| Safe Haskell | Unsafe |
|---|
Quipper.Libraries.ClassicalOptim.AlgExp
Description
This module contains an efficient representation of algebraic boolean formulas.
Synopsis
- mapOfSet :: Ord a => Set a -> Map a Int
- setOfMap :: Ord a => Map a Int -> Set a
- split_even :: [a] -> ([a], [a])
- type Exp = Set IntSet
- listOfExp :: Exp -> [[Int]]
- expOfList :: [[Int]] -> Exp
- exp_and :: Exp -> Exp -> Exp
- exp_xor :: Exp -> Exp -> Exp
- exp_false :: Exp
- exp_true :: Exp
- exp_not :: Exp -> Exp
- exp_var :: Int -> Exp
- vars_of_exp :: Exp -> [Int]
- exp_eval :: Exp -> Map Int Bool -> Bool
- valuations_of_vars :: [Int] -> [Map Int Bool]
- truth_table_of_exp :: [Int] -> Exp -> [Bool]
- exp_of_truth_table :: Int -> [Bool] -> Exp
Auxiliary functions
split_even :: [a] -> ([a], [a]) Source #
Split a list in the middle.
Expressions
type Exp = Set IntSet Source #
The type of algebraic boolean expressions.
We represent boolean expressions using "and" and "xor" as the primitive connectives. Equivalently, we can regard booleans as the elements of the two-element field F2, with operations "*" (times) and "+" (plus).
An algebraic expression
x1*x2*x3 + y1*y2*y3 + z1*z2
is encoded as
{{x1,x2,x3},{y1,y2,y3},{z1,z2}}.
In particular,
{} == False == 0 and
{{}} == True == 1.
Properties of expressions
The important property of expressions is that two formulas have the same truth table iff they are syntactically equal. This makes the equality test of wires theoretically straightforward.
Truth tables
A valuation on a set of variables is a map from variables to booleans. This can be thought of as a row in a truth table. A truth table is a map from valuations to booleans, but we just represent this as a list of booleans, listed in lexicographically increasing order of valuations.
vars_of_exp :: Exp -> [Int] Source #
Get the variables used in an expression.
exp_eval :: Exp -> Map Int Bool -> Bool Source #
Evaluate the expression with respect to the given valuation. A valuation is a map from variables to booleans, i.e., a row in a truth table.
valuations_of_vars :: [Int] -> [Map Int Bool] Source #
Construct the list of all 2n valuations for a given list of n variables.
truth_table_of_exp :: [Int] -> Exp -> [Bool] Source #
Build the truth table for the given expression, on the given list of variables. The truth table is returned as a list of booleans in lexicographic order of valuations. For example, if
1 2 | exp F F | f1 F T | f2 T F | f3 T T | f4
then the output of the function is [f1,f2,f3,f4].
exp_of_truth_table :: Int -> [Bool] -> Exp Source #
Return an expression realizing the given truth table. Uses variables starting with the given number.