cmu (empty) → 1.0
raw patch · 9 files changed
+1359/−0 lines, 9 filesdep +arraydep +basedep +containerssetup-changed
Dependencies added: array, base, containers
Files
- ChangeLog +3/−0
- Makefile +24/−0
- Setup.hs +2/−0
- cmu.cabal +34/−0
- license.txt +675/−0
- readme.txt +18/−0
- src/Algebra/CommutativeMonoid/HomLinDiaphEq.hs +143/−0
- src/Algebra/CommutativeMonoid/Main.hs +105/−0
- src/Algebra/CommutativeMonoid/Unification.hs +355/−0
+ ChangeLog view
@@ -0,0 +1,3 @@+2009-09-17 John D. Ramsdell <ramsdell@mitre.org>++ * cmu.cabal (Version): Released as version 1.0.
+ Makefile view
@@ -0,0 +1,24 @@+# Haskell/Cabal Makefile+# Requires GNU Make+# The all target creates a default configuration if need be.++PACKAGE := $(wildcard *.cabal)+CONFIG = dist/setup-config+SETUP = runhaskell Setup.hs++all: $(CONFIG)+ $(SETUP) build++Makefile:+ @echo make $@++$(PACKAGE):+ @echo make $@++$(CONFIG): $(PACKAGE)+ $(SETUP) configure --ghc --user --prefix="${HOME}"++%: force+ $(SETUP) $@++.PHONY: all force
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ cmu.cabal view
@@ -0,0 +1,34 @@+Name: cmu+Version: 1.0+Maintainer: ramsdell@mitre.org+Cabal-Version: >= 1.2+License: GPL+License-File: license.txt+Synopsis: Unification in a Commutative Monaid+Description: The unification problem is given the problem+ statement t =? t\', find a most general+ substitution s such that s(t) = s(t\') modulo+ the axioms of a commutative monoid.+ Substitition s is more general than s\' if+ there is a substitition s\" such that s\' =+ s\" o s.+Category: Algebra+Build-Type: Simple+Extra-Source-Files: readme.txt ChangeLog Makefile++Library+ Build-Depends: base <= 4.1.0.0, containers, array+ Exposed-Modules: Algebra.CommutativeMonoid.Unification+ Algebra.CommutativeMonoid.HomLinDiaphEq+ Hs-Source-Dirs: src+ GHC-Options:+ -Wall -fno-warn-name-shadowing -fwarn-unused-imports++Executable cmu+ Main-Is: Algebra/CommutativeMonoid/Main.hs+ Build-Depends: base <= 4.1.0.0, containers, array+ Other-Modules: Algebra.CommutativeMonoid.Unification+ Algebra.CommutativeMonoid.HomLinDiaphEq+ Hs-Source-Dirs: src+ GHC-Options:+ -Wall -fno-warn-name-shadowing -fwarn-unused-imports
+ license.txt view
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Of course, your program's commands+might be different; for a GUI interface, you would use an "about box".++ You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU GPL, see+<http://www.gnu.org/licenses/>.++ The GNU General Public License does not permit incorporating your program+into proprietary programs. If your program is a subroutine library, you+may consider it more useful to permit linking proprietary applications with+the library. If this is what you want to do, use the GNU Lesser General+Public License instead of this License. But first, please read+<http://www.gnu.org/philosophy/why-not-lgpl.html>.
+ readme.txt view
@@ -0,0 +1,18 @@+This package contains a library for unification in+commutative monoid and a program that exercises the library.++$ cmu+Commutative monoid unification -- :? for help+cmu> 2x+y=3z+Problem: 2x + y = 3z+Unifier: [x : g0 + 3g2,y : g0 + 3g1,z : g0 + g1 + 2g2]++cmu> 2x=x+y+Problem: 2x = x + y+Unifier: [x : g0,y : g0]++cmu> 64x=41y+a+Problem: 64x = a + 41y+Unifier: [a : 5g0 + 2g1 + 23g2 + g3 + 64g4,x : 2g0 + 9g1 + g2 + 25g3 + g4 + 41g5,y : 3g0 + 14g1 + g2 + 39g3 + 64g5]++cmu> :quit
+ src/Algebra/CommutativeMonoid/HomLinDiaphEq.hs view
@@ -0,0 +1,143 @@+-- Homogeneous Linear Diaphantine Equation solver+--+-- Copyright (C) 2009 John D. Ramsdell+--+-- This program is free software: you can redistribute it and/or modify+-- it under the terms of the GNU General Public License as published by+-- the Free Software Foundation, either version 3 of the License, or+-- (at your option) any later version.++-- This program is distributed in the hope that it will be useful,+-- but WITHOUT ANY WARRANTY; without even the implied warranty of+-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+-- GNU General Public License for more details.++-- You should have received a copy of the GNU General Public License+-- along with this program. If not, see <http://www.gnu.org/licenses/>.++-- |+-- Module : Algebra.CommutativeMonoid.HomLinDiaphEq+-- Copyright : (C) 2009 John D. Ramsdell+-- License : GPL+--+-- Homogeneous Linear Diaphantine Equation solver.+--+-- The solver uses the algorithm of Contejean and Devie as specified+-- by David Papp and Bela Vizari in \"Effective Solutions of Linear+-- Diophantine Equation Systems with an Application to Chemistry\",+-- Rutcor Research Report RRR 28-2004, September, 2004,+-- <http://rutcor.rutgers.edu/pub/rrr/reports2004/28_2004.ps>, after+-- modification so as to ensure every basis vector is considered.+--+-- The algorithm for systems of homogeneous linear Diophantine+-- equations follows. Let e[k] be the kth basis vector for 1 <= k <=+-- n. To find the minimal, non-negative solutions M to the system of+-- equations sum(i=1,n,a[i]*v[i]) = 0, the algorithm of Contejean and+-- Devie is:+--+-- 1. [init] A := {e[k] | 1 <= k <= n}; M := {}+--+-- 2. [new minimal results] M := M + {a in A | a is a solution}+--+-- 3. [unnecessary branches] A := {a in A | all m in M : some+-- 1 <= k <= n : m[k] < a[k]}+--+-- 4. [test] If A = {}, stop+--+-- 5. [breadth-first search] A := {a + e[k] | a in A, 1 <= k <= n,+-- \<sum(i=1,n,a[i]*v[i]),v[k]> \< 0}; go to step 2++module Algebra.CommutativeMonoid.HomLinDiaphEq (homLinDiaphEq) where++import Data.Array+import Data.Set (Set)+import qualified Data.Set as S++{-- Debugging hack+import System.IO.Unsafe++z :: Show a => a -> b -> b+z x y = seq (unsafePerformIO (print x)) y+--}++type Vector a = Array Int a++vector :: Int -> [a] -> Vector a+vector n elems =+ listArray (0, n - 1) elems++-- | The 'homLinDiaphEq' function takes a list of integers that+-- specifies a homogeneous linear Diophantine equation, and returns+-- the equation's minimal, non-negative solutions.+homLinDiaphEq :: [Int] -> [[Int]]+homLinDiaphEq [] = []+homLinDiaphEq v =+ newMinimalResults (vector n v) (basis n) S.empty+ where n = length v++-- Construct the basis vectors for an n-dimensional space+basis :: Int -> Set (Vector Int)+basis n =+ foldl (flip S.insert) S.empty+ [ z // [(k, 1)] |+ k <- indices z ]+ where z = vector n $ replicate n 0++-- The main loop has been reorganized to ensure every basis vector is+-- considered. The breadth-first search step is now the last step.++-- Add elements of a that solve the equation to m and the output+newMinimalResults :: Vector Int -> Set (Vector Int) ->+ Set (Vector Int) -> [[Int]]+newMinimalResults v a m =+ loop m (S.toList a) -- Test each element in a+ where+ loop m [] =+ nextSearch v a m -- Generate new a and try again+ loop m (x:xs)+ | prod v x == 0 && S.notMember x m =+ elems x:loop (S.insert x m) xs -- Answer found+ | otherwise =+ loop m xs++-- Generate the next set of test vectors--if there aren't any, your done+nextSearch :: Vector Int -> Set (Vector Int) ->+ Set (Vector Int) -> [[Int]]+nextSearch v a m =+ if S.null a' then+ []+ else+ newMinimalResults v (breadthFirstSearch v a') m+ where+ a' = unnecessaryBranches a m++-- Remove unnecessary branches. A test vector is not necessary if all+-- of its elements are greater than or equal to the elements of some+-- minimal solution.+unnecessaryBranches :: Set (Vector Int) -> Set (Vector Int) -> Set (Vector Int)+unnecessaryBranches a m =+ S.filter f a+ where+ f x = all (g x) (S.toList m)+ g x y = not (lessEq y x)++-- Compare vectors element-wise.+lessEq :: Vector Int -> Vector Int -> Bool+lessEq x y =+ all (\i-> x!i <= y!i) (indices x)++-- Breadth-first search using the algorithm of Contejean and Devie+breadthFirstSearch :: Vector Int -> Set (Vector Int) -> Set (Vector Int)+breadthFirstSearch v a =+ S.fold f S.empty a+ where+ f x acc =+ foldl (flip S.insert) acc+ [ x // [(k, x!k + 1)] |+ k <- indices x,+ prod v x * v!k < 0 ] -- Contejean-Devie contribution++-- Inner product+prod :: Vector Int -> Vector Int -> Int+prod x y =+ sum [ x!i * y!i | i <- indices x ]
+ src/Algebra/CommutativeMonoid/Main.hs view
@@ -0,0 +1,105 @@+-- A top-level loop for unification in a Commutative Moniod+--+-- Copyright (C) 2009 John D. Ramsdell+--+-- This program is free software: you can redistribute it and/or modify+-- it under the terms of the GNU General Public License as published by+-- the Free Software Foundation, either version 3 of the License, or+-- (at your option) any later version.++-- This program is distributed in the hope that it will be useful,+-- but WITHOUT ANY WARRANTY; without even the implied warranty of+-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+-- GNU General Public License for more details.++-- You should have received a copy of the GNU General Public License+-- along with this program. If not, see <http://www.gnu.org/licenses/>.++module Main (main, test) where++import System.IO (isEOF, hFlush, stdout)+import Algebra.CommutativeMonoid.Unification++-- Test Routine++-- Given an equation, display a unifier.+test :: String -> IO ()+test prob =+ case readM prob of+ Err err -> putStrLn err+ Ans (Equation (t0, t1)) ->+ do+ putStr "Problem: "+ print $ Equation (t0, t1)+ putStr "Unifier: "+ let subst = unify $ Equation (t0, t1)+ print subst+ putStrLn ""++readM :: (Read a, Monad m) => String -> m a+readM s =+ case [ x | (x, t) <- reads s, ("", "") <- lex t ] of+ [x] -> return x+ [] -> fail "no parse"+ _ -> fail "ambiguous parse"++data AnsErr a+ = Ans a+ | Err String++instance Monad AnsErr where+ (Ans x) >>= k = k x+ (Err s) >>= _ = Err s+ return = Ans+ fail = Err++-- Main loop++main :: IO ()+main =+ do+ putStrLn "Commutative monoid unification -- :? for help"+ loop++loop :: IO ()+loop =+ do+ putStr "cmu> "+ hFlush stdout+ done <- isEOF+ case done of+ True ->+ do+ putStrLn ""+ return ()+ False ->+ do+ line <- getLine+ case () of+ _ | line == ":?" || line == ":help" ->+ do+ help+ loop+ | line == ":quit" ->+ return ()+ | otherwise ->+ do+ test line+ loop++help :: IO ()+help =+ mapM_ putStrLn mesg++mesg :: [String]+mesg =+ [ "Pose a question as an equation such as",+ " 2x + y = 3z, or",+ " 2x = x + y, or",+ " 64x - 41y = a.",+ "The cmu programs shows the result of unification.",+ "",+ "The unification problem is given two terms t and t', find a most",+ "general unifier s such that s(t) = s(t').",+ "",+ ":quit quits the program, :? and :help print this message."]
+ src/Algebra/CommutativeMonoid/Unification.hs view
@@ -0,0 +1,355 @@+-- Unification in a commutative monoid+--+-- Copyright (C) 2009 John D. Ramsdell+--+-- This program is free software: you can redistribute it and/or modify+-- it under the terms of the GNU General Public License as published by+-- the Free Software Foundation, either version 3 of the License, or+-- (at your option) any later version.++-- This program is distributed in the hope that it will be useful,+-- but WITHOUT ANY WARRANTY; without even the implied warranty of+-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+-- GNU General Public License for more details.++-- You should have received a copy of the GNU General Public License+-- along with this program. If not, see <http://www.gnu.org/licenses/>.++-- |+-- Module : Algebra.CommutativeMonoid.Unification+-- Copyright : (C) 2009 John D. Ramsdell+-- License : GPL+--+-- This module provides unification in a commutative monoid.+--+-- In this module, a commutative monoid is a free algebra over a+-- signature with two function symbols:+--+-- * the binary symbol +, the group operator,+--+-- * a constant 0, the identity element, and+--+-- The algebra is generated by a set of variables. Syntactically, a+-- variable is an identifer such as x and y (see 'isVar').+--+-- The axioms associated with the algebra are:+--+-- [Communtativity] x + y = y + x+--+-- [Associativity] (x + y) + z = x + (y + z)+--+-- [Group Identity] x + 0 = x+--+-- A substitution maps variables to terms. A substitution s is+-- applied to a term as follows.+--+-- * s(0) = 0+--+-- * s(t + t\') = s(t) + s(t\')+--+-- The unification problem is given the problem statement t =? t\',+-- find a most general substitution s such that s(t) = s(t\') modulo+-- the axioms of the algebra. Substitition s is more general than s\'+-- if there is a substitition s\" such that s\' = s\" o s.++module Algebra.CommutativeMonoid.Unification+ (+ -- * Terms+ Term, ide, isVar, var, mul, add, assocs,+ -- * Equations and Substitutions+ Equation(..), Substitution, subst, maplets, apply,+ -- * Unification+ unify) where++import Data.Char (isSpace, isAlpha, isAlphaNum, isDigit)+import Data.List (transpose)+import Data.Map (Map)+import qualified Data.Map as Map+import Algebra.CommutativeMonoid.HomLinDiaphEq++-- Chapter 8, Section 5 of the Handbook of Automated Reasoning by+-- Franz Baader and Wayne Snyder describes unification in+-- commutative/monoidal theories. This module refines the described+-- algorithms for the special case of a commutative monoid.++-- In this module, a commutative monoid is a free algebra over a signature+-- with two function symbols:+--+-- * the binary symbol +, the group operator,+-- * a constant 0, the identity element, and+--+-- The algebra is generated by a set of variables. Syntactically, a+-- variable is an identifer such as x and y.++-- The axioms associated with the algebra are:+--+-- * x + y = y + x Commutativity+-- * (x + y) + z = x + (y + z) Associativity+-- * x + 0 = x Group identity++-- A substitution maps variables to terms. A substitution s is+-- extended to a term as follows.+--+-- s(0) = 0+-- s(t + t') = s(t) + s(t')++-- The unification problem is given the problem statement t =? t',+-- find a most general substitution s such that s(t) = s(t') modulo+-- the axioms of the algebra. Substitition s is more general than s'+-- if there is a substitition s" such that s' = s" o s.++-- A term is represented by the group identity, or as the sum of+-- factors. A factor is the product of a positive integer coefficient+-- and a variable. In this representation, no variable occurs twice.+-- Thus a term is represented by a finite map from variables to+-- non-negative integers.++-- | A term in a commutative monoid is represented by the group+-- identity element, or as the sum of factors. A factor is the+-- product of a positive integer coefficient and a variable. No+-- variable occurs twice in a term. For the show and read methods,+-- zero is the group identity, the plus sign is the group operation.+newtype Term = Term (Map String Int) deriving Eq++-- Constructors++-- | 'ide' represents the identity element (zero).+ide :: Term+ide = Term Map.empty++-- | A variable is an alphabetic Unicode character followed by a+-- sequence of alphabetic or numeric digit Unicode characters. The+-- show method for a term works correctly when variables satisfy+-- the 'isVar' predicate.+isVar :: String -> Bool+isVar [] = False+isVar (c:s) = isAlpha c && all isAlphaNum s++-- | Return a term that consists of a single variable.+var :: String -> Term+var x = Term $ Map.singleton x 1++-- | Multiply every coefficient in a term by an non-negative integer.+mul :: Int -> Term -> Term+mul 0 (Term _) = ide+mul 1 t = t+mul n (Term t)+ | n < 0 = error "Negative coefficient found"+ | otherwise = Term $ Map.map (* n) t++-- Invert a term by negating its coefficients.+neg :: Term -> Term+neg (Term t) =+ Term $ Map.map negate t++-- | Add two terms.+add :: Term -> Term -> Term+add (Term t) (Term t') =+ Term $ Map.foldWithKey f t' t -- Fold over the mappings in t+ where+ f x c t = -- Alter the mapping of+ Map.alter (g c) x t -- variable x in t+ g c Nothing = -- Variable x not currently mapped+ Just c -- so add a mapping+ g c (Just c') -- Variable x maps to c'+ | c + c' == 0 = Nothing -- Delete the mapping+ | otherwise = Just $ c + c' -- Adjust the mapping++-- | Return all variable-coefficient pairs in the term in ascending+-- variable order.+assocs :: Term -> [(String, Int)]+assocs (Term t) = Map.assocs t++-- | Convert a list of variable-coefficient pairs into a term.+term :: [(String, Int)] -> Term+term assoc =+ foldr f ide assoc+ where+ f (x, c) t = add t $ mul c $ var x++-- Equations and Substitutions++-- | An equation is a pair of terms. For the show and read methods,+-- the two terms are separated by an equal sign.+newtype Equation = Equation (Term, Term) deriving Eq++-- | A substitution maps variables into terms. For the show and read+-- methods, the substitution is a list of maplets, and the variable+-- and the term in each element of the list are separated by a colon.+newtype Substitution = Substitution (Map String Term) deriving Eq++-- | Construct a substitution from a list of variable-term pairs.+subst :: [(String, Term)] -> Substitution+subst assocs =+ Substitution $ foldl f Map.empty assocs+ where+ f t (x, n) = Map.insert x n t++-- | Return all variable-term pairs in ascending variable order.+maplets :: Substitution -> [(String, Term)]+maplets (Substitution s) = Map.assocs s++-- | Return the result of applying a substitution to a term.+apply :: Substitution -> Term -> Term+apply (Substitution s) (Term t) =+ Map.foldWithKey f ide t+ where+ f x n t =+ add (mul n (Map.findWithDefault (var x) x s)) t++-- Unification++-- | Given 'Equation' (t0, t1), return a most general substitution s+-- such that s(t0) = s(t1) modulo the equational axioms of a+-- commutative monoid.+unify :: Equation -> Substitution+unify (Equation (t0, t1)) =+ case assocs (add t0 (neg t1)) of+ [] -> Substitution Map.empty+ t ->+ let basis = homLinDiaphEq (map snd t) in+ mgu (map fst t) basis++-- Construct a most general unifier the minimal non-negative solutions+-- to a linear equation. The function adds the variables back into+-- terms, and generates fresh variables as needed.+mgu :: [String] -> [[Int]] -> Substitution+mgu vars basis =+ subst (zip vars terms)+ where+ terms = map (term . zip genSyms) (transpose basis)+ genSyms = genSymsAvoiding vars++-- Generated variables start with this character.+genChar :: Char+genChar = 'g'++-- Generated symbols are the gen start char followed by a number.+genSym :: Int -> String+genSym i = genChar : show i++-- Produce a stream of generated identifiers avoiding what's in vars.+genSymsAvoiding :: [String] -> [String]+genSymsAvoiding vars =+ genSymStream 0+ where+ seen = filter genStr vars+ genStr (c:_) = c == genChar+ genStr _ = False+ genSymStream n+ | elem (genSym n) seen = genSymStream (n + 1)+ | otherwise = genSym n : genSymStream (n + 1)++-- So why solve linear equations? Consider the matching problem+--+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] =?+-- d[0]*a[0] + d[1]*a[1] + ... + d[m-1]*a[m-1]+--+-- with n variables and m constants. We seek a most general unifier s+-- such that+--+-- s(c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1]) =+-- d[0]*a[0] + d[1]*a[1] + ... + d[m-1]*a[m-1]+--+-- which is the same as+--+-- c[0]*s(x[0]) + c[1]*s(x[1]) + ... + c[n-1]*s(x[n-1]) =+-- d[0]*a[0] + d[1]*a[1] + ... + d[m-1]*a[m-1]+--+-- Notice that the number of occurrences of constant a[0] in s(x[0])+-- plus s(x[1]) ... s(x[n-1]) must equal d[0]. Thus the mappings of+-- the unifier that involve constant a[0] respect non-negative integer+-- solutions of the following linear equation.+--+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = d[0]+--+-- To compute a most general unifier, the set of minimal non-negative+-- integer solutions to a linear equation must be found. See module+-- Algebra.CommutativeMonoid.HomLinDiaphEq.++-- Input and Output++instance Show Term where+ showsPrec _ t =+ case assocs t of+ [] -> showString "0"+ (t:ts) -> showFactor t . showl ts+ where+ showFactor (x, 1) = showString x+ showFactor (x, c) = shows c . showString x+ showl [] = id+ showl (t:ts) = showString " + " . showFactor t . showl ts++instance Read Term where+ readsPrec _ s0 =+ [ (t1, s2) | (t0, s1) <- readFactor s0,+ (t1, s2) <- readRest t0 s1 ]+ where+ readPrimary s0 =+ [ (t0, s1) | (x, s1) <- scan s0, isVarToken x,+ let t0 = var x ] +++ [ (t0, s1) | ("0", s1) <- scan s0,+ let t0 = ide ] +++ [ (t0, s3) | ("(", s1) <- scan s0,+ (t0, s2) <- reads s1,+ (")", s3) <- scan s2 ]+ readFactor s0 =+ [ (t0, s1) | (t0, s1) <- readPrimary s0 ] +++ [ (t1, s2) | (n, s1) <- scan s0, isNumToken n,+ (t0, s2) <- readPrimary s1,+ let t1 = mul (read n) t0 ]+ readRest t0 s0 =+ [ (t2, s3) | ("+", s1) <- scan s0,+ (t1, s2) <- readFactor s1,+ (t2, s3) <- readRest (add t0 t1) s2 ] +++ [ (t0, s0) | (s, _) <- scan s0, s /= "+" ]++isNumToken :: String -> Bool+isNumToken (c:_) = isDigit c+isNumToken _ = False++isVarToken :: String -> Bool+isVarToken (c:_) = isAlpha c+isVarToken _ = False++scan :: ReadS String+scan "" = [("", "")]+scan (c:s)+ | isSpace c = scan s+ | isAlpha c = [ (c:part, t) | (part,t) <- [span isAlphaNum s] ]+ | isDigit c = [ (c:part, t) | (part,t) <- [span isDigit s] ]+ | otherwise = [([c], s)]++instance Show Equation where+ showsPrec _ (Equation (t0, t1)) =+ shows t0 . showString " = " . shows t1++instance Read Equation where+ readsPrec _ s0 =+ [ (Equation (t0, t1), s3) | (t0, s1) <- reads s0,+ ("=", s2) <- scan s1,+ (t1, s3) <- reads s2 ]++-- This datatype is used only in the read and show methods for+-- substitutions.+newtype Maplet = Maplet (String, Term) deriving Eq++instance Show Maplet where+ showsPrec _ (Maplet (x, t)) =+ showString x . showString " : " . shows t++instance Read Maplet where+ readsPrec _ s0 =+ [ (Maplet (x, t), s3) | (x, s1) <- scan s0, isVarToken x,+ (":", s2) <- scan s1,+ (t, s3) <- reads s2 ]++instance Show Substitution where+ showsPrec _ s =+ shows $ map Maplet $ maplets s++instance Read Substitution where+ readsPrec _ s0 =+ [ (subst $ map pair ms, s1) | (ms, s1) <- reads s0 ]+ where+ pair (Maplet (x, t)) = (x, t)