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Bisimulation
In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa. Intuitively two systems are bisimilar if they, assuming we view them as playing a ''game'' according to some rules, match each other's moves. In this sense, each of the systems cannot be distinguished from the other by an observer. Formal definition Given a labeled state transition system , where is a set of states, \Lambda is a set of labels and → is a set of labelled transitions (i.e., a subset of S \times \Lambda \times S), a bisimulation is a binary relation R \subseteq S \times S, such that both and its converse R^T are simulations. From this follows that the symmetric closure of a bisimulation is a bisimulation, and that each symmetric simulation is a bisimulation. Thus some authors define bisimulation as a symmetric simulation. Equivalently, is a bisimulatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition System
In theoretical computer science, a transition system is a concept used in the study of computation. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled with labels chosen from a set; the same label may appear on more than one transition. If the label set is a singleton, the system is essentially unlabeled, and a simpler definition that omits the labels is possible. Transition systems coincide mathematically with abstract rewriting systems (as explained further in this article) and directed graphs. They differ from finite-state automata in several ways: * The set of states is not necessarily finite, or even countable. * The set of transitions is not necessarily finite, or even countable. * No "start" state or "final" states are given. Transition systems can be represented as directed graphs. Formal definition Formally, a transition system is a pair (S, T) where S is a set of states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulation Preorder
In theoretical computer science a simulation is a relation between state transition systems associating systems that behave in the same way in the sense that one system ''simulates'' the other. Intuitively, a system simulates another system if it can match all of its moves. The basic definition relates states within one transition system, but this is easily adapted to relate two separate transition systems by building a system consisting of the disjoint union of the corresponding components. Formal definition Given a labelled state transition system (S, \Lambda, →), where S is a set of states, \Lambda is a set of labels and → is a set of labelled transitions (i.e., a subset of S \times \Lambda \times S), a relation R \subseteq S \times S is a simulation if and only if for every pair of states (p,q) in R and all labels λ in \Lambda: : if p \overset p', then there is q \overset q' such that (p',q') \in R Equivalently, in terms of relational composition: :R^\,;\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stutter Bisimulation
In theoretical computer science, a stutter bisimulation is a relationship between two transition systems, abstract machines that model computation. It is defined coinductively and generalizes the idea of bisimulations. A bisimulation matches up the states of a machine such that transitions correspond; a stutter bisimulation allows transitions to be matched to finite path fragments. Definition In '' Principles of Model Checking'', Baier and Katoen define a stutter bisimulation for a single transition system and later adapt it to a relation on two transition systems. A stutter bisimulation for \text=(S, \text, \to, I, \text, L) is a binary relation ''R'' on ''S'' such that for all (''s''1,''s''2) in ''R'': # s_1, s_2 have the same labels # If s_1\to s_1' is a valid transition and (s_1',s_2)\not\in R then there exists a finite path fragment s_2u_1\cdots u_n s_2' (n\ge 0) such that each pair (s_1, u_i) is in ''R'' and (s_1',s_2') is in ''R'' # If s_2\to s_2' is a valid transition and ( ... [...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] |
Product (category Theory)
In category theory, the product of two (or more) object (category theory), objects in a category (mathematics), category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of set (mathematics), sets, the direct product of group (mathematics), groups or ring (mathematics), rings, and the product topology, product of topological spaces. Essentially, the product of a indexed family, family of objects is the "most general" object which admits a morphism to each of the given objects. Definition Product of two objects Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following universal property: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Diagram
350px, The commutative diagram used in the proof of the five lemma In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. Description A commutative diagram often consists of three parts: * objects (also known as ''vertices'') * morphisms (also known as ''arrows'' or ''edges'') * paths or composites Arrow symbols In algebra texts, the type of morphism can be denoted with different arrow usages: * A monomorphism may be labeled with a \hookrightarrow or a \rightarrowtail. * An epimorphism may be labeled with a \twoheadrightarrow. * An isomorphism may be labeled with a \overset. * The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as \e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lecture Notes In Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorials, state-of-the-art surveys, and "hot topics" are increasingly being included. The series is indexed by DBLP. See also *'' Monographiae Biologicae'', another monograph series published by Springer Science+Business Media *'' Lecture Notes in Physics'' *'' Lecture Notes in Mathematics'' *'' Electronic Workshops in Computing'', published by the British Computer Society image:Maurice Vincent Wilkes 1980 (3).jpg, Sir Maurice Wilkes served as the first President of BCS in 1957. The British Computer Society (BCS), branded BCS, The Chartered Institute for IT, since 2009, is a professional body and a learned ... References External links * Academic journals established in 1973 Computer science books Series of non-fiction books ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principles Of Model Checking
''Principles of Model Checking'' is a textbook on model checking, an area of computer science that automates the problem of determining if a machine meets specification requirements. It was written by Christel Baier and Joost-Pieter Katoen, and published in 2008 by MIT Press. Synopsis The introduction and first chapter outline the field of model checking: a model of a machine or process can be analysed to see if desirable properties hold. For instance, a vending machine might satisfy the property "the balance can never fall below €0,00". A video game might enforce the rule "if the player has 0 lives then the game ends in a loss". Both the vending machine and video game can be modelled as transition systems. Model checking is the process of describing such requirements in mathematical language, and automating proofs that the model satisfies the requirements, or discovery of counterexamples if the model is faulty. The second chapter focuses on creating an appropriate model for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, " is unique up to " means that all objects under consideration are in the same equivalence class with respect to the relation . Moreover, the equivalence relation is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation that relates two lists if one can be obtained by reordering (permutation, permuting) the other. As another example, the statement "the solution to an indefinite integral is , up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
