Satisfiability modulo theories in the context of "Undecidable problem"

In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula ${\displaystyle x+3=y}$ is satisfiable because it is true when ${\displaystyle x=3}$ and ${\displaystyle y=6}$, while the formula ${\displaystyle x+1=x}$ is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is valid if every assignment of values to its variables makes the formula true. For example, ${\displaystyle x+3=3+x}$ is valid over the integers, but ${\displaystyle x+3=y}$ is not.

Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it relates to the meaning of the symbols, for example, the meaning of ${\displaystyle +}$ in a formula such as ${\displaystyle x+1=x}$. Formally, we define an interpretation (or model) to be an assignment of values to the variables and an assignment of meaning to all other non-logical symbols, and a formula is said to be satisfiable if there is some interpretation which makes it true. While this allows non-standard interpretations of symbols such as ${\displaystyle +}$, one can restrict their meaning by providing additional axioms. The satisfiability modulo theories problem considers satisfiability of a formula with respect to a formal theory, which is a (finite or infinite) set of axioms.

Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables, which is solved by constraint satisfaction methods. CSPs are the subject of research in both artificial intelligence and operations research, since the regularity in their formulation provides a common basis to analyze and solve problems of many seemingly unrelated families. CSPs often exhibit high complexity, requiring a combination of heuristics and combinatorial search methods to be solved in a reasonable time. Constraint programming (CP) is the field of research that specifically focuses on tackling these kinds of problems. Additionally, the Boolean satisfiability problem (SAT), satisfiability modulo theories (SMT), mixed integer programming (MIP) and answer set programming (ASP) are all fields of research focusing on the resolution of particular forms of the constraint satisfaction problem.

Examples of problems that can be modeled as a constraint satisfaction problem include: