Satisfiability modulo theoriesIn 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).
Existential quantificationIn predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.
Modular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00.
Model checkingIn computer science, model checking or property checking is a method for checking whether a finite-state model of a system meets a given specification (also known as correctness). This is typically associated with hardware or software systems, where the specification contains liveness requirements (such as avoidance of livelock) as well as safety requirements (such as avoidance of states representing a system crash). In order to solve such a problem algorithmically, both the model of the system and its specification are formulated in some precise mathematical language.
Boolean satisfiability problemIn logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable.
Static program analysisIn computer science, static program analysis (or static analysis) is the analysis of computer programs performed without executing them, in contrast with dynamic program analysis, which is performed on programs during their execution. The term is usually applied to analysis performed by an automated tool, with human analysis typically being called "program understanding", program comprehension, or code review. In the last of these, software inspection and software walkthroughs are also used.
Descriptive complexity theoryDescriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic.
NegationIn logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", standing for " is not true", written , or . It is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa).
Program synthesisIn computer science, program synthesis is the task to construct a program that provably satisfies a given high-level formal specification. In contrast to program verification, the program is to be constructed rather than given; however, both fields make use of formal proof techniques, and both comprise approaches of different degrees of automatization. In contrast to automatic programming techniques, specifications in program synthesis are usually non-algorithmic statements in an appropriate logical calculus.
Alternating Turing machineIn computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer and independently by Kozen in 1976, with a joint journal publication in 1981. The definition of NP uses the existential mode of computation: if any choice leads to an accepting state, then the whole computation accepts.
