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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] |
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] |
Kripke Structure
A Kripke structure is a variation of the transition system, originally proposed by Saul Kripke, used in model checking to represent the behavior of a system. It consists of a graph whose nodes represent the reachable states of the system and whose edges represent state transitions, together with a labelling function which maps each node to a set of properties that hold in the corresponding state. Temporal logics are traditionally interpreted in terms of Kripke structures. Formal definition Let be a set of ''atomic propositions'', i.e. boolean-valued expressions formed from variables, constants and predicate symbols. Clarke et al. define a Kripke structure over as a 4-tuple consisting of * a finite set of states . * a set of initial states . * a transition relation such that is left-total, i.e., such that . * a labeling (or ''interpretation'') function . Since is left-total, it is always possible to construct an infinite path through the Kripke structure. A deadlock sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-state Machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of ''State (computer science), states'' at any given time. The FSM can change from one state to another in response to some Input (computer science), inputs; the change from one state to another is called a ''transition''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—Deterministic finite automaton, deterministic finite-state machines and Nondeterministic finite automaton, non-deterministic finite-state machines. For any non-deterministic finite-state machine, an equivalent deterministic one can be constructed. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions d ... [...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] |
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] |
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] |
Semigroup Action
In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are ''acting'' as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs. An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation Monoid
In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation group. A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they are equal. An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set. In automata theory, some authors use the term ''transformation semigroup'' to refer to a semigroup acting faithfully on a set of "states" different from the semigroup's base set. There is a correspondence between the two notions. Transformation semigroups and monoids A transformation semigroup is a pair (''X'',''S''), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Monoid
In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set ''Q'' of states, a set Σ called the input alphabet, and a function ''T'': ''Q'' × Σ → ''Q'' called the transition function. Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states ''Q''. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of ''Q''. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors. Transformation semigroups and monoid acts : A transformation semigroup or transformation monoid is a pair (M,Q) consisting of a set ''Q'' (often called the "set of states") and a semigroup or monoid ''M'' of functions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of ''pairs'', i.e. a subset of the Cartesian product of some sets ''A'' and ''B'', so a ternary relation is a set of triples, forming a subset of the Cartesian product of three sets ''A'', ''B'' and ''C''. An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident with) the line. Examples Binary functions A function in two variables, mapping two values from sets ''A'' and ''B'', respectively, to a value in ''C'' ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action Language
In computer science, an action language is a language for specifying state transition systems, and is commonly used to create formal models of the effects of actions on the world. Action languages are commonly used in the artificial intelligence and robotics domains, where they describe how actions affect the states of systems over time, and may be used for automated planning. Action languages fall into two classes: action description languages and action query languages. Examples of the former include STRIPS, PDDL, Language A (a generalization of STRIPS; the propositional part of Pednault's ADL), Language B (an extension of A adding ''indirect effects'', distinguishing static and dynamic laws) and Language C (which adds indirect effects also, and does not assume that every fluent is automatically "inertial"). There are also the Action Query Languages P, Q and R. Several different algorithms exist for converting action languages, and in particular, action language C, to answ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
