Satisfiability

In mathematical logic, satisfiability and validity are elementary concepts of semantics. A formula is "satisfiable" if it is possible to find an interpretation (model) that makes the formula true. A formula is "valid" if all interpretations make the formula true. The opposites of these concepts are unsatisfiability and invalidity, that is, a formula is "unsatisfiable" if none of the interpretations make the formula true, and "invalid" if some such interpretation makes the formula false. These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition.
The four concepts can be raised to apply to whole theories: a theory is satisfiable (valid) if one (all) of the interpretations make(s) each of the axioms of the theory true, and a theory is unsatisfiable (invalid) if all (one) of the interpretations make(s) one of the axioms of the theory false.
It is also possible to consider only interpretations that make all of the axioms of a second theory true. This generalization is commonly called satisfiability modulo theories.
The question whether a sentence in propositional logic is satisfiable is a decidable problem. In general, the question whether sentences in first-order logic are satisfiable is not decidable. In universal algebra and equational theory, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability. Whether a particular theory is decidable or not depends whether the theory is variable-free or on other conditions.
Reduction of validity to satisfiability.
For classical logics, it is generally possible to reexpress the question of the validity of a formula to one involving satisfiability, because of the relationships between the concepts expressed in the above square of opposition. In particular φ is valid if and only if ¬φ is unsatisfiable, which is to say it is not true that ¬φ is satisfiable. Put another way, φ is satisfiable if and only if ¬φ is invalid.
For logics without negation, such as the positive propositional calculus, the questions of validity and satisfiability may be unrelated. In the case of the positive propositional calculus, the satisfiability problem is trivial, as every formula is satisfiable, while the validity problem is co-NP complete.
Propositional satisfiability.
In the case of classical propositional logic, satisfiability is decidable for propositional formulae. In particular, satisfiability is an NP-complete problem, and is one of the most intensively studied problems in computational complexity theory.
Satisfiability in first-order logic.
Satisfiability is undecidable and indeed it isn't even a semidecidable property of formulae in first-order logic (FOL). This fact has to do with the undecidability of the validity problem for FOL. The question of the status of the validity problem was posed firstly by David Hilbert, as the so-called Entscheidungsproblem. The universal validity of a formula is a semi-decidable problem. If satisfiability were also a semi-decidable problem, then the problem of the existence of counter-models would be too (a formula has counter-models iff its negation is satisfiable). So the problem of logical validity would be decidable, which contradicts the Church–Turing theorem, a result stating the negative answer for the Entscheidungsproblem.
Satisfiability in model theory.
In model theory, an atomic formula is satisfiable if there is a collection of elements of a structure that render the formula true. If "A" is a structure, φ is a formula, and "a" is a collection of elements, taken from the structure, that satisfy φ, then it is commonly written that
If φ has no free variables, that is, if φ is an atomic sentence, and it is satisfied by "A", then one writes
In this case, one may also say that "A" is a model for φ, or that φ is "true" in "A". If "T" is a collection of atomic sentences (a theory) satisfied by "A", one writes
Finite satisfiability.
A problem related to satisfiability is that of finite satisfiability, which is the question of determining whether a formula admits a "finite" model that makes it true. For a logic that has the finite model property, the problems of satisfiability and finite satisfiability coincide, as a formula of that logic has a model if and only if it has a finite model. This question is important in the mathematical field of finite model theory.
Nevertheless, finite satisfiability and satisfiability need not coincide in general. For instance, consider the first-order logic formula obtained as the conjunction of the following sentences, where formula_1 and formula_2 are constants:
The resulting formula has the infinite model formula_7, but it can be shown that it has no finite model (starting at the fact formula_8 and following the chain of formula_9 atoms that must exist by the third axiom, the finiteness of a model would require the existence of a loop, which would violate the fourth axiom, whether it loops back on formula_10 or on a different element).
The computational complexity of deciding satisfiability for an input formula in a given logic may differ from that of deciding finite satisfiability; in fact, for some logics, only one of them is decidable.
Numerical constraints.
Numerical constraints often appear in the field of mathematical optimization, where one usually wants to maximize (or minimize) an objective function subject to some constraints. However, leaving aside the objective function, the basic issue of simply deciding whether the constraints are satisfiable can be challenging or undecidable in some settings. The following table summarizes the main cases.
"Table source: Bockmayr and Weispfenning".
For linear constraints, a fuller picture is provided by the following table.
"Table source: Bockmayr and Weispfenning".