In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") together with a binary relation, traditionally denoted with ; this definition can be further refined if we index (label) subsets of the binary relation. Despite its simplicity, an ARS is sufficient to describe important properties of rewriting systems like normal forms, termination, and various notions of confluence.
Historically, there have been several formalizations of rewriting in an abstract setting, each with its idiosyncrasies. This is due in part to the fact that some notions are equivalent, see below in this article. The formalization that is most commonly encountered in monographs and textbooks, and which is generally followed here, is due to Gérard Huet (1980).
An abstract reduction system (ARS) is the most general (unidimensional) notion about specifying a set of objects and rules that can be applied to transform them. More recently, authors use the term abstract rewriting system as well. (The preference for the word "reduction" here instead of "rewriting" constitutes a departure from the uniform use of "rewriting" in the names of systems that are particularizations of ARS. Because the word "reduction" does not appear in the names of more specialized systems, in older texts reduction system is a synonym for ARS.)
An ARS is a set A, whose elements are usually called objects, together with a binary relation on A, traditionally denoted by →, and called the reduction relation, rewrite relation or just reduction. This (entrenched) terminology using "reduction" is a little misleading, because the relation is not necessarily reducing some measure of the objects.
In some contexts it may be beneficial to distinguish between some subsets of the rules, i.e. some subsets of the reduction relation →, e.
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In computer science, confluence is a property of rewriting systems, describing which terms in such a system can be rewritten in more than one way, to yield the same result. This article describes the properties in the most abstract setting of an abstract rewriting system. The usual rules of elementary arithmetic form an abstract rewriting system. For example, the expression (11 + 9) × (2 + 4) can be evaluated starting either at the left or at the right parentheses; however, in both cases the same result is eventually obtained.
In abstract rewriting, an object is in normal form if it cannot be rewritten any further, i.e. it is irreducible. Depending on the rewriting system, an object may rewrite to several normal forms or none at all. Many properties of rewriting systems relate to normal forms. Stated formally, if (A,→) is an abstract rewriting system, x∈A is in normal form if no y∈A exists such that x→y, i.e. x is an irreducible term. An object a is weakly normalizing if there exists at least one particular sequence of rewrites starting from a that eventually yields a normal form.
In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent. Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph.
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