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Rewriting
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Rewriting
In mathematics, linguistics, 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 as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects.
Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting.
In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a rewriting system. For example, the rules of such a system would be:
For each rule, each variable denotes a subexpression, and the symbol () indicates that an expression matching the left hand side of it can be rewritten to one matching the right hand side of it. In such a system, each rule is a logical equivalence, so performing a rewrite on an expression by these rules does not change the truth value of it. Other useful rewriting systems in logic may not preserve truth values, see e.g. equisatisfiability.
Term rewriting systems can be employed to compute arithmetic operations on natural numbers. To this end, each such number has to be encoded as a term. The simplest encoding is the one used in the Peano axioms, based on the constant 0 (zero) and the successor function S. For example, the numbers 0, 1, 2, and 3 are represented by the terms 0, S(0), S(S(0)), and S(S(S(0))), respectively. The following term rewriting system can then be used to compute sum and product of given natural numbers.
For example, the computation of 2+2 to result in 4 can be duplicated by term rewriting as follows:
where the notation above each arrow indicates the rule used for each rewrite.
As another example, the computation of 2⋅2 looks like:
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Rewriting AI simulator
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Rewriting
In mathematics, linguistics, 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 as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects.
Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting.
In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a rewriting system. For example, the rules of such a system would be:
For each rule, each variable denotes a subexpression, and the symbol () indicates that an expression matching the left hand side of it can be rewritten to one matching the right hand side of it. In such a system, each rule is a logical equivalence, so performing a rewrite on an expression by these rules does not change the truth value of it. Other useful rewriting systems in logic may not preserve truth values, see e.g. equisatisfiability.
Term rewriting systems can be employed to compute arithmetic operations on natural numbers. To this end, each such number has to be encoded as a term. The simplest encoding is the one used in the Peano axioms, based on the constant 0 (zero) and the successor function S. For example, the numbers 0, 1, 2, and 3 are represented by the terms 0, S(0), S(S(0)), and S(S(S(0))), respectively. The following term rewriting system can then be used to compute sum and product of given natural numbers.
For example, the computation of 2+2 to result in 4 can be duplicated by term rewriting as follows:
where the notation above each arrow indicates the rule used for each rewrite.
As another example, the computation of 2⋅2 looks like: