A Type-Coercion Problem in Computer Algebra

Transcription

1 Lecture Notes in Computer Science 737, pages A Type-Coercion Problem in Computer Algebra Andreas Weber Wilhelm-Schickard-Institut Universität Tübingen W-7400 Tübingen, Germany hweber@informatik.uni-tuebingen.dei Abstract. An important feature of modern computer algebra systems is the support of a rich type system with the possibility of type inference. Basic features of such a type system are polymorphism and coercion between types. Recently the use of order-sorted rewrite systems was proposed as a general framework. We will give a quite simple example of a family of types arising in computer algebra whose coercion relations cannot be captured by a finite set of first-order rewrite rules. Keywords: Computer algebra, type systems, subtyping, type coercion, type inference, order-sorted rewriting, universal algebra. 1 Introduction Early computer algebra systems did not have a sophisticated type system. This is mainly due to the fact that the types occurring in traditional programming languages are not fully appropriate for algebraic computations. With the progress in computer algebra the number of applications and so of computational domains grew. As a result of not having language constructs that aid in organization the larger systems are considered to be at or near the ceiling of their extendability. Therefore modern languages for symbolic computation come with a type concept [1], [2], and [3]. An important feature that has to be accomplished by a type system is the possibility of an automatic type inference. As an example, the user of a system would like to write down x x + 2 and the system should infer that this is a polynomial over the rationals. Since the problem of type inference in computer algebra is largely predefined by mathematical practice, finding a general mechanism for doing type inference in computer algebra has turned out to be difficult. Some suggestions are given in [4], [5], [6], and [7]. The suggestions given in [4], [6], and [7] are also an attempt to give a safe theoretical foundation for the type inference facilities of an existing system as AXIOM 1 [1]. 1 AXIOM is a trademark of The Numerical Algorithms Group Ltd.

2 189 A notion that is common to all these approaches is that of coercion. It should be possible that the system automatically coerces one type to another, e. g. an integer into a rational number or into an integral polynomial. In [6] the coercion rules are interpreted as rewrite rules over an order-sorted algebra. Types are terms in an order-sorted algebra and term rewriting techniques are used for type inference. This approach seems to be very promising. By the use of type variables many coercion problems arising in a computer algebra system can be handled in a uniform way. Moreover, the use of rules guarantees an easy extendability for future applications. Since term rewriting techniques are well established, a transfer of useful results should be possible to gain practically applicable systems. However, in a certain sense the approach in [6] is too general to be applicable to an important range of problems. If arbitrary (first-order) terms are allowed in the rewrite rules, the general type inference problem becomes undecidable. Therefore we study typical examples from computer algebra in order to find restrictions on the rewrite rules under which the type inference problem becomes decidable. Unfortunately there are not only examples which would suggest such restrictions. There is also a quite simple example of a family of types arising in computer algebra whose coercion relations cannot be captured by a finite set of first-order rewrite rules at all! We will present this example in Sect. 4. A technical result which is needed to set up the example will be proved in Sect Preliminaries We will assume that the reader is familiar with the basic notions of rewrite systems as can be found in [8]. We will also need some concepts of universal algebra. A comprehensive reference is [9]. The set of non-negative integers will be denoted by IN. We will 0 for the first infinite cardinal, the cardinality of IN. The set of strings over an alphabet L will be L, where " is the empty string. We will use to denote a first-order signature. E, E 0 will be sets of equations over, in which we will use t x y v 0 v 1 ::: as variables. The size of a term is the number of function symbols occurring in it. If an algebra A is a model of a set of equations E, we will write Aj= E. A set of equations E over is axiomatized by a set of equations E 0 iff for any algebra A of the type given by, Aj= E implies Aj= E 0. In this case we will write E 0 j= E. E is finitely based iff there is a finite set of equations E 0 which axiomatizes it. Recall that for any cardinal and any set of equations E the free algebra on generators for the equational class defined by E exists (see [9, p. 167]). It can be constructed as the term algebra over the set of generators modulo the equivalence relation given by the equations and is unique up to isomorphism. We will call this algebra the free model of generators over E. We will use the notation of [6]. Since we are basically dealing with terms of the same sort in the example (integral domains), the order-sorted framework collapses to a single sorted one. Thus we will just write down first-order terms to simplify notation.

3 190 3 A Technical Result Definition 1. Let f : fp Fg ;!fp Fg be the function, which is defined by the following algorithm: If no F is occurring in the input string, then return the input string as output string. Otherwise, remove any F except the leftmost occurrence from the input string and return the result as output string. Let be the binary relation on fp Fg which is defined by 8v w 2fP Fg : v w () f (v) =f (w): Obviously, the function f can be computed in linear time and the relation is an equivalence relation on fp Fg. Let be the first-order signature consisting of the two unary function Symbols F and P. We will now lift the equivalence relation to a set of equations over. Definition 2. Let E be the following set of equations: E = f S 1 (S 2 (S k (x) )=S k+1(s k+2(s r (x) )) : S i 2fF Pg (1 i r) and S 1 S 2 S k S k+1s k+2 S r g Theorem 3. E is not finitely based, i. e. there is no finite set of axioms for E. Proof. Assume towards a contradiction that there is such a finite set E 0. Let M be the free model 0 generators over E and let M 0 be the free model of one generator over E 0. Except for a possible renaming of the variable symbol x, E 0 has to be a subset of E. Otherwise, E 0 would contain an equation of the form or of the form S 1 (S 2 (S k (x) )=S k+1(s k+2(s r (y) )) S 1 (S 2 (S k (x) )=S k+1(s k+2(s r (x) )) S 1 S 2 S k 6 S k+1s k+2 S r : However, none of these equations holds in M. Now let n 2 IN be the maximal size of a term in E 0. Then the equation F (P (P ((P {z } n (x)) ))) = F (P (F (P (P ((P (x)) ))))) {z } n;1 holds in M, but it does not hold in M 0. ut

4 191 4 A Type-Coercion Problem If R is an integral domain, we can form the field of fractions FF(R). We can also built the ring of univariate polynomials in the indeterminate x which we will denote by UP(R x) the ring of polynomials R[x] in the standard mathematical notation which is again an integral domain by a Lemma of Gauß. Thus we can also built the field of fractions of UP(R x), FF(UP(R x)) the field of rational functions R(x). Starting from an integral domain R we will always get an integral domain and can repeatedly built the field of fractions and the ring of polynomials in a new indeterminate. Thus if a computer algebra system has a fixed integral domain R and names for symbols x 0 x 1 x 2 :::, it should also provide types of the form 1. R, 2. FF(R), 3. UP(R x 0 ), 4. UP(FF(R) x 0 ), 5. FF(UP(R x 0 )), 6. UP(UP(R x 0 ) x 1 ), 7. UP(FF(UP(R x 0 )) x 1 ), 8. FF(UP(UP(R x 0 ) x 1 )), 9. FF(UP(FF(UP(R x 0 )) x 1 ), 10. UP(UP(UP(R x 0 ) x 1 ) x 2 ),. It is convenient to use the same symbols for a mathematical object and the symbolic expression which denotes the object. In order to clarify things we will sometimes use additional hhii for the mathematical objects. There are canonical embeddings from an integral domain into its field of fractions and into the ring of polynomials in one indeterminate (an element is mapped to the corresponding constant polynomial). It is common mathematical practice to identify the integral domain with its image under these embeddings. Thus the type system should also provide a coercion between these types. If t is a type variable which ranges over integral domains and x is a symbol, this property can be expressed by the rules and t! FF(t) t! UP(t x) in the framework of [6]. However, not all of the types built by the type constructors FF and UP should be regarded to be different. If the integral domain R happens to be a field, then R will be isomorphic to its field of fractions. Especially, for any integral domain R, hhff(r)ii and hhff(ff(r))ii are isomorphic.

5 192 The fact that also hhff(ff(r))ii can be embedded in hhff(r)ii can be expressed by a rule FF(FF(t))! FF(t) which is one of the examples given in [6, p. 354]. But there are more isomorphisms which govern the relations of this family of types. If we assume that an application of the type constructor UP always uses a new indeterminate as its second argument, any application of the type constructor FF except the outermost one application is redundant. This observation will be captured by the following formal treatment. In order to avoid the technical difficulty of introducing new indeterminates, we will use an unary type constructor up instead the binary UP. The intended meaning of up(t) is UP(t x n ), where x n is a new symbol, i. e. not occurring in t. Definition 4. Define a function trans from ff Pg into the set of types recursively by the following equations. For w 2fF Pg, trans(") =R, trans(fw)=ff(trans(w)), trans(pw)=up(trans(w)). If we take hhrii to be the ring of integers, the following lemma will be an exercise in elementary calculus. 2 Lemma 5. Let hhrii be the ring of integers. For any v w 2fF Pg, the integral domains hhtrans(v)ii and hhtrans(w)ii are isomorphic iff v w. Moreover, hhtrans(v)ii can be embedded in hhtrans(w)ii and hhtrans(w)ii can be embedded in hhtrans(v)ii iff hhtrans(v)ii and hhtrans(w)ii are isomorphic. The simplifications of rational expressions are some of the most frequently used operations in many computer algebra systems. Very often, they are just concrete implementations of the embeddings of Lemma 5. Thus it would be of practical interest if these coercions between types could also be captured within the type system. However, this cannot be done by the use of a term rewriting system over the corresponding signature, since the finite set of rules corresponds to a finite set of equations only. Theorem 6. Let be the signature consisting of the unary function symbols FF and up and the constant R. Let hhrii be the ring of integers. Then there is no finite set of Equations E 0 over, such that for ground terms t 1 and t 2 the following holds. E 0 j= ft 1 = t 2 g()hht 1 ii and hht 2 ii are isomorphic. Proof. If t 1 and t 2 are ground terms, then there are v w 2fF Pg such that t 1 = trans(v) and t 2 = trans(w). Now we are done by Lemma 5 and Theorem 3. ut 2 If we started with the ring of polynomials in infinitely many indeterminates over some domain, then there would be additional isomorphisms.

6 193 The problem is that the equational theory which describes the coercion relations in the example we gave is not finitely based. Since this property of an equational theory is equivalence-invariant in the sense of [9, p. 382], the use of another signature for describing the types does not help. 5 Conclusion The use of rewrite rules seems to be a promising way to describe the coercions between types which occur in a computer algebra system. Important examples can be nicely described in a very short way and it is possible to extend a system by simply adding new rules. Unfortunately there are simple examples of families of types arising in computer algebra whose coercion relations cannot be captured by a finite set of first-order rewrite rules at all. In the present paper we have given a quite simple example of such a system, whose behavior on the object-level can be handled by many existing computer algebra systems. It is of major practical importance to have a safe and reliable interaction between various parts of a computer algebra system. Since many parts of the type system of such a system can be nicely described by rewrite rules, it would be useful if the example that we have given could be incorporated in this framework. This seems to be possible. The relations of the types can be described by means of the equational theory E of Definition 2. Since equivalence under E is decidable, it is possible to use methods of class-rewriting (see [8, Sect. 2.5]). For most practical purposes it will be sufficient to distinguish only between the integers, the rationals, polynomials (in arbitrary many indeterminates) over the integers, polynomials (in arbitrary many indeterminates) over the rationals, and rational functions (in arbitrary many indeterminates) over the integers. This approach is taken in the type system of AXIOM. Acknowledgments. I am indebted to R. Loos, R. Bündgen, and F. Haug for helpful discussions. The suggestions of an anonymous referee helped to improve the paper. References 1. Robert S. Sutor and Richard D. Jenks. The type inference and coercion facilities in the Scratchpad II interpreter. ACM SIGPLAN Notices, 22(7):56 63, SIGPLAN 87 Symposium on Interpreters and Interpretive Techniques. 2. J. H. Davenport and B. M. Trager. Scratchpad s view of algebra I: Basic commutative algebra. In Miola [10], pages S. Kamal Abdali, Guy W. Cherry, and Neil Soiffer. A Smalltak system for algebraic manipulation. ACM SIGPLAN Notices, 21(11): , November OOPSLA 86 Conference Proceedings, Portland, Oregon. 4. D. L. Rector. Semantics in algebraic computation. In Erich Kaltofen and Stephen M. Watt, editors, Computers and Mathematics, pages , Massachusetts Institute of Technology, June Springer-Verlag.

7 Gerald Baumgartner and Ryan Stansifer. A proposal to study type systems for computer algebra. Technical Report , Research Institute for Symbolic Computation Linz, A Linz, Austria, March H. Comon, D. Lugiez, and Philippe Schnoebelen. A rewrite-based type discipline for a subset of computer algebra. Journal of Symbolic Computation, 11: , Albrecht Fortenbacher. Efficient type inference and coercion in computer algebra. In Miola [10], pages Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Formal Models and Semantics, volume B of Handbook of Theoretical Computer Science, chapter 6, pages Elsevier, Amsterdam, George Grätzer. Universal Algebra. Springer-Verlag, New York, second edition, A. Miola, editor. Design and Implementation of Symbolic Computation Systems (DISCO 90), volume 429 of Lecture Notes in Computer Science, Capri, Italy, April Springer-Verlag. This article was processed using the LAT E X macro package with LLNCS style