# Category: Rewriting systems

Critical pair (term rewriting)
A critical pair arises in a term rewriting system when two rewrite rules overlap to yield two different terms. In more detail, (t1, t2) is a critical pair if there is a term t for which two different
Director string
In mathematics, in the area of lambda calculus and computation, directors or director strings are a mechanism for keeping track of the free variables in a term. Loosely speaking, they can be understoo
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 qui
Reduction (abstract rewriting)
No description available.
Canonical rewriting system
No description available.
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 no
Termination (term rewriting)
No description available.
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known
Path ordering (term rewriting)
In theoretical computer science, in particular in term rewriting, a path ordering is a well-founded strict total order (>) on the set of all terms such that f(...) > g(s1,...,sn) if f .> g and f(...)
Orthogonality (term rewriting)
Orthogonality as a property of term rewriting systems (TRSs) describes where the reduction rules of the system are all left-linear, that is each variable occurs only once on the left hand side of each
Divergence (computer science)
In computer science, a computation is said to diverge if it does not terminate or terminates in an exceptional state. Otherwise it is said to converge. In domains where computations are expected to be
Rewrite order
In theoretical computer science, in particular in automated reasoning about formal equations, reduction orderings are used to prevent endless loops. Rewrite orders, and, in turn, rewrite relations, ar
Word problem (mathematics)
In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word p
Congruence closure
No description available.
Overlap (term rewriting)
In mathematics, computer science and logic, overlap, as a property of the reduction rules in term rewriting system, describes a situation where a number of different reduction rules specify potentiall
Term (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language,
Symmetric closure
In mathematics, the symmetric closure of a binary relation on a set is the smallest symmetric relation on that contains For example, if is a set of airports and means "there is a direct flight from ai
Reduction strategy
In rewriting, a reduction strategy or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. Some authors use the term to refer to a
Explicit substitution
In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution. This is in contrast to the standard lambd
Buchberger's algorithm
In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same c
Dershowitz–Manna ordering
In mathematics, the Dershowitz–Manna ordering is a well-founded ordering on multisets named after Nachum Dershowitz and Zohar Manna. It is often used in context of termination of programs or term rewr
Reduction ordering
No description available.
Reflexive closure
In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R. For example, if X is a set of distinct numbers and x R y means "x is le
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
Church–Rosser theorem
In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does not make a difference to the eventual result. Mo
Encompassment ordering
In theoretical computer science, in particular in automated theorem proving and term rewriting,the containment, or encompassment, preorder (≤) on the set of terms, is defined by s ≤ t if a subterm of
Convergence (logic)
In mathematics, computer science and logic, convergence is the idea that different sequences of transformations come to a conclusion in a finite amount of time (the transformations are terminating), a
Unification (computer science)
In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions. Depending on which expressions (also called terms) are allowed to occur in an eq
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal i
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
Markov algorithm
In theoretical computer science, a Markov algorithm is a string rewriting system that uses grammar-like rules to operate on strings of symbols. Markov algorithms have been shown to be Turing-complete,
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix) is a semi-decision algorithm for transforming a set of equations (over terms) into a confluent term rewriting system.
Semi-Thue system
In theoretical computer science and mathematical logic a string rewriting system (SRS), historically called a semi-Thue system, is a rewriting system over strings from a (usually finite) alphabet. Giv