Satisfiability and validity
Encyclopedia
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) each 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 or not a particular theory is decidable or not depends whether or not the theory is variable-free or on other conditions.
s, 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
.
problem, and is one of the most intensively studied problems in computational complexity theory
.
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 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 thesis.
, 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 variables, that is, if φ is a atomic sentence
, and it is satsified 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
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, satisfiability and validity
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
are elementary concepts of semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
. A formula is satisfiable if it is possible to find an interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
(model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
) 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
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's square of opposition
Square of opposition
In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
.
The four concepts can be raised to apply to whole theories
Theory (mathematical logic)
In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context. An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom...
: 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) each 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
Satisfiability Modulo Theories
In computer science, the Satisfiability Modulo Theories problem is a decision problem for logical formulas with respect to combinations of background theories expressed in classical first-order logic with equality...
.
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
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
and equational theory, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability. Whether or not a particular theory is decidable or not depends whether or not the theory is variable-free or on other conditions.
Reduction of validity to satisfiability
For classical logicClassical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
s, 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
Co-NP-complete
In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that they are the ones most likely not to be in P...
.
Propositional satisfiability
In the case of classical propositional logic, satisfiability is decidable for propositional formulae. In particular, satisfiability is an NP-completeNP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
problem, and is one of the most intensively studied problems in computational complexity theory
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
.
Satisfiability in first-order logic
Satisfiability is undecidableUndecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer....
and indeed it isn't even a semidecidable property of formulae in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
(FOL). This fact has to do with the undecidability of the validity problem for FOL. 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 thesis.
Satisfiability in model theory
In model theoryModel theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, an atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
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
- A ⊧ φ [a]
If φ has no variables, that is, if φ is a atomic sentence
Atomic sentence
In logic, an atomic sentence is a type of declarative sentence which is either true or false and which cannot be broken down into other simpler sentences...
, and it is satsified by A, then one writes
- A ⊧ φ
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
- A ⊧ T
See also
- 2-satisfiability2-satisfiabilityIn computer science, 2-satisfiability is the problem of determining whether a collection of two-valued variables with constraints on pairs of variables can be assigned values satisfying all the constraints...
- Boolean satisfiability problemBoolean satisfiability problemIn computer science, satisfiability is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE...
- Karp's 21 NP-complete problemsKarp's 21 NP-complete problemsOne of the most important results in computational complexity theory was Stephen Cook's 1971 demonstration of the first NP-complete problem, the boolean satisfiability problem...
- ValidityValidityIn logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
- Constraint satisfactionConstraint satisfactionIn artificial intelligence and operations research, constraint satisfaction is the process of finding a solution to a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a vector of variables that satisfies all constraints.The techniques used in...