Presburger Arithmetic and Its Use in Verification

CHAPTER 1.
INTRODUCTION
functional paradigm in the context of parallel programming. Functional programming
has its clear advantages of supporting parallel computing. First, functional
programming relies on data immutability which guarantees code execution without
side effects; therefore, different parts of algorithms could be parallelized without
introducing any synchronization construct. Second, the declarative way of programming
enables developers to describe what problems are rather than how to
solve them and consequently make them easier to break up and parallelize. Third,
functional constructs such as high-order functions and lambda expressions provide
convenient tools for clearly structuring the code, which eases the pain of prototyping
parallel programs. F# is chosen as the functional programming language for
development. Beside other advantages of a functional programming language, its
well-supported .NET framework provides rich libraries for developing applications
and efficient constructs for parallelism.
Later we review the idiom of functional paradigm and parallel execution along
with decision procedures for Presburger Arithmetic (PA). These algorithms are difficult
case studies of tool support; Presburger formulas are known to be decidable
but their decision procedures are doubly exponential lower bound and triply exponential
upper bound [23]. However, instances of PA keep appearing in compiler
optimization and model checking problems, which raises the need for practically
fast implementation of PA decision procedure. Some Presburger fragments are being
used in connection with a model checker for Duration Calculus (DC) [10]. For
example, power usage of nodes on a Wireless Sensor Network (WSN) is expressed
in DC and later converted into a Presburger fragment. To be able to deduce conclusions
about power usage, the Presburger formula which may appear to have rather
big size has to be decided. Therefore, we perform experiments with parallelism and
PA decision procedures using F# and .NET framework. Hopefully, these experiments
can help us to get closer to the goal of efficient tool support for PA.
1.2 Presburger Arithmetic and problems of parallel
decision procedures
Decision procedures for PA exist but they are quite expensive for practical usage
[28]. There are various attempts to optimize those decision procedures in many
aspects. However, those efforts only help to reduce memory usage and provide
fast response for a certain type of formulas; no attempt on employing extra CPU
power for PA algorithms is found in the academia. Although lack of reference for
related work on the problem brings us a new challenge, we enlarge the investigation
to parallel execution of decision procedures in general; hopefully understanding of
their approaches might be helpful. As it turns out, parallelization of SAT solvers
is a rather unexplored topic. Two main approaches are mainly used for parallel
SAT solving. The first one is Search Space Splitting where search space is broken
into independent parts and subproblems are solved in parallel. Typically in this
approach, if one thread completes its work early, it will be assigned other tasks by
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