Newman's lemma

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. Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980. Newman's original proof was considerably more complicated. In general, Newman's lemma can be seen as a combinatorial result about binary relations → on a set A (written backwards, so that a → b means that b is below a) with the following two properties: → is a well-founded relation: every non-empty subset X of A has a minimal element (an element a of X such that a → b for no b in X). Equivalently, there is no infinite chain a0 → a1 → a2 → a3 → .... In the terminology of rewriting systems, → is terminating. Every covering is bounded below. That is, if an element a in A covers elements b and c in A in the sense that a → b and a → c, then there is an element d in A such that b d and c d, where denotes the reflexive transitive closure of →. In the terminology of rewriting systems, → is locally confluent. The lemma states that if the above two conditions hold, then → is confluent: whenever a b and a c, there is an element d such that b d and c d. In view of the termination of →, this implies that every connected component of → as a graph contains a unique minimal element a, moreover b a for every element b of the component.

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Related concepts (5)

Abstract rewriting system

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.

Confluence (abstract rewriting)

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.

Normal form (abstract rewriting)

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.