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Free Monoid
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set ''A'' is usually denoted ''A''∗. The free semigroup on ''A'' is the sub semigroup of ''A''∗ containing all elements except the empty string. It is usually denoted ''A''+./ref> More generally, an abstract monoid (or semigroup) ''S'' is described as free if it is isomorphic to the free monoid (or semigroup) on some set. As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Star
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a Set (mathematics), set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize Automata theory, automata for regular expressions. In mathematics, it is more commonly known as the free monoid construction. Definition Given a set V, define :V^=\ (the set consists only of the empty string), :V^=V, and define recursively the set :V^=\ for each i>0. V^i is called the i-th power of V, it is a shorthand for the Concatenation#Concatenation of sets of strings, concatenation of V by itself i times. That is, ''V^i'' can be understood to be the set of all strings that can be represented as the concatenation of i members from V. The definition of Kleene star on V is : V^*=\bigcup_V^i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Monoid
In computer science, a trace is an equivalence class of strings, wherein certain letters in the string are allowed to commute, but others are not. Traces generalize the concept of strings by relaxing the requirement for all the letters to have a definite order, instead allowing for indefinite orderings in which certain reshufflings could take place. In an opposite way, traces generalize the concept of sets with multiplicities by allowing for specifying some incomplete ordering of the letters rather than requiring complete equivalence under all reorderings. The trace monoid or free partially commutative monoid is a monoid of traces. Traces were introduced by Pierre Cartier and Dominique Foata in 1969 to give a combinatorial proof of MacMahon's master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Monoid
In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendezvous points between a set of independently executing processes or threads. History monoids occur in the theory of concurrent computation, and provide a low-level mathematical foundation for process calculi, such as CSP the language of communicating sequential processes, or CCS, the calculus of communicating systems. History monoids were first presented by M.W. Shields.M.W. Shields "Concurrent Machines", ''Computer Journal'', (1985) 28 pp. 449–465. History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thread Join
In computer science, a thread of execution is the smallest sequence of programmed instructions that can be managed independently by a scheduler, which is typically a part of the operating system. In many cases, a thread is a component of a process. The multiple threads of a given process may be executed concurrently (via multithreading capabilities), sharing resources such as memory, while different processes do not share these resources. In particular, the threads of a process share its executable code and the values of its dynamically allocated variables and non- thread-local global variables at any given time. The implementation of threads and processes differs between operating systems. History Threads made an early appearance under the name of "tasks" in IBM's batch processing operating system, OS/360, in 1967. It provided users with three available configurations of the OS/360 control system, of which Multiprogramming with a Variable Number of Tasks (MVT) was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutex
In computer science, a lock or mutex (from mutual exclusion) is a synchronization primitive that prevents state from being modified or accessed by multiple threads of execution at once. Locks enforce mutual exclusion concurrency control policies, and with a variety of possible methods there exist multiple unique implementations for different applications. Types Generally, locks are ''advisory locks'', where each thread cooperates by acquiring the lock before accessing the corresponding data. Some systems also implement ''mandatory locks'', where attempting unauthorized access to a locked resource will force an exception in the entity attempting to make the access. The simplest type of lock is a binary semaphore. It provides exclusive access to the locked data. Other schemes also provide shared access for reading data. Other widely implemented access modes are exclusive, intend-to-exclude and intend-to-upgrade. Another way to classify locks is by what happens when the lock stra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lock (computer Science)
In computer science, a lock or mutex (from mutual exclusion) is a synchronization primitive that prevents state from being modified or accessed by multiple threads of execution at once. Locks enforce mutual exclusion concurrency control policies, and with a variety of possible methods there exist multiple unique implementations for different applications. Types Generally, locks are ''advisory locks'', where each thread cooperates by acquiring the lock before accessing the corresponding data. Some systems also implement ''mandatory locks'', where attempting unauthorized access to a locked resource will force an exception in the entity attempting to make the access. The simplest type of lock is a binary semaphore. It provides exclusive access to the locked data. Other schemes also provide shared access for reading data. Other widely implemented access modes are exclusive, intend-to-exclude and intend-to-upgrade. Another way to classify locks is by what happens when the lock st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concurrent Computation
Concurrent computing is a form of computing in which several computations are executed ''Concurrency (computer science), concurrently''—during overlapping time periods—instead of ''sequentially—''with one completing before the next starts. This is a property of a system—whether a computer program, program, computer, or a computer network, network—where there is a separate execution point or "thread of control" for each process. A ''concurrent system'' is one where a computation can advance without waiting for all other computations to complete. Concurrent computing is a form of modular programming. In its programming paradigm, paradigm an overall computation is decomposition (computer science), factored into subcomputations that may be executed concurrently. Pioneers in the field of concurrent computing include Edsger Dijkstra, Per Brinch Hansen, and C.A.R. Hoare. Introduction The concept of concurrent computing is frequently confused with the related but distinct c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Finite Automaton
In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state automaton (DFSA)—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by the string. ''Deterministic'' refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943. The figure illustrates a deterministic finite automaton using a state diagram. In this example automaton, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiautomaton
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] |
Regular Language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are Regular expression#Patterns for non-regular languages, augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognised by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by regular grammar, Type-3 grammars. Formal definition The collection of regular languages over an Alphabet (formal languages), alphabet Σ is defined recursively as follows: * The empty language ∅ is a regular language. * For each ''a'' ∈ Σ (''a'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
