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Introduction to Term Rewriting: Techniques and Applications

Introduction to Term Rewriting: Techniques and
Applications
´
´
Departamento de Sistemas Informaticos y Computacion (DSIC)
´cnica de Valencia (UPV)
Universidad Polite

Salvador Lucas http://www.dsic.upv.es/˜slucas Summer School on Foundations of Information
Technologies – FIT 2009
1

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Equations
Equations, that is, expressions of the form s=t are very common in mathematics, logic, artificial intelligence, and programming. Equations are frequently used for:
1

writing programs as sets of equations,

2

defining interpreters of programming languages,

3

requirements specification in software engineering,

4

reasoning about program and system properties and implementing its
(automated) verification,

5

modeling computational systems.
2

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Equational reasoning
The following equations : x +0 = x x + (−x) = 0

(x + y ) + z x +y

= x + (y + z)
= y +x

express the axioms of Abelian groups (zero element, inverse element, associativity, and commutativity).
Variables x, y , z are universally quantified here.
Given this set of equations E , we are interested in answering questions like: is 0 + x = x (more precisely: (∀x) 0 + x = x) a consequence of E ?

Remark
The application of rewriting techniques to this kind of problems provide a powerful framework for equational reasoning.
3

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Equational programming
The following set of equations can be used to describe a sorting function over lists of natural numbers (in Peano’s notation: 0 = 0, 1 = s(0),
2 = s(s(0)), . . .) max(0, X ) = X

min(0, X ) = 0

max(X , 0) = X

min(X , 0) = 0

max(s(X ), s(Y )) = s(max(X , Y ))

min(s(X ), s(Y )) = s(min(X , Y ))

sort(nil) = nil sort(cons(X , L)) = insert(X , sort(L)) insert(X , nil) = cons(X , nil) insert(X , cons(Y , L)) = cons(min(X , Y ), insert(max(X , Y ), L))
4

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Equational programming
We can use these equations to arrange the elements of the list [2,0,1] in increasing ordering: in Peano’s notation, this list is encoded as follows: cons(s(s(0)), cons(0, cons(s(0), nil)))
The evaluation of sort(cons(s(s(0)), cons(0, cons(s(0), nil)))) would ‘yield’ the ordered list [0, 1, 2], i.e., cons(0, cons(s(0), cons(s(s(0)), nil)))

Remark
From a logical point of view, we would rather say that the expression sort(cons(s(s(0)), cons(0, cons(s(0), nil)))) is equivalent to cons(0, cons(s(0), cons(s(s(0)), nil))) in the equational theory generated by the previous set of equations.
5

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Rewrite rules and equations
Equations are often thought or used in a given direction. This is natural in equational programming, where the focus is in computation, i.e., a state-transformation perspective is usually assumed.
For instance, we can impose a left-to-right orientation to the previous equations to obtain rewrite rules as follows: max(0, X ) → X

min(0, X ) → 0

max(X , 0) → X

min(X , 0) → 0

max(s(X ), s(Y )) → s(max(X , Y ))

min(s(X ), s(Y )) → s(min(X , Y ))

sort(nil) → nil sort(cons(X , L)) → insert(X , sort(L)) insert(X , nil) → cons(X , nil) insert(X , cons(Y , L)) → cons(min(X , Y ), insert(max(X , Y ), L))
6

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Rewrite rules, Term Rewriting Systems and Term Rewriting
When using
Rewrite Rules instead of equations, we obtain a
Term Rewriting System (TRS)
The intended computational mechanism is
Term Rewriting which replaces, within a given expression, instances of the left-hand side of a rule by the corresponding instance of the right-hand side.

Remark
The evaluation of an expression like 3*(2+2) involves the application of rewriting techniques: subexpressions like 2+2 are rewritten into equivalent but usually simpler ones (like 4) until no further simplification is possible.
A set of equations is (silently) used to specify equivalent expressions (for instance, 2+2=4 and 3*4=12).
Salvador Lucas

FIT’09

7

Introduction to Term Rewriting: Techniques and Applications

Rewriting computations
Example With the following TRS: max(0, X ) → X

(1)

max(X , 0) → X

(2)

max(s(X ), s(Y )) → s(max(X , Y ))

(3)

we can compute the maximum among three numbers: max(s(s(s(0))), max(s(0), s(s(0))))
→(3) max(s(s(s(0))), s(max(0, s(0))))
→(1) max(s(s(s(0))), s(s(0)))
→(3) s(max(s(s(0)), s(0)))
→(3) s(s(max(s(0), 0)))
→(2) s(s(s(0)))
8

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Rewriting and programming languages
In programming languages whose operational principle is (some variant of) term rewriting:
• Programs are Term Rewriting Systems.
• The computational mechanism is Term Rewriting (which is

Turing-complete).
• A program execution consists of the application of rewriting steps to

a given initial expression until no further step is possible (this corresponds to a normalization semantics).
Examples
Functional languajes (Haskell, ML),
Algebraic languages (Maude),
Equational languages
9

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Modeling computational systems
The following set of rules provides a partial representation of the conectivity of the (old) WWV’05 web site (held in Valencia):

wwv 05(U) wwv 05(U) wwv 05(U) wwv 05(U) submission(U) sbmlink(U) sbmlink(U) speakers(U) org (U) login(U) blogin(U)













submission(U) org (U) valencia(U) travelling (U) entcs(U) login(U) register (U) krishnamurthi(U) ballis(U) vlogin(U) submit(U)

wwv 05(U) wwv 05(U) wwv 05(U) submission(U) submission(U) sbmlink(U) speakers(U) org (U) org (U) vlogin(slucas) →










speakers(U) cfp accomodation(U) sbmlink(U) entcswwv 05(U) forgotten(U) finkelstein(U) alpuente(U) escobar (U) blogin(slucas) 10

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Modeling computational systems
One can use this rewrite system to ask questions about the reachability of some WWW pages for some users (depending on their registration status).
Maude> search wwv05(slucas) =>+ submit(slucas) . search in WebWWV05safe : wwv05(slucas) =>+ submit(slucas) .
Solution 1 (state 21)
No more solutions. states: 22 rewrites: 21 in 0ms cpu (1ms real) (~ rewrites/second)

Other users are not allowed to visit some pages:
Maude> search wwv05(smith) =>+ submit(smith) . search in WebWWV05safe : wwv05(smith) =>+ submit(smith) .
No solution. states: 20 rewrites: 19 in 0ms cpu (0ms real) (~ rewrites/second)
11

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Rewriting and programming languages
By using the concepts, results and methods of Term Rewriting we can investigate: 1

the deterministic character of computations (confluence),

2

how to prove that no infinite computation is possible (termination),

3

how to ensure that only finite computational sequences are followed, even when infinite sequences exist (normalizing strategies),

4

how to follow optimal computation paths with respect to some feasible notion of efficiency (optimal strategies),

5

methods and strategies to compute appropriate canonical forms
(normalizing, root-normalizing and infinitary normalizing strategies),

6

properties of combinations of programs (modularity).

7

...
12

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

Summary of the course

The aim of this course is providing a taste of rewriting and having a deeper contact with some essential topics like confluence and termination.
The course is structured as follows:
1

Introduction

2

Rewrite Systems and Term Rewriting
Termination of Rewriting

3

1
2
4

The Recursive Path Ordering
Termination with Polynomial Orderings

Confluence

13

Salvador Lucas

FIT’09

Introduction to Term Rewriting: Techniques and Applications

References
F. Baader and T. Nipkow.
Term Rewriting and All That.
Cambridge University Press, 1998.
N. Dershowitz and J.-P. Jouannaud.
Rewrite Systems.
In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, pages 243-320.
Elsevier, Amsterdam and The MIT Press, Cambridge, MA, 1990.
E. Ohlebusch.
Advanced Topics in Term Rewriting.
Springer-Verlag, Berlin, 2002.
TeReSe, editor,
Term Rewriting Systems, Cambridge University Press, 2003.
14

Salvador Lucas

FIT’09…...

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