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# Boolean Satisfiability: From Theoretical Hardness to Practical Success

There are many practical situations where we need to satisfy several potentially conflicting constraints. Simple examples of this abound in daily life, for example, determining a schedule for a series of games that resolves the availability of players and venues, or finding a seating assignment at dinner consistent with various rules the host would like to impose. This also applies to applications in computing, for example, ensuring that a hardware/software system functions correctly with its overall behavior constrained by the behavior of its components and their composition, or finding a plan for a robot to reach a goal that is consistent with the moves it can make at any step. While the applications may seem varied, at the core they all have variables whose values we need to determine (for example, the person sitting at a given seat at dinner) and constraints that these variables must satisfy (for example, the host's seating rules).

In its simplest form, the variables are *Boolean* valued
(true/false, often represented using 1/0) and *propositional
logic formulas* can be used to express the constraints on the
variables.^{15} In propositional logic
the *operators* AND, OR, and NOT (represented by the symbols
, , and ¬ respectively) are used to construct
formulas with variables. If *x* is a Boolean variable and
*f, f _{1}* and

*x*is a formula that evaluates to 1 when*x*is 1, and evaluates to 0 when*x*is 0- ¬
*f*is a formula that evaluates to 1 when*f*evaluates to 0, and 0 when*f*evaluates to 1 *f*_{1}*f*_{2}is a formula that evaluates to 1 when*f*and_{1}*f*both evaluate to 1, and evaluates to 0 if either_{2}*f*or_{1}*f*_{2}evaluate to 0*f*_{1}*f*_{2}is a formula that evaluates to 0 when*f*_{1}and*f*_{2}both evaluate to 0, and evaluates to 1 if either*f*_{1}or*f*_{2}evaluate to 1

(*x*_{1} ¬*x*_{2})
*x*_{3} is an example formula constructed using
these rules. Given a valuation of the variables, these rules can
be used to determine the valuation of the formula. For example:
when (*x*_{1} = 0, *x*_{2} = 0,
*x*_{3} = 1), this formula evaluates to 1 and when
(*x*_{3} = 0), this formula evaluates to 0,
regardless of the values of *x*_{1} and
*x*_{2}. This example also illustrates how the
operators in the formula provide constraints on the variables. In
this example, for this formula to be true (evaluate to 1),
*x*_{3} must be 1.

A satisfying assignment for a formula is an assignment of the
variables such that the formula evaluates to 1. It simultaneously
satisfies the constraints imposed by all the operators in the
formula. Such an assignment may not always exist. For example the
formula (¬ *x*_{1}
¬*x*_{2}) (*x*_{1}
*x*_{2}) (¬*x*_{1}
*x*_{2}) (*x*_{1}
¬*x*_{2}) cannot be satisfied by any of the four
possible assignments 0/0, 0/1, 1/0, 1/1 to *x*_{1}
and *x*_{2}. In this case the problem is
overconstrained. This leads us to a definition of the Boolean
Satisfiability problem (also referred to as Propositional
Satisfiability or just Satisfiability, and abbreviated as SAT):
*Given a formula, find a satisfying assignment or prove that
none exists.* This is the constructive version of the problem,
and one used in practice. A simpler decision version, often used
on the theoretical side, just needs to determine if there exists
a satisfying assignment for the formula (a yes/no answer). It is
easy to see that a solver for the decision version of the problem
can easily be used to construct a solution to the constructive
version, by solving a series of *n* decision problems where
*n* is the number of variables in a formula.

Many constraint satisfaction problems dealing with non-Boolean
variables can be relatively easily translated to SAT. For
example, consider an instance of the classic graph coloring
problem where an *n*-vertex graph needs to be checked for
4-colorability, that is, determining whether each vertex can be
colored using one of four possible colors such that no two
adjacent vertices have the same color. In this case, the
variables are the colors {*c*_{0},
*c*_{1}, *c*_{2}, ...,
*c*_{n}_{-1}} for the *n*
vertices, and the constraints are that adjacent vertices must
have different colors. For this problem the variables are not
Boolean and the constraints are not directly expressed with the
operators {, , ¬}. However, the variables and
constraints can be encoded into a propositional formula as
follows. Two Boolean variables,
*c*_{i}_{0},
*c*_{i}_{1}, are used in a two-bit
encoding of the four possible values of the color for vertex
*i.* Let *i* and *j* be adjacent vertices. The
constraint *c _{i}* ≠

Encodings have been useful in translating problems from a wide
range of domains to SAT, for example, scheduling basketball
games,^{40} planning in artificial
intelligence,^{20} validating software
models,^{17} routing field programmable
gate arrays,^{28} and synthesizing
consistent network configurations.^{29}
This makes SAT solvers powerful engines for solving constraint
satisfaction problems. However, SAT solvers are not always the
best engines—there are many cases where specialized
techniques work better for various constraint problems, including
graph coloring (for example, Johnson et
al.^{19}). Nonetheless, it is often much
easier and more efficient to use off-the-shelf SAT solvers than
developing specialized tools from scratch.

One of the more prominent practical applications of SAT has
been in the design and verification of digital circuits. Here,
the translation to a formula is very straightforward. The
functionality of digital circuits can be expressed as
compositions of basic logic gates. A logic gate has Boolean input
signals and produces Boolean output signals. The output of a gate
can be used as an input to another gate. The functions of the
basic logic gates are in direct correspondence to the operators
{, , ¬}. Thus various properties regarding the
functionality of logic circuits can be easily translated to
formulas. For example, checking that the values of two signals
*s*_{1} and *s*_{2} in the logic
circuit are always the same is equivalent to checking that their
corresponding formulas *f*_{1} and
*f*_{2} never differ, that is, (*f*_{1}
¬*f*_{2}) (¬*f*_{1}
*f*_{2}) is not satisfiable.

This technique can be extended to handle more complex
properties involving values on sequences of signals, for example,
a request is eventually acknowledged. For such problems,
techniques that deal with temporal properties of the system, such
as model checking, are used.^{6} Modern
SAT solvers have also been successfully applied for such
tasks.^{3, 26} One of
the main difficulties of applying SAT in checking such properties
is to find a way to express the concept of "*eventually.*"
In theory, there is no tractable way to express this using
propositional logic. However, in practice it is often good enough
to just set a *bound* on the number of steps. For example,
instead of asking whether a response to a request will eventually
occur, we ask whether there will be a response within *k*
clock cycles, where *k* is a small fixed number. Similar
techniques have also been used in AI
planning,^{20} for example, instead of
determining if a goal is reachable, we ask whether we can reach
the goal in *k* steps. This unrolling technique has been
widely adopted in practice, since we often only care about the
behavior of the system within a small bounded number of
steps.

The decision version of SAT, that is, determining if a given
formula has a satisfying solution, belongs to the class of
problems known as
*NP-complete.*^{8,12}
An instance of any one of these problems can be relatively easily
transformed into an instance of another. For example, both graph
coloring and SAT are NP-complete, and earlier we described how to
transform a graph coloring instance to a SAT instance.

All currently known solutions for NP-Complete problems, in the
worst case, require runtime that grows exponentially with the
size of the instance. Whether there exist subexponential
solutions to NP-Complete problems is arguably the most famous
open question in computer science.^{a}
Although there is no definitive conclusion, the answer is widely
believed to be in the negative. This exponential growth in time
complexity indicates the difficulty of scaling solutions to
larger instances.

However, an important part of this characterization is "worst
case." This holds out some hope for the "typical case," and more
importantly the typical case that might arise in specific problem
domains. *In fact, it is exactly the non-adversarial nature of
practical instances that is exploited by SAT solvers.*

Most SAT solvers work with a restricted representation of
formulas in conjunctive normal form (CNF), defined as follows. A
literal *l* is either a positive or a negative occurrence of
a variable (for example, *x* or ¬*x*). A clause,
*c*, is the OR of a set of literals, such as
(*l*_{1} *l*_{2}
*l*_{3} ... *l _{n}*). A CNF
formula is the AND of a set of clauses, such as
(

The restriction to CNF is an active choice made by SAT solvers
as it enables their underlying algorithms. Further, this is not a
limitation in terms of the formulas that can be handled. Indeed,
with the addition of new auxiliary variables; it is easy to
translate any formula into CNF with only a linear increase in
size.^{36} However, this representation
is not used exclusively and there has been recent success with
solvers for non-clausal representations (for example,
NFLSAT^{18}).

Most practically successful SAT solvers are based on an
approach called systematic search. Figure 2
depicts the search space of a formula. The search space is a tree
with each vertex representing a variable and the out edges
representing the two decision choices for this variable. For a
formula with *n* variables, there are 2* ^{n}*
leaves in the tree. Each path from the root to a leaf corresponds
to a possible assignment to the

The NP-completeness of the problem indicates that we will
likely need to visit an exponential number of vertices in the
worst case. The only hope for a practical solver is that by being
smart in the search, almost all of the tree can be pruned away
and only a minuscule fraction is actually visited in most cases.
For an instance with a million variables, which is considered
within the reach of modern solvers, the tree has
2^{106} leaves, and in reasonable computation time
(about a day), we may be able to visit a billion (about
2^{30}) vertices as part of the
search—a numerically insignificant fraction of the tree
size!

Most search-based SAT solvers are based on the so called
*DPLL approach* proposed by Davis, Logemann, and Loveland in
a seminal *Communications* paper published in
1962.^{9} (This research builds on the
work by Davis and Putnam^{10} and thus
Putnam is often given shared credit for it.). Given a CNF
formula, the DPLL algorithm first heuristically chooses an
unassigned variable and assigns it a value: either 1 or 0. This
is called branching or the *decision* step. The solver then
tries to deduce the consequences of the variable assignment using
deduction rules. The most widely used deduction rule is the
*unitclause* rule, which states that if a clause in the
formula has all but one of its literals assigned 0 and the
remaining one is unassigned, then the only way for the clause to
evaluate to true, and thus the formula to evaluate to true, is
for this last unassigned literal to be assigned to 1. Such
clauses are called unit clauses and the forced assignments are
called implications. This rule is applied iteratively until no
unit clause exists. Note that this deduction is enabled by the
CNF representation and is the main reason for SAT solvers
preferring this form.

If at some point there is a clause in the formula with all of
its literals evaluating to 0, then the formula cannot be true
under the current assignment. This is called a *conflict*
and this clause is referred to as a conflicting clause. A
conflict indicates that some of the earlier decision choices
cannot lead to a satisfying solution and the solver has to
backtrack and try a different branch value. It accomplishes this
by finding the most recent decision variable for which both
branches have not been taken, flip its value, undo all variable
assignments after that decision, and run the deduction process
again. Otherwise, if no such conflicting clause exists, the
solver continues by branching on another unassigned variable. The
search stops either when all variables are assigned a value, in
which case we have hit a green leaf and the formula is
satisfiable, or when a conflicting clause exists when all
branches have been explored, in which case the formula is
unsatisfiable.

Consider the application of the algorithm to the formula shown
in Figure 2. At the beginning the solver branches
on variable *x*_{1} with value 1. After branching,
the first clause becomes unit and the remaining free literal
¬*x*_{2} is implied to 1, which means
*x*_{2} must be 0. Now the second clause becomes
unit and ¬*x*_{3} is implied to 1. Then
¬*x*_{4} is implied to 1 due to the third
clause. At this point the formula is satisfied, and the
satisfying assignment corresponds to the 8th leaf node from the
left in the search tree. (This path is marked in bold in the
figure.) As we can see, by applying the unit-clause rule, a
single branching leads directly to the satisfying solution.

Many significant improvements in the basic DPLL algorithm have
been proposed over the years. In particular, a technique called
conflict-driven learning and non-chronological
backtracking^{2, 24} has
greatly enhanced the power of DPLL SAT solvers on problem
instances arising from real applications, and has become a key
element of modern SAT solvers. The technique is illustrated in
Figure 3. The column on the left lists the
clauses in the example formula. The colors of the literals show
the current assignments during the search (red representing 0,
green 1, and black representing unassigned). The middle graph
shows the branching and implications at the current point in the
search. At each vertex the branching assignment is shown in blue
and the implications in gray. The first branching is on
*x*_{1}, and it implies *x*_{4}=1
(because of the first clause), the second branching is on
*x*_{3}, and it implies *x*_{8}=0 and
*x*_{12}=1 and so on. The right graph shows the
implication relationships between variables. For example,
*x*_{4}=1 is implied because of
*x*_{1}=0, so there is a directed edge from node
*x*_{1}=0 to node *x*_{4}=1.
*x*_{8}=0 is implied because of both
*x*_{1}=0 and *x*_{3}=1 (the red
literals in the second clause), therefore, these nodes have edges
leading to *x*_{8}=0.

After branching on *x*_{7} and implying
*x*_{9}=1 because of the 5th clause, we find that
the 6th clause becomes a conflicting clause and the solver has to
backtrack. Instead of flipping the last decision variable
*x*_{7} and trying *x*_{7}=0, we can
learn some information from the conflict. From the implication
graph, we see that there is a conflict because
*x*_{9} is implied to be both 1 and 0. If we
consider a cut (shown as the orange line) separating the
conflicting implications from the branching decisions, we know
that once the assignments corresponding to the cut edges are
made, we will end up with a conflict, since no further decisions
are made. Thus, the edges that cross the cut are, in some sense,
responsible for the conflict. In the example,
*x*_{3}, *x*_{7}, and
*x*_{8} have edges cross the cut, thus the
combination of *x*_{3}=1, *x*_{7}=1,
and *x*_{8}=0 results in the conflict. We can learn
from this and ensure that this assignment combination is not
tried in the future. This is accomplished by recording the
condition (¬*x*_{3}
¬*x*_{7} *x*_{8}). This
clause, referred to as a learned clause, can be added to the
formula. While it is redundant in the sense that it is implied by
the formula, it is nonetheless useful as it prevents search from
ever making the assignment (*x*_{3}=1,
*x*_{7}=1, *x*_{8}=0) again.

Further, because of this learned clause, *x*_{7}
= 1 is now implied after the second decision, and we can
backtrack to this earlier decision level as the choice of
*x*_{2} = 0 is irrelevant to the current conflict.
Since such backtracking skips branches, it is called
non-chronological backtracking and helps prune away unsatisfiable
parts of the search space.

Recent work has exposed several significant areas of
improvement now integral to modern SAT
solvers.^{22} The first deals with
efficient implementation of the unit-clause rule using a
technique called two-literal watching. The second area relates to
improvements in the branching step by focusing on exhausting
local sub-spaces before moving to new spaces. This is
accomplished by placing increased emphasis on variables present
in recently added conflict clauses. Another commonly used
technique is random restart,^{13} which
periodically restarts the search while retaining the learned
clauses from the current search to avoid being stuck in a search
sub-space for too long. Other recent directions include formula
preprocessing for clause and variable
elimination,^{11} considering algorithm
portfolios that use empirical hardness models to choose among
their constituent solvers on a per-instance
basis^{39} and using learning techniques
to adjust parameters of heuristics.^{16}
With the advent of multicore processing, there is emerging
interest in efficient multi-core implementations of parallel SAT
solvers.^{14}

The original Davis Putnam
algorithm^{10} based on resolution is
often regarded as the first algorithm for SAT and has great
theoretical and historical significance. However, this algorithm
suffers from a space growth problem that makes it impractical.
Reduced Ordered Binary Decision Diagrams
(ROBDDs)^{5} are a canonical
representation of logic functions, that is, each function has a
unique representation for a fixed variable ordering. Thus, ROBDDS
can be used directly for SAT. However, ROBDDs also face space
limitations with increasing instance size. Stålmarck's
algorithm^{35} uses breadth-first search
instead of depth-first search as in DPLL, and has been shown to
be practically useful. Its performance relative to DPLL based
solvers is unclear as public versions of efficient
implementations of Stålmarck's algorithm are not available
due to its proprietary nature.

When represented in CNF, SAT can be regarded as a discrete
optimization problem with the objective to maximize the number of
satisfied clauses. If this max value is equal to the total number
of clauses, then the instance is satisfiable. Many discrete
optimization techniques have been explored in the SAT context,
including simulated annealing,^{33} tabu
search,^{25} neural
networks,^{34} and genetic
algorithms.^{23}

A variation of the optimization approach, first proposed in
the early 1990s, solves SAT using local search (for example,
GSAT^{31}). The algorithm first randomly
selects a value for each variable, and calculates how many
clauses are satisfied. If not all clauses are satisfied, it
repeatedly flips the value of a variable to increase the number
of the clauses satisfied. If no such variable is available, it
accepts a decrease in the objective function by either flipping a
random variable, or restarting from a fresh set of variable
assignments. This is accelerated further, by confining the flips
to literals in clauses not satisfied by the current
assignment.^{30} This simple algorithm,
when carefully implemented, is surprisingly effective on certain
classes of SAT instances. Unfortunately, this algorithm is
incomplete in the sense that while it may be able to find an
assignment for a satisfiable SAT instance, it cannot prove an
instance to be unsatisfiable. More recently, incomplete solvers
based on a technique called survey
propagation^{4} have been found to be
very effective for certain classes of SAT instances and have
attracted much attention in the theory community.

It is important to note the role of practical benchmarks in
the development of modern SAT solvers. These benchmarks are
critical in tuning the solvers to various classes of practical
instances (that is, instances generated from real-world
applications). While we do not have deep insight into how these
solvers exploit the special structure found in these instances,
we do know that the structure is critical in our ability to
tackle them. (There exists some recent work that provides initial
insights into the effect of structure on DPLL
search.^{37,38})
Experimental research in SAT solvers has been enabled in large
part by benchmarks put forward collectively by the research
community, and the challenge in the form of a SAT solver
competition that is held regularly with the International
Conference on Theory and Applications of Satisfiability Testing
(SAT).^{b} The community has also
benefited from the SATLive portal, which has provided widespread
dissemination of links to SAT articles and
software.^{c}

One of the more prominent practical applications of SAT has been in the design and verification of digital circuits. The functionality of digital circuits can be expressed as compositions of basic logic gates.

Figure 4 provides some data on the
improvements in SAT solvers at the SAT Competition in recent
years.^{d} It plots the relative solving
times for a set of solvers developed over the last 10 years. This
includes solvers that placed first in the industrial benchmarks
category of the SAT competitions. The solvers were run on a set
of benchmarks from hardware and software verification (not used
in the competitions).^{32} This is
normalized to the best solver in the 2007 competition (RSAT with
the SatElite preprocessor). The slow-down of the Grasp solver is
a lower bound, since it could not complete some of the benchmarks
in the 10,000-second time limit. While this study is limited to a
specific set of benchmarks, it is indicative of the progress in
SAT solvers since 2000.

SAT solvers are maturing to the point that developers are
using them in a range of application domains, much like
mathematical programming tools or linear equation solvers. Early
use of SAT was seen in planning in artificial intelligence with
practical use in space exploration.^{27}
Recent increases in the capacity of commercial solvers has
enabled widespread use in the electronic design automation (EDA)
industry as the reasoning engine behind verification and testing
tools such as automatic test pattern
generators,^{21} equivalence checkers,
and property checkers. SAT-based bounded model checkers have been
used in industrial microprocessor
verification.^{7} More recently, SAT has
also been used in tools for software verification and debugging,
for example, industrial verification of device drivers using
SAT-based model checking,^{e} as well as
SAT-based static analysis.^{f} Outside
of verification and testing, SAT techniques have also been
applied in configuration management such as resolving software
package dependencies.^{g}

The success with SAT solvers has emboldened researchers to consider problems related to, but more difficult than SAT. The most promising of these is Satisfiability Modulo Theories (SMT) that has received significant attention in recent years.

In SAT, the variables are assumed to be constrained only by
the clauses in the formula. SMT extends SAT by considering the
case when the variables may be connected by one or more
underlying theories. For example, consider the formula
(*x*_{1} ¬*x*_{2}
*x*_{3}). This formula is clearly satisfiable with
(*x*_{1} = 1, *x*_{2}=0,
*x*_{3}=1). However, if *x*_{1},
*x*_{2} and *x*_{3} represent the
following relationships among the real variables
*y*_{1} and *y*_{2}:

Then, in fact, there is no assignment to *y*_{1}
and *y*_{2} for which (*x*_{1} = 1,
*x*_{2}=0, *x*_{3}=1), i.e.,
*y*_{1} and *y*_{2} cannot be both
negative and their sum at least one. Thus, the original formula
is unsatisfiable given this underlying relationship. In this
example, the specific theory used to determine the validity of a
satisfying assignment is Linear Real Arithmetic. Emerging SMT
solvers can incorporate reasoning for a range of theories such as
Linear Integer Arithmetic, Difference Logic, Arrays, Lists,
Uninterpreted Functions and many others, including their
combinations.^{1} The theoretical
difficulty depends on the specific theories considered. SMT is
seeing rapid progress and initial commercial use in software
verification.

The success with SAT has led to its widespread commercial use in certain domains such as design and verification of hardware and software systems. There is even a sense in parts of the computer science community that this problem has been successfully tamed in practice. This is probably too optimistic a view. There are still enough instances that are difficult for current solvers, and it is unclear if they will be able to handle the change in scale/nature of instances from yet unseen domains. However, there is definitely a sense of confidence that we will be able to continue to strengthen our solvers.

Given its theoretical hardness, the practical success of SAT has come as a surprise to many in the computer science community. The combination of strong practical drivers and open competition in this experimental research effort created enough momentum to overcome the pessimism based on theory. Can we take these lessons to other problems and domains?

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b. http://www.satcompetition.org//

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DOI: http://doi.acm.org/10.1145/1536616.1536637

Figure 1. Encoding of graph coloring.

Figure 2. Search space of a formula.

Figure 3. Conflict-driven learning and non-chronological backtracking.

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