# Term Rewriting

## Contents

## General

We use an adaption of the ARI format, so TRSs are represented as S-Expressions (see here, Sec. 3.1). Our format differs from the format used at CoCo as follows:

- We do not impose any variable conditions, so a rewrite rule is a pair of arbitrary terms.
- Identifiers must be valid SMT-LIB symbols (see here, Sec. 3.2), whereas CoCo uses a more liberal definition. Both simple and quoted symbols are allowed. As in SMT-LIB, simple symbols must not be equal to reserved words. In our case, the reserved words are:
fun, rule, format, sort, theory, define-fun, prule, ->

To ease parsing, we impose the following additional restrictions:- Quoted identifiers must be non-empty.
- Quoted identifiers must not contain whitespace, parantheses, or semicolons.
- All occurrences of the same identifier must be either quoted or unquoted. So
(rule |a| |a|)

is valid, but(rule a |a|)

is invalid. - Use sane identifiers. For example,
a->a

is a valid identifier, but it will cause confusion, so using it is a bad idea.

- Rules are annotated with optional costs, which are natural numbers. This allows, e.g., to model relative rewriting (by setting the cost of relative rules to 0). Categories may disallow costs. So rules are defined as follows:

rule ::= ( rule term term cost? ) cost ::= :cost number

In contrast to the former XTC format, the goal of the analysis is implicitly specified by the category.

## Termination

All termination categories consider termination w.r.t. arbitrary start terms.

### Relative Termination

All categories for relative termination consider full rewriting. See here for the format of all categories for relative termination.

#### TRS Relative

- no further restrictions

#### SRS Relative

- just unary function symbols

### Non-Relative Termination

All categories for non-relative termination disallow costs.

#### TRS Standard

- full rewriting
- see here

#### SRS Standard

- full rewriting
- just unary function symbols
- see here

#### TRS Contextsensitive

- context-sensitive rewriting
- see here

#### TRS Equational

Here, a *theory* may be added to declarations of function symbols.

fun ::= ( fun identifier number theory? ) theory ::= :theory [A | C | AC]

- full rewriting modulo associativity / commutativity / associativity and commutativity

#### TRS Innermost

- innermost rewriting
- see here

#### TRS Outermost

- outermost rewriting
- see here

#### TRS Conditional

- full conditional rewriting
- see here
- currently, we only support the condition-type
*oriented*

#### TRS Conditional - Operational Termination

- TODO clarify the difference to
*TRS Conditional* - see here
- currently, we only support the condition-type
*oriented*

## Complexity

See here for all complexity categories.

### Runtime Complexity

All categories for runtime complexity consider basic start terms only.

#### Runtime Complexity Innermost

- innermost rewriting

#### Runtime Complexity Full

- full rewriting

### Derivational Complexity

All categories for derivational complexity consider arbitrary start terms.

#### Derivational Complexity Innermost

- innermost rewriting

#### Derivational Complexity Full

- full rewriting