Building Bridges between Symbolic Computation and Satisfiability Checking

Abstract

The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. For first-order arithmetic theories, in the early 20th century some novel solutions in form of decision procedures were developed in the area of mathematical logic. With the advent of powerful computer architectures, a new research line started to develop practically feasible implementations of such decision procedures. Since then, symbolic computation has grown to an extremely successful scientific area, supporting all kinds of scientific computing by efficient computer algebra systems.Independently, around 1960 a new technology called SAT solving started its career. Restricted to propositional logic, SAT solvers showed to be very efficient when employed by formal methods for verification. It did not take long till the power of SAT solving for Boolean problems had been extended to cover also different theories. Nowadays, fast SAT-modulo-theories (SMT) solvers are available also for arithmetic problems.Due to their different roots, symbolic computation and SMT solving tackle the satisfiability problem differently. We discuss differences and similarities in their approaches, highlight potentials of combining their strengths, and discuss the challenges that come with this task.

A nonexistence certificate for projective planes of order ten with weight 15 codewords

“…In particular, an off-the-shelf version of MapleSAT completed the search in 6.3 minutes, while a programmatic version of MapleSAT augmented with our CAS symmetry breaking method completed the search in 6.8 seconds. This demonstrates the utility of the SAT+CAS method and provides further evidence (as originally argued by [1] and independently by [53]) for the power of the method.…”

What Does “Without Loss of Generality” Mean, and How Do We Detect It

“…Table 2 CAD of R n for numerators of (4)- (7) Chen and Moreno Maza [3] England et al [5] The really surprising effect was the difference between (6) and (5). As far as the author could tell, the code was still projecting when interrupted after 2 1 2 h: at least it had produced no warnings about orientation. This needs further investigation.…”

“…At its most basic, a symmetry is some transformation of an object that leaves the object (or some aspect of the object) unchanged. [11] That quotation comes from a major survey of symmetry in purely Boolean satisfiability problems, but our setting is "Satisfiability Modulo Theories" over the real numbers, and the desire to enhance this with techniques from Computer Algebra, notably Cylindrical Algebraic Decomposition: see [1,2]. Hence we need to worry about symmetry in the underlying theory as well as in the Boolean formulation, and ask what alignment there is between symmetry in the underlying theory and in the Boolean satisfiability problem that encodes the problem.…”

A Survey of Symbolic Execution Techniques

“…Consider the code shown in Figure 5. The write operation at line 4 affects either a[0] or a [1], depending on the unknown value of array index i. State forking creates two states after executing the memory assignment to explicitly consider both possible scenarios ( Figure 6).…”