Presburger Arithmetic and Its Use in Verification

CHAPTER 7.
PARALLEL EXECUTION OF DECISION PROCEDURES
parallel elimination is based on following rules where formulas are in NNF:
∃ x 1 ...x n . ∨ i
∀ x 1 ...x n . ∧ i
F i ≡ ∨ i
F i ≡ ∧ i
∃ x 1 ...x n .F i (7.1)
∀ x 1 ...x n .F i (7.2)
However, for arbitrary NNF formulas, the degree of parallelism may be limited
because of small disjunctions (or conjunctions). One way to enhance concurrency is
converting formulas into DNF. Normally a DNF formula is in a form of a huge disjunction
of many inner conjunctions and quantifier elimination can be distributed to
inner conjunctions immediately. Because conversion into DNF causes the formula’s
size to grow very fast, we only do DNF conversion once at the outermost level of
quantifiers.
Because there are still some cases where DNF formulas do not expose enough
concurrency, we seek concurrency inside the procedure. First, due to recursive
nature of Presburger formulas, these formulas are represented by tree data structures.
Certainly we can manipulate these formulas by doing operations on tree in
aparallelmanner,forexample,doingparallelevaluationoftreebranches.Second,
certain parts in quantifier elimination could be done in parallel. As can be seen
from Figure 7.1, Get Coefficients, Get A-Terms and Get B-Terms have no order
of execution. We are able to create three Tasks to run them concurrently. Similarly
Least Satisfying Assignment and Small Satisfying Assignments could
be calculated in parallel. The figure preserves relative ratios between sizes of different
tasks where Small Satisfying Assignments is the largest task (it consists of
|B| (or |A|) timessubstitutionofavariablebyatermintheformula)andEliminate
Variable is a very lightweight task where we assemble different results. Assuming
that the number of B-Terms are always smaller than that of A-Terms, we have
aroughestimationofparallelismbasedontheDAG model of multithreading as
follows:
•AssumethatEliminate Variable is an insignificant task and omit it.
•EstimateeachtaskbythenumberoftraversalsthroughthewholePresburger
formula.
• Work = 1 + 1 + 1 + 1 + 1 + |B| = |B| + 5
• Span = 1 + 1 + 1 = 3
• Parallelism Factor =(|B| + 5)/3
Roughly speaking, Parallelism Factor is bounded by (|B| + 5)/3, sowecanhardly
achieve a good speedup if the number of B-Terms (or A-Terms) is too small. Actually
the number of terms is quite small due to symbolic representation of big
disjuncts and the choice of the smallest number of terms in projection which leads
to limited concurrency in each elimination step. One interesting thing is after each
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