ }
+ };
+ this.has_free = function (v) {
+ var left_res = this.left.has_free(v);
+ var right_res = this.right.has_free(v);
+ var left_bool = left_res[0];
+ var right_bool = right_res[0];
+ var left_tag = left_res[1].tag;
+ var right_tag = right_res[1].tag;
+ var res;
+ if (left_tag > right_tag) {
+ res = left_res[1];
+ } else {
+ res = right_res[1];
+ }
+ return [left_bool || right_bool, res];
+ };
+ this.subst = function (v, new_term) {
+ return new Lambda_app(subst(v, new_term, this.left), subst(v, new_term, this.right));
+ };
+ this.check_eta = function () {
+ return this;
+ };
+ this.eval_loop = function (stack, eta) {
+ var new_stack = stack.slice(0);
+ new_stack.unshift(this.right);
+ return this.left.eval_loop(new_stack, eta);
+ };
+ this.eval_cbv = function (aggressive) {
+ var left = this.left.eval_cbv(aggressive);
+ var right = this.right.eval_cbv(aggressive);
+ if (left.body) {
+ return subst(left.bound, right, left.body).eval_cbv(aggressive);
+ } else {
+ return new Lambda_app(left, right);
+ }
+ };
}
function Lambda_lam(variable, body) {
- this.bound = variable;
- this.body = body;
- this.debruijn = function (seq) {
- var new_seq = seq.slice(0);
- new_seq.unshift(this.bound);
- return new Db_lam(this.body.debruijn(new_seq));
- };
- this.to_string = function (as_atom) {
- var base = "\\" + this.to_string_funct();
- if (as_atom) {
- return "(" + base + ")";
- } else {
- return base;
- }
- };
- this.to_string_funct = function () {
- if (this.body.to_string_funct) {
- return this.bound.to_string() + " " + this.body.to_string_funct();
- } else {
- return this.bound.to_string() + ". " + this.body.to_string();
- }
- };
- this.has_free = function (v) {
- if (this.bound.equal(v)) {
- return [false, v];
- } else {
- return this.body.has_free(v);
- }
- };
- this.subst = function (v, new_term) {
- function bump_tag(v1, v2) {
- var max;
- if (v1.tag > v2.tag) {
- max = v1.tag;
- } else {
- max = v2.tag;
- }
- return new Variable(v1.name, max + 1);
- }
- function bump_tag2(v1, v2) {
- if (v1.name !== v2.name) {
- return v1;
- } else {
- return bump_tag(v1, v2);
- }
- }
- if (this.bound.equal(v)) {
- return this;
- } else {
- var res = free_in(this.bound, new_term);
+ this.bound = variable;
+ this.body = body;
+ this.debruijn = function (seq) {
+ var new_seq = seq.slice(0);
+ new_seq.unshift(this.bound);
+ return new Db_lam(this.body.debruijn(new_seq));
+ };
+ this.to_string = function (as_atom) {
+ var base = "\\" + this.to_string_funct();
+ if (as_atom) {
+ return "(" + base + ")";
+ } else {
+ return base;
+ }
+ };
+ this.to_string_funct = function () {
+ if (this.body.to_string_funct) {
+ return this.bound.to_string() + " " + this.body.to_string_funct();
+ } else {
+ return this.bound.to_string() + ". " + this.body.to_string();
+ }
+ };
+ this.has_free = function (v) {
+ if (this.bound.equal(v)) {
+ return [false, v];
+ } else {
+ return this.body.has_free(v);
+ }
+ };
+ this.subst = function (v, new_term) {
+ function bump_tag(v1, v2) {
+ var max;
+ if (v1.tag > v2.tag) {
+ max = v1.tag;
+ } else {
+ max = v2.tag;
+ }
+ return new Variable(v1.name, max + 1);
+ }
+ function bump_tag2(v1, v2) {
+ if (v1.name !== v2.name) {
+ return v1;
+ } else {
+ return bump_tag(v1, v2);
+ }
+ }
+ if (this.bound.equal(v)) {
+ return this;
+ } else {
+ var res = free_in(this.bound, new_term);
// if x is free in the inserted term new_term, a capture is possible
- if (res) {
- // this.bound is free in new_term, need to alpha-convert
- var uniq_x = bump_tag2(bump_tag(this.bound, res), v);
- var res2 = free_in(uniq_x, this.body);
- if (res2) {
- uniq_x = bump_tag(uniq_x, res2);
- }
- var body2 = subst(this.bound, new Lambda_var(uniq_x), this.body);
- return new Lambda_lam(uniq_x, subst(v, new_term, body2));
- } else {
- // this.bound not free in new_term, can substitute new_term for v without any captures
- return new Lambda_lam(this.bound, subst(v, new_term, this.body));
- }
- }
- };
- this.check_eta = function () {
- if (this.body.right && this.body.right.variable && this.bound.equal(this.body.right.variable) && !free_in(this.bound, this.body.left)) {
- return this.body.left;
- } else {
- return this;
- }
- };
- this.eval_loop = function (stack, eta) {
- if (stack.length === 0) {
- var term = new Lambda_lam(this.bound, this.body.eval_loop([], eta));
- if (eta) {
- return term.check_eta();
- } else {
- return term;
- }
- } else {
- var x = stack[0];
- var xs = stack.slice(1);
- return subst(this.bound, x, this.body).eval_loop(xs, eta);
- }
- };
- this.eval_cbv = function (aggressive) {
- if (aggressive) {
- return new Lambda_lam(this.bound, this.body.eval_cbv(aggressive));
- } else {
- return this;
- }
- };
- this.to_int = function (sofar) {
- if (this.body.body && this.body.body.variable && this.body.bound.equal(this.body.body.variable)) {
- return 0 + sofar;
- } else if (this.body.variable && this.bound.equal(this.body.variable)) {
- return 1 + sofar;
- } else if (this.body.body && this.body.body.left && this.body.body.left.variable && this.bound.equal(this.body.body.left.variable)) {
- var new_int = new Lambda_lam(this.bound, new Lambda_lam(this.body.bound, this.body.body.right));
- return new_int.to_int(1 + sofar);
- } else {
- return "not a church numeral";
- }
- };
+ if (res) {
+ // this.bound is free in new_term, need to alpha-convert
+ var uniq_x = bump_tag2(bump_tag(this.bound, res), v);
+ var res2 = free_in(uniq_x, this.body);
+ if (res2) {
+ uniq_x = bump_tag(uniq_x, res2);
+ }
+ var body2 = subst(this.bound, new Lambda_var(uniq_x), this.body);
+ return new Lambda_lam(uniq_x, subst(v, new_term, body2));
+ } else {
+ // this.bound not free in new_term, can substitute new_term for v without any captures
+ return new Lambda_lam(this.bound, subst(v, new_term, this.body));
+ }
+ }
+ };
+ this.check_eta = function () {
+ if (this.body.right && this.body.right.variable && this.bound.equal(this.body.right.variable) && !free_in(this.bound, this.body.left)) {
+ return this.body.left;
+ } else {
+ return this;
+ }
+ };
+ this.eval_loop = function (stack, eta) {
+ if (stack.length === 0) {
+ var term = new Lambda_lam(this.bound, this.body.eval_loop([], eta));
+ if (eta) {
+ return term.check_eta();
+ } else {
+ return term;
+ }
+ } else {
+ var x = stack[0];
+ var xs = stack.slice(1);
+ return subst(this.bound, x, this.body).eval_loop(xs, eta);
+ }
+ };
+ this.eval_cbv = function (aggressive) {
+ if (aggressive) {
+ return new Lambda_lam(this.bound, this.body.eval_cbv(aggressive));
+ } else {
+ return this;
+ }
+ };
+ this.to_int = function (sofar) {
+ if (this.body.body && this.body.body.variable && this.body.bound.equal(this.body.body.variable)) {
+ return 0 + sofar;
+ } else if (this.body.variable && this.bound.equal(this.body.variable)) {
+ return 1 + sofar;
+ } else if (this.body.body && this.body.body.left && this.body.body.left.variable && this.bound.equal(this.body.body.left.variable)) {
+ var new_int = new Lambda_lam(this.bound, new Lambda_lam(this.body.bound, this.body.body.right));
+ return new_int.to_int(1 + sofar);
+ } else {
+ return "not a church numeral";
+ }
+ };
}
// cbv is 1: use call-by-value, don't descend inside lambda
// cbv is 2: applicative order
function reduce(expr, eta, cbv) {
- if (cbv) {
- return expr.eval_cbv(cbv > 1);
- } else {
- return expr.eval_loop([], eta);
- }
+ if (cbv) {
+ return expr.eval_cbv(cbv > 1);
+ } else {
+ return expr.eval_loop([], eta);
+ }
}
function make_var(name) {
- return new Variable(name);
+ return new Variable(name);
}
function make_app(aa, bb) {
- return new Lambda_app(aa, bb);
+ return new Lambda_app(aa, bb);
}
function make_lam(a, aa) {
- return new Lambda_lam(a, aa);
+ return new Lambda_lam(a, aa);
}
try {
- if (console && console.debug) {
- function print() {
- console.debug.apply(this, arguments);
- }
- }
+ if (console && console.debug) {
+ function print() {
+ console.debug.apply(this, arguments);
+ }
+ }
} catch (e) {}
-
+/*
+let true = K in
+let false = \x y. y in
+let and = \l r. l r false in
+let or = \l r. l true r in
+let pair = \u v f. f u v in
+let triple = \u v w f. f u v w in
+let succ = \n s z. s (n s z) in
+let pred = \n s z. n (\u v. v (u s)) (K z) I in
+let ifzero = \n. n (\u v. v (u succ)) (K 0) (\n withp whenz. withp n) in
+let add = \m n. n succ m in
+let mul = \m n. n (\z. add m z) 0 in
+let mul = \m n s. m (n s) in
+let sub = (\mzero msucc mtail. \m n. n mtail (m msucc mzero) true) (pair 0 I) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let min = \m n. sub m (sub m n) in
+let max = \m n. add n (sub m n) in
+let lt = (\mzero msucc mtail. \n m. n mtail (m msucc mzero) true (\x. true) false) (pair 0 I) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let leq = (\mzero msucc mtail. \m n. n mtail (m msucc mzero) true (\x. false) true) (pair 0 I) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let eq = (\mzero msucc mtail. \m n. n mtail (m msucc mzero) true (\x. false) true) (pair 0 (K (pair 1 I))) (\d. d (\a b. pair (succ a) (K d))) (\d. d false d) in
+let divmod = (\mzero msucc mtail. \n divisor.
+ (\dhead. n (mtail dhead) (\sel. dhead (sel 0 0)))
+ (divisor msucc mzero (\a b c. c x))
+ (\d m a b c. pair d m) )
+ (triple succ (K 0) I)
+ (\d. triple I succ (K d))
+ (\dhead d. d (\dz mz df mf drest sel. drest dhead (sel (df dz) (mf mz)))) in
+let div = \n d. divmod n d true in
+let mod = \n d. divmod n d false in
+let Y = \f. (\y. f(y y)) (\y. f(y y)) in
+let Z = (\u f. f(u u f)) (\u f. f(u u f)) in
+let fact = \y. y (\f n. ifzero n (\p. mul n (f p)) 1) in
+fact Z 3
+*/