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 (x1 ∧ Øx2 ∧ x3). This
formula is clearly satisfiable with (x1 =
1, x2=0, x3= 1). However, if x1, x2 and x3
represent the following relationships
among the real variables y1 and y2:
x1: y1 <0
x2: y1 + y2 < 1
x3: y2 < 0
Then, in fact, there is no assignment
to y1 and y2 for which (x1 = 1, x2=0, x3= 1),
i.e., y1 and y2 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
1 The theoretical
difficulty depends on the specific theories considered. SMT is seeing rapid
progress and initial commercial use in
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/na-ture 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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Sharad Malik ( firstname.lastname@example.org) is a professor in
the Department of electrical engineering at Princeton
university, Princeton, nJ.
Lintao Zhang ( email@example.com) is a researcher at
Microsoft research asia, beijing, China.