Satisfiability Modulo Theory Task
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A Satisfiability Modulo Theory Task is a decision problem that determines whether a given logical formula can be satisfied concerning certain background theories such as real numbers, integers, array data structures, or bit vectors.
- Context:
- It can (typically) extend the Boolean satisfiability problem (SAT) by incorporating various mathematical theories, making it more expressive for handling complex structures.
- It can (often) be expressed as a first-order logic formula and then evaluated within a specific formal theory (mathematical logic), such as Presburger arithmetic or linear real arithmetic.
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- It can range from handling decidable problems to undecidable problems depending on the choice of underlying theories and their combinations.
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- It can be applied in fields such as formal verification, program analysis, and automated theorem proving for reasoning about systems and software correctness.
- It can involve the use of specialized SMT solvers like Z3 Theorem Prover and cvc5 to handle specific combinations of theories.
- It can be used for constraint programming to express and solve problems involving multiple constraints over different types of variables.
- It can be studied in the context of computational complexity to understand the resource requirements for different classes of SMT problems.
- It can be reduced to a constraint satisfaction problem, where the goal is to satisfy constraints defined over a combination of logical and mathematical theories.
- It can be employed in software testing techniques such as white-box testing to generate test cases that meet specific logical conditions.
- It can encounter issues like NP-hardness and undecidability, making efficient solution methods critical for large-scale problems.
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- Example(s):
- an SMT Solver like Z3 that determines the satisfiability of formulas with theories of arrays, bit-vectors, and linear integer arithmetic.
- an Automated Theorem Prover that uses SMT for reasoning about logical properties in programs.
- a Formal Verification System that checks whether a given software program satisfies its specification using SMT-based techniques.
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- Counter-Example(s):
- Boolean Satisfiability Problem (SAT)s, which only consider formulas over Boolean variables and do not account for richer mathematical structures.
- Quantifier-Free Theorys, which may restrict the complexity of the input formulas but simplify the decision process.
- See: Constraint Programming, Mathematical Logic, Decision Problem, Theory (Mathematical Logic), First-Order Logic, Real Numbers, Integers, Data Structure, List (Computing), Array Data Structure, Bit Vector.
References
2024
- (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Satisfiability_modulo_theories Retrieved:2024-9-27.
- In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality (often disallowing quantifiers). SMT solvers are tools that aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing.
Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. Researchers study which theories or subsets of theories lead to a decidable SMT problem and the computational complexity of decidable cases. The resulting decision procedures are often implemented directly in SMT solvers; see, for instance, the decidability of Presburger arithmetic. SMT can be thought of as a constraint satisfaction problem and thus a certain formalized approach to constraint programming.
- In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality (often disallowing quantifiers). SMT solvers are tools that aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing.
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/satisfiability_modulo_theories Retrieved:2017-12-30.
- In computer science and mathematical logic, the satisfiability modulo theories (SMT) problem is a decision problem for logical formulas with respect to combinations of background theories expressed in classical first-order logic with equality. Examples of theories typically used in computer science are the theory of real numbers, the theory of integers, and the theories of various data structures such as lists, arrays, bit vectors and so on. SMT can be thought of as a form of the constraint satisfaction problem and thus a certain formalized approach to constraint programming.