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Matching Logic
Matching logic is a family of formal systems that were created mainly to specify and reason about computer programs and their correctness. Compared to classical logics such as first-order logic, matching logic's formulas, called ''patterns'', are interpreted as, not elements, but power sets of the underlying carrier set(s), with the intuition that a pattern is matched by the set of elements that "match" it''. ''This way, matching logic is said to admit a semantics based on pattern matching. Matching logic was initially coined by Grigore Rosu and finalized with Xiaohong Chen in 2019. Matching logic is the logical foundation of the K framework. History Early development The term "matching logic" was coined in 2009. and has been used to refer to a couple of formal systems since then. In the early days, matching logic was represented in the literature as a formal system to specify and reason about computer programs' configurations. Together with a set of rules, it was used to spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Systems
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Concepts A formal system has the following: * Formal language, which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules). * Deductive system, deductive apparatus, or proof system, which has rules of inference that take axioms and infers theorems, both of which are part of the formal language. A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigore Rosu
Grigore, the equivalent of Gregory, is a Romanian-language first name. It may refer to: *Grigore Alexandrescu (1810–1885), Romanian poet and translator *Grigore Antipa (1866–1944), Romanian Darwinist biologist, ichthyologist, ecologist, oceanologist *Grigore Băjenaru (1907–1986), Romanian writer *Grigore Bălan (1896–1944), Romanian Brigadier General during World War II *Grigore Vasiliu Birlic (1905–1970), Romanian actor *Grigore Brișcu (1984–1965), Romanian engineer and inventor *Grigore Cobălcescu (1831–1892), founder of Romanian geology and paleontology * Grigore Constantinescu (1875–1932), priest and journalist from Romania *Grigore Cugler (1903–1972), Romanian avant-garde short story writer, poet, and humorist *Grigore Eremei (b. 1935), Moldovan politician, final First secretary of the Communist Party of Moldavia *Grigore Gafencu (1892–1957), Romanian politician, diplomat and journalist *Grigore Alexandru Ghica (1803 or 1807–1857), Prince of Moldavia *Gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by A^c (or ), is the set of Element (mathematics), elements not in . When all elements in the Universe (set theory), universe, i.e. all elements under consideration, are considered to be Element (mathematics), members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by A \operatorname\Delta B (alternatively, A \operatorname\vartriangle B), A \oplus B, or A \ominus B. It can be viewed as a form of addition modulo 2. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. Properties The symmetric difference is equivalent to the union of both relative complements, that is: :A\, \Delta\,B = \left(A \setminus B\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operational Semantics
Operational semantics is a category of formal programming language semantics in which certain desired properties of a program, such as correctness, safety or security, are verified by constructing proofs from logical statements about its execution and procedures, rather than by attaching mathematical meanings to its terms (denotational semantics). Operational semantics are classified in two categories: structural operational semantics (or small-step semantics) formally describe how the ''individual steps'' of a computation take place in a computer-based system; by opposition natural semantics (or big-step semantics) describe how the ''overall results'' of the executions are obtained. Other approaches to providing a formal semantics of programming languages include axiomatic semantics and denotational semantics. The operational semantics for a programming language describes how a valid program is interpreted as sequences of computational steps. These sequences then ''are'' the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Verification
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods. It represents an important dimension of analysis and verification in electronic design automation and is one approach to software verification. The use of formal verification enables the highest Evaluation Assurance Level ( EAL7) in the framework of common criteria for computer security certification. Formal verification can be helpful in proving the correctness of systems such as: cryptographic protocols, combinational circuits, digital circuits with internal memory, and software expressed as source code in a programming language. Prominent examples of verified software systems include the CompCert verified C compiler and the seL ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoare Logic
Hoare logic (also known as Floyd–Hoare logic or Hoare rules) is a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in 1969 by the British computer scientist and logician Tony Hoare, and subsequently refined by Hoare and other researchers. The original ideas were seeded by the work of Robert W. Floyd, who had published a similar system for flowcharts. Hoare triple The central feature of Hoare logic is the Hoare triple. A triple describes how the execution of a piece of code changes the state of the computation. A Hoare triple is of the form : \ C \ where P and Q are '' assertions'' and C is a ''command''.Hoare originally wrote "P\Q" rather than "\C\". P is named the '' precondition'' and Q the '' postcondition'': when the precondition is met, executing the command establishes the postcondition. Assertions are formulae in predicate logic. Hoare logic provides axioms and inference rules for all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satisfiability Modulo Theories
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. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Logic
In computer science, separation logic is an extension of Hoare logic, a way of reasoning about programs. It was developed by John C. Reynolds, Peter O'Hearn, Samin Ishtiaq and Hongseok Yang, drawing upon early work by Rod Burstall. The assertion language of separation logic is a special case of the logic of bunched implications (BI). A CACM review article by O'Hearn charts developments in the subject to early 2019. Overview Separation logic facilitates reasoning about: * programs that manipulate pointer data structures—including information hiding in the presence of pointers; * ''"transfer of ownership"'' (avoidance of semantic frame axioms); and * virtual separation (modular reasoning) between concurrent modules. Separation logic supports the developing field of research described by Peter O'Hearn and others as ''local reasoning'', whereby specifications and proofs of a program component mention only the portion of memory used by the component, and not the entire global ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
