Restrictions of rewriting can eventually achieve termination by pruning all infinite rewrite sequences issued from every term. However, proving that is challenging. Context-sensitive rewriting (CSR) is a restriction of rewriting which is useful for describing semantic aspects of programming languages (e.g., Maude, OBJ2, OBJ3, or CafeOBJ) and analyzing the computational properties (e.g., termination or completeness) of the corresponding programs [Luc01a, Luc01b, Luc03].

In CSR, a replacement map, i.e., a mapping μ from symbols into subsets of their argument positions, is used to discriminate the argument positions on which replacements are allowed; in this way, a restriction of rewriting is obtained.

Example 1. Consider the TRS from Example 6 in [Luc98]: first(0,X) -> nil first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) from(X) -> cons(X,from(s(X))) In order to avoid infinite rewritings with expressions involving calls to from, it is natural to forbid rewritings on the second argument of the list constructor cons (i.e., we let μ(cons) = {1}). It is possible to prove that such context-sensitive computations are terminating.

In Lazy Rewriting, a replacement map is also used to distinguish between arguments which can be freely eveluated from those which are evaluated only on-demand. Thus, in contrast to CSR, reductions are eventually allowed on non-replacing arguments of symbols f, i.e., arguments i which do not belong to μ(f). See [Luc02] for further details about the connections between CSR and Lazy Rewriting.

Formats

We provide differents formats to specify the rules and the associated replacement map μ.

TPDB Format

Termination of CSR is one of the subcategories of termination of rewriting which is included in the termination problem data base (TPDB). It is possible to specify a term rewriting system with context-sensitive replacement restrictions (given by a replacement map) by using the TPDB format:

spec ::= (decl) spec | e
decl ::= VAR idlist | THEORY listofthdecl | RULES listofrules
| STRATEGY strategydecl | id anylist
anylist ::= e | id anylist | string anylist
| (anylist) anylist | , anylist
idlist ::= e | id idlist
listofthdecl ::= e | (thdecl) listofthdecl
thdecl ::= id idlist | EQUATIONS eqlist
eqlist ::= e | equation eqlist
equation ::= term == term
listofrules ::= e | rule listofrules
rule ::= term -> term | term -> term | condlist
| term ->= term | term ->= term | condlist
condlist ::= cond | cond, condlist
cond ::= term -> term | term -><- term
term ::= id | id ( ) | id (termlist)
termlist ::= term | term, termlist
strategydecl ::= INNERMOST | OUTERMOST | CONTEXTSENSITIVE csstratlist
csstratlist ::= e | (id intlist) csstratlist
intlist ::= e | int intlist

where

• id are non-empty sequences of characters except space, '(', ')', '"' and ',', and excluding special sequences '->', '==', '->=', '-><-', '|' and keywords CONTEXTSENSITIVE, EQUATIONS, INNERMOST,OUTERMOST, RULES, STRATEGY, THEORY and VAR;
• string are sequences of any characters between double quotes; and
• int are non-empty sequences of digits.

Moreover, at least one VAR and one RULES sections are mandatory. If they are several, the union is taken and:

• a symbol declared in a VAR section must not have been used in previous declarations, and is assumed to denote a variable in remaining declarations (in particular must not be applied to arguments),
• a symbol occuring in a RULES section which has not been used before is assumed to denote a function symbol, and must be used afterwards always with the same arity.

OBJ/Maude Format

It provides a more compact description as an OBJ module (which is also compatible with the Maude syntax). Here, the replacement map is described by means of strategy annotations (i₁…i_n 0) associated to each symbol; this means that μ(f) = {i₁,…,i_n}, if the arity of f is k. (see [Luc03]). Symbols f without any strategy OBJ module have no replacement restriction, i.e., μ(f) = {1,…,k}.

Structure and functionality

MU-TERM provides a simple way to deal with Term Rewriting Systems and replacement maps. The application is written in Haskell and has been developed with GHC, a popular Haskell compiler. With MU-TERM you can:

• Load TRSs, extract the corresponding signature, and associate a replacement map to the symbols of the signature.
• Modify a replacement map previously associated to the symbols of the signature.
• Prove termination of Context-Sensitive Rewriting for a given TRS R and replacement map μ. Most available techniques for proving termination of CSR (direct proofs via polynomial oderings, modular decompositions, transformations, etc.) have been implemented.
• Prove termination of Innermost Context-Sensitive Rewriting for a given TRS R and replacement map μ.
• Prove termination of Lazy Rewriting for a given TRS R and replacement map μ.
• Prove termination of Rewriting by using the techniques which have been developed for proving termination of CSR (polynomials over the rational numbers, weakly monotonic orderings together with the dependency pair framework, etc.).

Example 2. The TRS R and replacement map μ in Example 1 can be specified by the following TPDB module: (VAR X Y Z) (STRATEGY CONTEXTSENSITIVE (first 1 2) (0) (nil) (s 1) (cons 1) (from 1) ) (RULES first(0,X) -> nil first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) from(X) -> cons(X,from(s(X))) ) You can also do it as an OBJ module: obj Ex6_Luc98 is sort S . op first : S S -> S . op 0 : -> S . op nil : -> S . op s : S -> S . op cons : S S -> S [strat (1 0)] . op from : S -> S . vars X Y Z : S . eq first(0,X) = nil . eq first(s(X),cons(Y,Z)) = cons(Y,first(X,Z)) . eq from(X) = cons(X,from(s(X))) . endo The following à la Maude version would also be accepted: fmod Ex6_Luc98 is sort S . op first : S S -> S . op 0 : -> S . op nil : -> S . op s : S -> S . op cons : S S -> S [strat (1 0)] . op from : S -> S . vars X Y Z : S . eq first(0,X) = nil . eq first(s(X),cons(Y,Z)) = cons(Y,first(X,Z)) . eq from(X) = cons(X,from(s(X))) . endfm

We refer to [AGIL07, Luc04a] for further details.

Termination of CSR with MU-TERM

A number of methods have been developed for proving the μ-termination of a TRS R (i.e., termination of CSR for R and the replacement map μ). Basically, we have:

• Direct methods, i.e., orderings > on terms which can be used to directly compare the left-hand sides and right-hand sides of the rules in order to conclude the μ-termination of the TRS, see [BLR02, Bor03, GL02b, Luc04b, Zan97]. MU-TERM implements the techniques described in [BLR02] (CS-RPO) and [Luc04b, Luc05] (polynomial orderings).
• Context-sensitive dependency pairs: We have also adapted Arts and Giesl’s dependency pairs method [AG00] to CSR leading to a context-sensitive dependency pairs approach [AGL06]. MU-TERM implements all techniques described in [AGL06].
• Transformations from TRSs R and replacement maps μ that obtain a TRS R’. If we are able to prove termination of R’ (using the standard methods), then the μ-termination of R is ensured, see [FR99, GM04, Luc96, Zan97]. Our tool, MU-TERM, implements these transformations using a common environment that eases their combination and comparison.

Examples of CS-TRSs and the corresponding termination proofs by using these techniques can be found here.

Termination of Lazy Rewriting with MU-TERM

In [Luc02], termination of lazy rewriting is proved by using a transformation which permits to use any available technique for proving termination of CSR:

Example 3. The termination of lazy rewriting for the following module: obj Ex1_2_Luc02c is sort S . op 2nd : S -> S . op cons : S S -> S [strat (1 0)] . op from : S -> S . op s : S -> S . vars X Y Z : S . eq 2nd(cons(X,cons(Y,Z))) = Y . eq from(X) = cons(X,from(s(X))) . endo can be proved by proving termination of CSR for the following system: obj Ex1_2_Luc02c_LR is sort S . op 2nd : S -> S . op cons : S S -> S [strat (1 0)] . op from : S -> S . op s : S -> S . op cons1 : S S -> S . vars X Y Z X1 : S . eq 2nd(cons1(X,cons(Y,Z))) = Y . eq 2nd(cons(X,X1)) = 2nd(cons1(X,X1)) . eq from(X) = cons(X,from(s(X))) . endo which is obtained by applying the transformation in [Luc02] (implemented in MU-TERM).

Termination of rewriting with MU-TERM

Since termination of rewriting is a particular case of termination of CSR (where μ(f) = {1,…,k} for each k-ary symbol f), we can also use MU-TERM to prove termination of rewriting.
Nowadays, MU-TERM is the only tool which implements the automatic generation of polynomial orderings based on polynomial intepretations which use rational coefficients. This can be used for proving termination of rewriting.

Example 4. The following TRS R f(X) -> u1(X,X) u1(h(Y),X) -> u2(i(X),X,Y) u2(Y,X,Y) -> g(Y) i(X) -> a a -> u(c) u(d) -> b c -> d can be automatically proved terminating by using the following polynomial interpretation computed by MU-TERM: [f](X) = 3.X + 1 [u1](X1,X2) = X1 + 2.X2 [h](X) = X + 3 [u2](X1,X2,X3) = X1 + X2 + X3 + 1 [i](X) = X + 3/2 [g](X) = X [a] = 1 [u](X) = X + 1/2 [c] = 1/3 [d] = 0 [b] = 0

On the other hand, μ-reduction orderings based on polynomial interpretations over the rationals are also used together with Arts and Giesl’s dependency pairs approach to prove termination of rewriting.

Example 5. For example, μ-termination of Ex1_2_Luc02c_LR in Example 3 above can be proved by using Ferreira and Ribeiro’s transformation. The TRS Ex1_2_Luc02c_LR_FR: 2nd(cons1(X,cons(Y,Z))) -> Y 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X can be proved terminating by using the following polynomial interpretation computed by MU-TERM: [2nd](X) = 3.X [cons](X1,X2) = X1 + 1/3.X2 + 1 [from](X) = 2.X + 3 [s](X) = X + 1 [cons1](X1,X2) = 1/3.X2 [n__from](X) = 2.X + 3 [n__s](X) = X + 1 [activate](X) = X [nF_2nd](X) = X [nF_from](X) = 0 [nF_activate](X) = 1/3.X where nF_2nd, nF_from, and nF_activate are new symbols which have been automatically introduced to compute the dependency pairs that correspond to Ex1_2_Luc02c_LR_FR.

A more detailed justification of these techniques is given in this note: [Luc05].

Termination of Order-Sorted Rewriting with MU-TERM

Order-sorted TRSs (OS-TRSs) are TRSs together with a set of sorts S which comes with a subsort ordering, which is interpreted in the models as set inclusion. Furthermore, the arguments of function symbols have sorts and can be subsort overloaded, like
+ : Nat Nat -> Nat
+ : Int Int -> Int

where Nat < Int, supporting so-called subtype polymorphism. Since Maude provides an appropriate (self-explanatory) notation for OS-TRSs, the OS-TRSs that MU-TERM accepts are presented as Maude modules.

Example 6. The following example computes the factorial of a number. It is order-sorted terminating, but if you remove the sorts then the system is non-terminating. fmod Factorial is sorts Nat NzNat . subsorts NzNat < Nat . op 0 : -> Nat . op s : Nat -> NzNat . op p : NzNat -> Nat . op _+_ : Nat Nat -> Nat . op _+_ : NzNat Nat -> NzNat . op _+_ : NzNat NzNat -> NzNat . op _*_ : Nat Nat -> Nat . op _*_ : NzNat NzNat -> NzNat . op fact : Nat -> NzNat . vars x y : Nat . vars x' : NzNat . eq x + 0 = x . eq x + s(y) = s(x + y) . eq x * 0 = 0 . eq x * s(y) = x + (x * y) . eq fact(0) = s(0) . eq fact(x') = x' * fact(p(x')) . eq p(s(x)) = x . endfm

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