Logic Solver
Source code of released version  Source code of development version
Introduction
Logic Solver is a boolean satisfiability solver written in JavaScript. Given a problem expressed as logical constraints on boolean (true/false) variables, it either provides a possible solution, or tells you definitively that there is no possible assignment of the variables that satisfies the constraints.
Many kinds of logic problems can be expressed in terms of constraints on boolean variables, including Sudoku puzzles, scheduling problems, and the package dependency problem faced by package managers that automatically resolve version conflicts.
Logic Solver can handle complex problems with thousands of variables, and has some powerful features such as incremental solving and solving under temporary assumptions. It also supports smallinteger sums and inequalities, and can minimize or maximize an integer expression.
Logic Solver contains a copy of MiniSat, an industrialstrength SAT solver, compiled from C++ to JavaScript using Emscripten. See About MiniSat for more information.
Logic Solver can solve a hard Sudoku in under a second in a web browser, with very cleanlooking code compared to many constraint solvers. Try this demo
On NPM
var Logic = require('logicsolver');
https://www.npmjs.com/package/logicsolver
Table of Contents
 Introduction
 On NPM
 Example: Dinner Guests
 Example: Magic Squares
 Variables
 Terms
 Formulas
 Logic.Solver
 Logic.Solution
 Optimization
 Bits (integers)
 new Logic.Bits(formulas)
 Logic.isBits(value)
 Logic.constantBits(wholeNumber)
 Logic.variableBits(baseName, N)
 Logic.equalBits(bits1, bits2)
 Logic.lessThan(bits1, bits2)
 Logic.lessThanOrEqual(bits1, bits2)
 Logic.greaterThan(bits1, bits2)
 Logic.greaterThanOrEqual(bits1, bits2)
 Logic.sum(operands...)
 Logic.weightedSum(formulas, weights)
 About MiniSat
Example: Dinner Guests
We are trying to decide what combination of Alice, Bob, and Charlie to invite over to dinner, subject to the following constraints:
 Don't invite both Alice and Bob
 Invite either Bob or Charlie
Setting up these constraints in code:
1var solver = new Logic.Solver(); 2 3solver.require(Logic.atMostOne("Alice", "Bob")); 4solver.require(Logic.or("Bob", "Charlie"));
Solving now will give us one possible solution, chosen arbitrarily:
1var sol1 = solver.solve(); 2sol1.getTrueVars() // => ["Bob"]
Let's see what happens if we invite Alice. By using solveAssuming
, we
can look for a solution that makes an additional logical expression true
over the ones we have required so far:
1var sol2 = solver.solveAssuming("Alice"); 2sol2.getTrueVars() // => ["Alice", "Charlie"]
Aha! It seems that inviting Alice means we can't invite Bob, but then we must invite Charlie! If our reasoning is correct, it is impossible to invite Alice and not invite Charlie. We can confirm this:
1solver.solveAssuming(Logic.and("Alice", "Charlie")) // => null
(Note that "Charlie"
is shorthand for Logic.not("Charlie")
.)
Let's write some code to list all possible solutions:
1var solutions = []; 2var curSol; 3while ((curSol = solver.solve())) { 4 solutions.push(curSol.getTrueVars()); 5 solver.forbid(curSol.getFormula()); // forbid the current solution 6} 7 8solutions 9// => [["Alice", "Charlie"], ["Charlie"], ["Bob", "Charlie"], ["Bob"]]
As you can see, there are four possible solutions to the original problem.
After running the above code, all possible solutions are now
forbidden, so the solver is in an unsatisfiable state. Calls to
solver.require
and solver.forbid
are permanent, so we cannot
return to a satisfiable state, and any call to solve
or
solveAssuming
henceforth will return no solution:
1solver.solve() // => null
It's informative to look at the clauses generated by Logic Solver during
this example. In this notation, v
is the boolean "OR" operator:
Alice v Bob (at most one of Alice, Bob) Bob v Charlie (at least one of Bob, Charlie) Alice v $assump1 (solve assuming Alice) $and1 v $assump2 (solve assuming Alice and not Charlie) Alice v $and1 Charlie v $and1 Alice v Bob v Charlie (forbid ["Alice", "Charlie"]) Alice v Bob v Charlie (forbid ["Charlie"]) Alice v Bob v Charlie (etc.) Alice v Bob v Charlie
These clauses are sent to MiniSat using variable numbers in place of names, making the entire problem quite compact:
[[3,4], [4,5], [3,6], [8,7], [3,8], [5,8], [3,4,5], [3,4,5], [3,4,5], [3,4,5]]
Example: Magic Squares
A 3x3 "magic square" is an arrangement of the digits 1 through 9 into a square such that the digits in each row, column, and diagonal add up to the same number. Here is an example from Wikipedia:
2 7 6 9 5 1 4 3 8
Each row, column, and threedigit diagonal adds up to 15, as you can verify. (There are many 3x3 magic squares, but the magic sum is always 15, because all the digits together add up to 45!)
Let's use Logic Solver to find magic squares. We could be fancy about it and write code that would generalize to NxN magic squares, but let's keep it simple and name the digit locations as follows:
A B C D E F G H I
Because each location holds an integer, we must use integer variables instead of boolean variables. An integer in Logic Solver is represented as a group of bits, where each bit is a boolean variable, or an entire boolean formula. Let's create a 4bit group of variables for each digit location:
1var A = Logic.variableBits('A', 4); 2var B = Logic.variableBits('B', 4); 3var C = Logic.variableBits('C', 4); 4var D = Logic.variableBits('D', 4); 5var E = Logic.variableBits('E', 4); 6var F = Logic.variableBits('F', 4); 7var G = Logic.variableBits('G', 4); 8var H = Logic.variableBits('H', 4); 9var I = Logic.variableBits('I', 4); 10 11var locations = [A, B, C, D, E, F, G, H, I]; 12 13A.bits // => ["A$0", "A$1", "A$2", "A$3"]
Let's also assign the number 15, in bit form, to a variable for convenience.
1var fifteen = Logic.constantBits(15); 2fifteen.bits // => ["$T", "$T", "$T", "$T"]
The binary representation of 15 is "1111", so its bit form consists of
four copies of Logic.TRUE
or "$T"
. We didn't have to know that,
though, because Logic.constantBits
generated it for us.
Now, we create a Solver and express our sum constraints:
1var solver = new Logic.Solver(); 2 3_.each([[A,B,C], [D,E,F], [G,H,I], [A,D,G], [B,E,H], [C,F,I], 4 [A,E,I], [G,E,C]], 5 function (terms) { 6 solver.require(Logic.equalBits(Logic.sum(terms), fifteen)); 7 });
Let's see what solution we get!
1var sol1 = solver.solve(); 2sol1.evaluate(A) // => 3 3sol1.evaluate(B) // => 10 (uh oh) 4_.map(locations, function (loc) { return sol1.evaluate(loc); }) 5// => [3, 10, 2, 6// 4, 5, 6, 7// 8, 0, 7]
Oops, it looks like we forgot to specify that each "digit" is between
1 and 9! There is no harm done, because we have only underspecified
the problem. We can continue to use the same solver
instance.
Now we add inequalities to make each location A through I hold a number between 1 and 9 inclusive, and solve again:
1_.each(locations, function (loc) { 2 solver.require(Logic.greaterThanOrEqual(loc, Logic.constantBits(1))); 3 solver.require(Logic.lessThanOrEqual(loc, Logic.constantBits(9))); 4}); 5 6var sol2 = solver.solve(); 7_.map(locations, function (loc) { return sol2.evaluate(loc); }) 8// => [8, 1, 6, 9// 3, 5, 7, 10// 4, 9, 2]
Now we have a proper magic square!
However, it just so happens that we also forgot to specify that the
numbers be distinct. To demonstrate that this is an important missing
constraint, we can use solveAssuming
to ask for a solution where A
and B are equal:
1var sol3 = solver.solveAssuming(Logic.equalBits(A, B)); 2_.map(locations, function (loc) { return sol3.evaluate(loc); }) 3// => [4, 4, 7, 4// 8, 5, 2, 5// 3, 6, 6]
Or where A, B, and C are equal:
1var sol4 = solver.solveAssuming(Logic.and(Logic.equalBits(A, B), 2 Logic.equalBits(B, C))); 3_.map(locations, function (loc) { return sol4.evaluate(loc); }) 4// => [5, 5, 5, 5// 5, 5, 5, 6// 5, 5, 5]
A good way to enforce that all locations hold different digits is to generate a requirement about each pair of different locations:
1_.each(locations, function (loc1, i) { 2 _.each(locations, function (loc2, j) { 3 if (i !== j) { 4 solver.forbid(Logic.equalBits(loc1, loc2)); 5 } 6 }); 7});
Solving now gives us a proper magic square again:
1var sol5 = solver.solve(); 2_.map(locations, function (loc) { return sol5.evaluate(loc); }) 3// => [6, 7, 2, 4// 1, 5, 9, 5// 8, 3, 4]
If we wished to continue interrogating the solver, we could try asking for a magic square with a 1 in the upperleft corner, or proceed to enumerate a list of magic squares.
Finally, let's demonstrate that our "integers" are really just groups of boolean variables:
1sol5.getTrueVars() 2// => ["A$1", "A$2", "B$0", "B$1", "B$2", "C$1", "D$0", "E$0", "E$2", 3// "F$0", "F$3", "G$3", "H$0", "H$1", "I$2"] 4 5_.map(A.bits, function (v) { return sol5.evaluate(v); }) 6// => [false, true, true, false]
You may be wondering whether it's bad that we generated 72 constraints as part of finding a 3x3 magic square. While there are certainly much faster ways to calculate magic squares, it is perfectly reasonable when setting up a logic problem to generate a complete set of pairwise constraints over N variables. In fact, having more constraints often improves performance in realworld problems, so it is worth generating extra constraints even when they are technically redundant. More constraints means more deductions can be made at each step, meaning fewer possibilities need to be tried that ultimately won't work out. In this case, it's important that when the solver assigns a digit to a particular location, it immediately be able to deduce that the same number does not appear at any other location.
Variables
Variable names are Strings which can contain spaces and punctuation:
1Logic.implies('it is raining', 'take an umbrella'); 2 3Logic.exactlyOne("1,1", "1,2", "1,3")
Restrictions: A variable name must not be empty, consist of only the
characters 0
through 9
, or start with 
. Variable names that
start with $
are reserved for internal use.
You do not need to declare or create your variables before using them
in formulas passed to require
and forbid
.
When you pass a variable name to a Solver for the first time, a
variable number is allocated, and that name and number become
synonymous for that Solver instance. You don't need to know about
variable numbers to use Logic Solver, but you can always use a
variable number in place of a variable name in terms and formulas, in
case that is useful. (It is useful internally, and would probably be
useful if you were to wrap Logic Solver in another library.) Examples
of Solver methods that may allocate new variables are require
,
forbid
, solveAssuming
, and getVarNum
.
If you want to add a free variable to a Solver but not require
anything about it, you can use getVarNum
to cause the variable to be
allocated. It will then appear in solutions.
Methods
Logic.Solver#getVarNum(variableName, [noCreate])
Returns the variable number for a variable name, allocating a number if
this is the first time this Solver has seen variableName
.
Parameters
variableName
 String  A valid variable name.noCreate
 Boolean  Optional. If true, this method will return 0 instead of allocating a new variable number ifvariableName
is new.
Returns
Integer  A positive integer variable number, or 0 if noCreate
is true
and there is no variable number allocated for variableName
.
Logic.Solver#getVarName(variableNum)
Returns the variable name for a given variable number. An error is thrown
if variableNum
is not an allocated variable number.
Parameters
variableNum
 Integer  An allocated variable number.
Returns
String  A variable name.
Terms
A Term is a variable name or number, optionally negated. To negate a
string Term, prefix it with ""
. Examples of valid Terms are
"foo"
, "foo"
, 5
, and 5
. In other solvers and papers, you may
see Terms referred to as "literals."
The following are equivalent:
1solver.require("A"); 2solver.require(Logic.not("A")); 3solver.forbid("A");
In fact, Logic.not("A")
returns "A"
. It is valid to have more
than one 
in a Term ("A"
), and the meaning will be what you'd
expect, but Logic.not
will never return you such a Term, so in
practice this case does not come up. Logic.not("A")
returns "A"
.
String Terms are called NameTerms, and numeric Terms are called NumTerms. You will not normally need to use numeric Terms, but if you do, note that it doesn't make sense to share them across Solver instances, because each Solver has its own variable numbers. See the Variables section for more information.
Constants
Logic.FALSE, Logic.TRUE
These Terms represent the constant boolean values false and true. You
may seem them appear as the internal variables $F
and $T
or 1
and 2
, which are automatically pinned to false and true.
Methods
Logic.isTerm(value)
Returns whether value
is a valid Term. A valid Term is either a
String consisting of a valid variable name preceded by zero or more

characters, or a nonzero integer.
Parameters
value
 Any
Returns
Boolean
Logic.isNameTerm(value)
Returns whether value
is a valid NameTerm (a Term that is a String).
Parameters
value
 Any
Returns
Boolean
Logic.isNumTerm(value)
Returns whether value
is a valid NumTerm (a Term that is a Number).
Parameters
value
 Any
Returns
Boolean
Logic.Solver#toNameTerm(term)
Converts a Term to a NameTerm if it isn't already. If term
is a
NumTerm, the variable number is translated into a variable name. An
error is thrown if the variable number is not an allocated variable
number of this Solver.
Parameters
term
 Term  The Term to convert, which may be a NameTerm or NumTerm.
Returns
NameTerm
Logic.Solver#toNumTerm(term, [noCreate])
Converts a Term to a NumTerm if it isn't already. If term
is a
NameTerm, the variable name is translated into a variable number. A
new variable number is allocated if the variable name has not been
seen before by this Solver, unless you pass true for noCreate
.
Parameters
term
 Term  The Term to convert, which may be a NameTerm or NumTerm.noCreate
 Boolean  Optional. If true, this method will not allocate a new variable number if it encounters a new variable name, but will return 0 instead.
Returns
NumTerm, or 0 (if noCreate
is true and a new variable name is encountered)
Formulas
A Formula is an object representing a boolean expression. Conceptually, a Formula is built out of Terms and operations that combine Terms.
Here are some examples of Formulas:
1// A and B 2Logic.and("A", "B") 3 4// If exactly one of (A, B, C) is true, then A does not equal D. 5Logic.implies(Logic.exactlyOne("A", "B", "C"), 6 Logic.not(Logic.equiv("A", "D"))) 7 8// More of (x1, x2, x3) are true than (y1, y2, y3) 9var xs = ["x1", "x2", "x3"]; 10var ys = ["y1", "y2", "y3"]; 11Logic.greaterThan(Logic.sum(xs), Logic.sum(ys))
Formulas are immutable. To be on the safe side, do not mutate any arrays you use to create a Formula.
Formulas are Solverindependent. They can be created without a Solver, and although Solvers keep track of Formula objects and recognize them (to avoid compiling the same Formula twice), a Formula object never becomes tied to one Solver object and can always be reused, as long as it doesn't contain any explicit variable numbers (NumTerms).
A Term is not a Formula, but you can always pass a Term anywhere a Formula is required.
Functions such as Logic.and
and Logic.greaterThan
are called
Formula constructor functions. One thing to note about them is that
they do not always return Formulas, but may return Terms as well.
Logic.and("A")
, for example, returns "A"
. Some constructor functions
take any number of arguments, which may be nested in arrays, so that
the following are equivalent:
1Logic.and("A", "B", "C") 2Logic.and(["A", "B", "C"]) 3Logic.and("A", [["B", "C"]], [])
To use a Formula, you must tell a Solver to require
or forbid
it.
Otherwise, the Formula does not take effect.
1var solver = new Logic.Solver(); 2solver.require("A"); 3 4Logic.exactlyOne("A", "B"); // no effect, just creates a Formula 5 6solver.require(Logic.exactlyOne("A", "B")); // this works 7 8var myFormula = Logic.exactlyOne("A", "B"); 9solver.require(myFormula); // this also works
You should save and reuse Formula objects whenever possible, because
the Solver will recognize the Formula object and not recompile it.
Internally, each Formula is replaced by a variable in the Solver, such
as $and1
for a Logic.and
, and clauses are generated that relate
the variable to the operands of the Formula. When you pass the same
Formula object again, it is replaced by the same variable, and the
Formula only needs to be compiled once.
Formulas that operate on integers are documented in the Bits section.
Methods
Logic.isFormula(value)
Returns true if value
is a Formula object. (A Term is not a Formula.)
Parameters
value
 Any
Returns
Boolean
Logic.not(operand)
Represents a boolean expression that is true when its operand is false, and vice versa.
When called on an operand that is a NameTerm, NumTerm, or Formula, returns a value of the same kind.
Parameters
operand
 Formula or Term
Returns
Formula or Term (same kind as operand
)
Examples
1Logic.not("A") // => "A" 2Logic.not("A") // => "A" 3Logic.not(Logic.and("A", "B")) // => a Formula object
Logic.or(operands...)
Represents a boolean expression that is true when at least one of its operands is true.
Parameters
operands...
 Zero or more Formulas, Terms, or Arrays
Returns
Formula or Term
Logic.and(operands...)
Represents a boolean expression that is true when all of its operands are true.
Parameters
operands...
 Zero or more Formulas, Terms, or Arrays
Returns
Formula or Term
Logic.xor(operands...)
Represents a boolean expression that is true when an odd number of its operands are true.
Parameters
operands...
 Zero or more Formulas, Terms, or Arrays
Returns
Formula or Term
Logic.implies(operand1, operand2)
Represents a boolean expression that is true unless operand1
is true and
operand2
is false. In other words, if this Formula is required to be true,
and operand1
is true, then operand2
must be true.
Parameters
operand1
 Formula or Termoperand2
 Formula or Term
Returns
Formula or Term
Logic.equiv(operand1, operand2)
Represents a boolean expression that is true when operand1
and operand2
are either both true or both false.
Parameters
operand1
 Formula or Termoperand2
 Formula or Term
Returns
Formula or Term
Parameters
operands...
 Zero or more Formulas, Terms, or Arrays
Returns
Formula or Term
Logic.exactlyOne(operands...)
Represents a boolean expression that is true when exactly one of its operands is true.
Parameters
operands...
 Zero or more Formulas, Terms, or Arrays
Returns
Formula or Term
Logic.atMostOne(operands...)
Represents a boolean expression that is true when zero or one of its operands are true.
Parameters
operands...
 Zero or more Formulas, Terms, or Arrays
Returns
Formula or Term
Logic.Solver
You create a Logic.Solver with new Logic.Solver()
.
A Solver maintains a list of Formulas that must be true (or false), which you can think of as a list of constraints. Each Solver instance embeds a selfcontained MiniSat instance, which learns and remembers facts that are derived from the constraints. At any time, you can ask the Solver for a solution that satisfies the current constraints, and it will either provide one (chosen arbitrarily) or report that none exists. You can then continue to add more constraints and solve again.
See Example: Dinner Guests for a good introduction to Solver.
Constraints are only ever added, never removed. If the current
constraints are not satisfiable, then solve()
will return null, and
adding additional constraints cannot make the problem solvable again.
However, using solveAssuming
, you can look for a solution with a
particular Formula temporarily in force. If solveAssuming
returns
null, there is no harm done, and you can continue to solve under other
assumptions or add more constraints.
Sometimes solve()
will take a long time! That is to be expected.
The best thing to do is to try expressing the problem in a different
way, with fewer variables, more sharing of common subexpressions, or
more constraints between variables so that the solver can make
important deductions in fewer steps. Also try wrapping your code in
Logic.disablingAssertions(function () { ... })
in case runtime type
checks are slowing down Formula compilation.
If you need an extra speed boost in Node, you could help me create a binary npm package containing a nativecompiled MiniSat.
Constructor
new Logic.Solver()
Methods
Logic.Solver#require(args...)
Requires that the Formulas and Terms listed in args
be true in order
for a solution to be valid.
Parameters
args...
 Zero or more Formulas, Terms, or Arrays
Logic.Solver#forbid(args...)
Requires that the Formulas and Terms listed in args
be false in
order for a solution to be valid.
Parameters
args...
 Zero or more Formulas, Terms, or Arrays
Logic.Solver#solve()
Finds a solution that satisfies all the constraints specified with
require
and forbid
, or determines that no such solution is
possible. A solution is an assignment of all the variables to boolean
values.
To find more than one solution, you can forbid the first solution
(using solver.forbid(solution.getFormula())
, and solve again.
Solving is fully incremental, and each call to solve()
has the
benefit of everything learned by previous calls to solve()
.
Resolving with one or two new constraints is typically very fast,
because no work is repeated.
There is no guarantee of which solution is found if there are more than one. However, some statements can be made about what to expect:

MiniSat starts by trying a solution where all variables are false, so underconstrained variables will tend to be set to false.

Calling
solve()
repeatedly, with no intervening method calls, will in practice return the same solution each time. On the other hand, if you callsolve
, thensolveAssuming
, thensolve
again, the call tosolveAssuming
will affect the solution returned by the secondsolve
. 
Logic Solver and MiniSat are deterministic, so the same series of calls on a new Solver will generally produce the same results. However, the results may not be stable across different versions of Logic Solver.
Returns
Logic.Solution, or null if no solution is possible
Logic.Solver#solveAssuming(assumption)
Like solve()
, but looks for a solution that additionally satisfies
assumption
. This is especially useful for testing whether a new
constraint would make the problem unsolvable before requiring it,
or for "querying" the solver about different types of solutions.
Note that any solution returned by solveAssuming
is also a valid
solution for solve
to return. If you call solve
, then
solveAssuming
, then solve
again, the second solve
will typically
return the same solution as solveAssuming
, because the internal
state of the solver has been changed (even though no new permanent
constraints have been introduced).
Parameters
assumption
 Formula or Term
Returns
Logic.Solution or null
Logic.disablingAssertions(func)
Calls func()
, disabling runtime type checks and assertions for the
duration. This speeds up the processing of complex Formulas,
especially when integers or large numbers of variables are involved,
at the price of not validating the arguments to most function calls.
It doesn't affect the time spent in MiniSat.
Parameters
func
 Function
Returns
Any  The return value of func()
.
Logic.Solution
A Solution represents an assignment or mapping of the Solver
variables to true/false values. Solution objects are returned by
Logic.Solver#solve
and Logic.Solver#solveAssuming
.
(Variables internal to the Solver, which start with $
and which
you'd probably only encounter while poking around in internals, are
not considered part of the assignment.)
Methods
Logic.Solution#getMap()
Returns a complete mapping of variables to their assigned values.
Returns
Object  Dictionary whose keys are variable names and whose values are booleans
Logic.Solution#getTrueVars()
Returns a list of all the variables that are assigned to true by this Solution.
Returns
Array of String  Names of the variables that are assigned to true
Logic.Solution#evaluate(expression)
Evaluates a Formula or Term under this Solution's assignment of the variables, returning a boolean value. For example:
1solution.evaluate('A') 2solution.evaluate('A') 3solution.evaluate(Logic.or('A', 'B')) 4solution.evaluate(myFormula) // Formula given to the Solver earlier
If expression
is a Bits, the result of evaluation is an integer:
1var x = Logic.variableBits('x', 3); // 3digit binary variable 2var y = Logic.variableBits('y', 3); 3var xySum = Logic.sum(x, y); 4var five = Logic.constantBits(5); 5 6var solver = new Logic.Solver; 7solver.require(Logic.equalBits(xySum, five)); 8var solution = solver.solve(); 9solution.evaluate(x) // 2 (for example) 10solution.evaluate(y) // 3 (for example) 11solution.evaluate(five) // 5
It is an error to try to evaluate an unknown variable or a variable that
did not exist at the time the Solution was created, unless you call
ignoreUnknownVariables()
first.
Parameters
expression
 Formula, Term, or Bits
Returns
Boolean or Integer
Logic.Solution#getFormula()
Creates a Formula (or Term) which can be used to require, or forbid, that variables are assigned to the exact values they have in this Solution.
To find all solutions to a logic problem:
1var solver = new Logic.Solver; 2solver.require(Logic.or('A', 'B')); 3 4var allSolutions = []; 5var curSolution = null; 6while ((curSolution = solver.solve())) { 7 allSolutions.push(curSolution.getTrueVars()); 8 solver.forbid(curSolution.getFormula()); 9} 10 11allSolutions // [["A"], ["A", "B"], ["B"]]
Adding a constraint and solving again in this way is quite efficient.
The Formula or Term returned may not be used with any other Solver instance besides the one that produced this Solution.
Returns
Formula or Term
Logic.Solution#getWeightedSum(formulas, weights)
Equivalent to evaluate(Logic.weightedSum(formulas, weights))
, but
much faster because the addition is done using integer arithmetic,
not boolean logic. Rather than constructing a Bits and evaluating it,
getWeightedSum
simply evaluates each of the Formulas to a boolean
value and then sums the weights corresponding to the Formulas that
evaluate to true.
See Logic.weightedSum
.
Parameters
formulas
 Array of Formula or Termweights
 Array of nonnegative integers, or a single nonnegative integer
Returns
Integer
Logic.Solution#ignoreUnknownVariables()
Causes all evaluation by this Solution instance, from now on, to treat variables that aren't part of this Solution as false instead of throwing an error. This includes unrecognized variable names and variables that were created after this Solution was created.
This method cannot be undone. Good style is to call it once when you first get the Solution object, or not at all.
Optimization
Logic Solver can perform basic integer optimization, using a combination of inequalities and incremental solving. The methods in this section are utilities for minimizing or maximizing the value of a weighted sum, which is a type of problem sometimes called pseudoboolean optimization.
To understand how these methods work, remember that if you have one
solution and want another solution to the same problem, a good
technique is to forbid the current solution and then resolve. In a
similar vein, if you have one solution and want another solution that
yields a larger or smaller value for an integer expression, you can
simply express this new constraint as an inequality and resolve. The
final wrinkle is to use solveAssuming
to test out each inequality
before requiring it, so that when the minimum or maximum value is
found, the solver is not put into an unsatisfiable state. The methods
in this section implement this technique for you.
This approach to integer optimization works surprisingly well, even when it takes many iterations to achieve the optimum value of a large cost function. However, depending on the structure of your problem, it may be quite a timeconsuming operation.
Methods
Logic.Solver#minimizeWeightedSum(solution, formulas, weights)
Finds a Solution that minimizes the value of
Logic.weightedSum(formulas, weights)
, and adds a requirement that
this minimum value is obtained (in the sense of calling
Solver#require
on this Solver).
To determine this minimum value, call
solution.getWeightedSum(formulas, weights)
on the returned Solution.
A currently valid Solution must be passed in as a starting point.
This starting Solution must have been obtained by calling solve
or
solveAssuming
on this Solver, and in addition, being "currently
valid" means that no calls to require
or forbid
have been made
since the Solution was produced that conflict with its assignments.
Note that while this method may add constraints to the Solver, the Solver is always in a satisfiable state both before and after this method is called.
Parameters
solution
 Logic.Solution  A currently valid Solution for this Solver.formulas
 Array of Formula or Termweights
 Array of nonnegative integers, or a single nonnegative integer
Returns
Logic.Solution  A valid Solution that achieves the minimum value of
the weighted sum. It may be solution
if no improvement on the original
value of the weighted sum is possible.
Logic.Solver#maximizeWeightedSum(solution, formulas, weights)
Finds a Solution that maximizes the value of
Logic.weightedSum(formulas, weights)
, and adds a requirement that
this maximum value is obtained (in the sense of calling
Solver#require
on this Solver).
To determine this maximum value, call
solution.getWeightedSum(formulas, weights)
on the returned Solution.
A currently valid Solution must be passed in as a starting point.
This starting Solution must have been obtained by calling solve
or
solveAssuming
on this Solver, and in addition, being "currently
valid" means that no calls to require
or forbid
have been made
since the Solution was produced that conflict with its assignments.
Note that while this method may add constraints to the Solver, the Solver is always in a satisfiable state both before and after this method is called.
Parameters
solution
 Logic.Solution  A currently valid Solution for this Solver.formulas
 Array of Formula or Termweights
 Array of nonnegative integers, or a single nonnegative integer
Returns
Logic.Solution  A valid Solution that achieves the maximum value of
the weighted sum. It may be solution
if no improvement on the original
value of the weighted sum is possible.
Bits (integers)
A Bits object represents an Ndigit binary number (nonnegative
integer) as an array of N Formulas. That is, it has a Formula
for the boolean value of each bit. The Formulas are stored in an
array called bits
with the least significant bit first, so bits[0]
is the ones digit, bits[1]
is the twos digit, bits[2]
is the fours
digit, and so on. (Note that this is the opposite order from how we
usually write numbers! It's much more convenient because the index
into the array is always the same as the power of two, with numbers
growing to the right as they gain largervalued digits.)
You usually don't construct a Bits using the constructor, but instead
using Logic.constantBits
, Logic.variableBits
, or an operation on
Formulas such as Logic.sum
. When you create an integer variable
using Logic.variableBits
, you specify the number of bits N, but in
other cases the number of bits is calculated automatically. For
example, Logic.sum()
with no arguments returns a 0length Bits.
Logic.sum('A', 'B')
returns a 2length Bits which is the equivalent
of new Logic.Bits([Logic.xor('A', 'B'), Logic.and('A', 'B')])
.
See Example: Magic Squares for a good example of using Bits.
To avoid confusion with NumTerms, there is no automatic promotion of
integers to Bits. If you want to use a constant like 5, you must
call Logic.constantBits(5)
to get a Bits object.
There is currently no explicit subtraction nor any negative numbers in Logic Solver.
Fields
bits
 Array of Formula or Term  Readonly.
Constructor
new Logic.Bits(formulas)
As previously mentioned, it's more common to create a Bits object
using Logic.constantBits
, Logic.variableBits
, Logic.sum
, or
Logic.weightedSum
than using this constructor.
Parameters
formulas
 Array of Formula or Term  Becomes the value of thebits
property of this Bits. The array is not copied, so don't mutate the original array. Unlike many Logic Solver methods, this constructor does not take a variable number of arguments, but requires exactly one array.
Methods
Logic.isBits(value)
Returns true if value
is a Bits object.
Parameters
value
 Any
Returns
Boolean
Logic.constantBits(wholeNumber)
Creates a constant Bits representing the given number.
For example, Logic.constantBits(4)
is equivalent to
new Logic.Bits([Logic.FALSE, Logic.FALSE, Logic.TRUE])
.
Parameters
wholeNumber
 nonnegative integer
Returns
Bits
Logic.variableBits(baseName, N)
Creates a Bits representing an Ndigit integer variable.
For example, Logic.variableBits('A', 3)
is equivalent to
new Logic.Bits(['A$0', 'A$1', 'A$2'])
.
Parameters
baseName
 StringN
 nonnegative integer
Returns
Bits
Logic.equalBits(bits1, bits2)
Represents a boolean expression that is true when bits1
and bits2
are the same integer.
Parameters
bits1
 Bitsbits2
 Bits
Returns
Formula or Term
Logic.lessThan(bits1, bits2)
Represents a boolean expression that is true when bits1
is less than
bits2
, interpreting each as a nonnegative integer.
Parameters
bits1
 Bitsbits2
 Bits
Returns
Formula or Term
Logic.lessThanOrEqual(bits1, bits2)
Represents a boolean expression that is true when bits1
is less than
or equal to bits2
, interpreting each as a nonnegative integer.
Parameters
bits1
 Bitsbits2
 Bits
Returns
Formula or Term
Logic.greaterThan(bits1, bits2)
Represents a boolean expression that is true when bits1
is greater than
bits2
, interpreting each as a nonnegative integer.
Parameters
bits1
 Bitsbits2
 Bits
Returns
Formula or Term
Logic.greaterThanOrEqual(bits1, bits2)
Represents a boolean expression that is true when bits1
is greater than
or equal to bits2
, interpreting each as a nonnegative integer.
Parameters
bits1
 Bitsbits2
 Bits
Returns
Formula or Term
Logic.sum(operands...)
Represents an integer expression that is the sum of the values of all the operands. Bits are interpreted as integers, and booleans are interpreted as 1 or 0.
As with Formula constructor functions that take a variable number of arguments, the operands may be nested in arrays arbitrarily and arbitrarily deeply.
Parameters
operands...
 Zero or more Formulas, Terms, Bits, or Arrays
Returns
Bits
Logic.weightedSum(formulas, weights)
Represents an integer expression that is a weighted sum of the given Formulas and Terms, after mapping false to 0 and true to 1.
In other words, the sum is:
(formulas[0] * weights[0]) + (formulas[1] * weights[1]) + ...
,
where formulas[0]
is replaced with 0 or 1 based on the boolean value
of that Formula.
weights
may either be an array of nonnegative integers, or a single
nonnegative integer, in which case that weight is used for all formulas.
If weights
is an array, it must have the same length as formulas
.
Parameters
formulas
 Array of Formula or Termweights
 Array of nonnegative integers, or a single nonnegative integer
Returns
Bits
About MiniSat
Solving satisfiability problems ("SATsolving") is notoriously difficult from an algorithmic perspective, but solvers such as MiniSat implement advanced techniques that have come out of years of research. You can read more about MiniSat on its web page at http://minisat.se/.
MiniSat accepts input in "conjunctive normal form," which is a fairly lowlevel representation of a logic problem. Logic Solver's main job is to take arbitrary boolean formulas that you specify, such as "exactly one of A, B, and C is true," and compile them into a list of statements that must all be satisfied  a conjunction of clauses  each of which is a simple disjunction such as: "A or B or C." "Not A, or not B."
Although MiniSat operates on a lowlevel representation of the problem and has no explicit knowledge of its overall structure, it is able to use sophisticated techniques to derive new clauses that are implied by the existing clauses. A naive solver would try assigning values to some of the variables until a conflict occurs, and then backtrack, but not really learn anything from the conflict. Even custom solvers written for a particular problem often work this way. Solvers such as MiniSat, on the other hand, employ ConflictDriven Clause Learning, which means that when they backtrack, they learn new clauses. These new clauses narrow the search space and cause subsequent trials to reach a conflict sooner, until the entire problem is found to be unsatisfiable or a valid assignment is found.
In principle, Logic Solver could be used as a clause generator for other SATsolver backends besides MiniSat, or for a backend consisting of MiniSat compiled to native machine code instead of JavaScript.