Satisfiability Modulo Theories (SMT) refers to the difficulty of establishing whether a first-order formula is satisfiable about a particular logical theory. Solvers based on SMT are employed as back-end engines in model-checking applications, including bounded, interpolation-based, and predicate-abstraction-based model checking. 

SMT expands the Boolean satisfiability problem (SAT) to include more complicated formulas with real numbers, integers, and data structures like lists, arrays, bit vectors, and strings. The name comes from the fact that these expressions are interpreted within ("modulo"). 

In addition, SMT solvers are pieces of software that try to solve the SMT problem for a valuable subset of inputs. Furthermore, SMT solvers like Z3 and cvc5 have been used as building blocks for various computers science applications, such as automated theorem proving, programme analysis, programme verification, and software testing.

Problem specific

SMT is based on three of the most critical problems in symbolic logic over the past 100 years: the decision problem, the problem of whether a logical theory is complete or not, and complexity theory. Most SMT problems are very hard to figure out how to solve. Combining special-purpose algorithms for each domain in a modular way is a deep and exciting problem, just like finding new algorithms that work well in the context of a combination. Linear programming, based on the theory of linear arithmetic, is a well-known theory that can be used in many ways. For example, we can use linear programming algorithms to check if two linear arithmetic inequalities can be satisfied, but they don't work directly for Boolean combinations. SMT solvers stand out because they can handle these kinds of combinations.

First-order logic is undecidable, and SAT is known to be NP-complete. Building a method that can solve every SMT problem is not viable due to the enormous computational complexity. Therefore, most procedures concentrate on the more practical objective of effectively resolving issues that arise in practice. They operate under the premise that, despite having the potential to be significant, most formulas generated by verification and analysis tools are brief. Only a tiny portion of a procedure is necessary to show satisfiability. The remaining material is filler.

Advancements

Recent advancements in fundamental algorithms, data structures, heuristics, and attention to implementation details have significantly increased the size of problems that We may solve. For example, modern SAT techniques can check formulas containing millions of clauses and variables. Similar advancements have been seen using SMT techniques for the most prevalent theories. The yearly competition for SAT and SMT techniques is crucial in fostering improvement. We briefly explain SMT and the essential technical concepts in this work.

Conclusion

SMT solvers can be used both for verification, proving that a programme is correct, and for synthesis, which is making new programme parts by searching through all possible agendas. In addition, SMT solvers have been used for things other than software verification. For example, they have been used for type inference and theoretic scenarios, such as modelling actors' beliefs in nuclear arms control.

Furthermore, SMT solvers are often used to check computer programmes with the help of computers. For example, a common way to check if all properties can hold is to turn preconditions, postconditions, loop conditions, and assertions into SMT formulas.