Deepak Kapur

Solving Polynomial Systems Using a Branch and Prune Approach

This paper presents {\tt Newton}, a branch and prune algorithm used to find all isolated solutions of a system of polynomial constraints. {\tt Newton} can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in {\tt Newton} consists of enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen--Sengupta narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. {\tt Newton} has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with \textit{state-of-the-art} continuation methods. Limitations of {\tt Newton} (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

An overview of Rewrite Rule Laboratory (RRL)

Abstract RRL ( Rewrite Rule Laboratory ) was originally developed as an environment for experimenting with automated reasoning algorithms for equational logic based on rewrite techniques. It has now matured into a full-fledged theorem prover which has been used to solve hard and challenging mathematical problems in automated reasoning literature as well as a research tool for investigating the use of formal methods in hardware and software design. We provide a brief historical account of development of RRL and its descendants, give an overview of the main capabilities of RRL and conclude with a discussion of applications of RRL .

Proof by consistency

Abstract Advances of the past decade in methods and computer programs for showing consistency of proof systems based on first-order equations have made it feasible, in some settings, to use proof by consistency as an alternative to conventional rules of inference. Musser described the method applied to proof of properties of inductively defined objects. Refinements of this inductionless induction method were discussed by Kapur, Goguen, Huet and Hullot, Huet and Oppen, Lankford, Dershowitz, Paul, and more recently by Jouannaud and Kounalis as well as by Kapur, Narendran and Zhang. This paper gives a very general account of proof by consistency and inductionless induction, and shows how previous results can be derived simply from the general theory. New results include a theorem giving characterizations of an unambiguity property that is key to applicability of proof by consistency, and a theorem similar to the Birkhoffs Completeness Theorem for equational proof systems, but concerning inductive proof.

Algebraic and geometric reasoning using Dixon resultants

Dixons method for computing multivariate resultants by simultaneously eliminating many variables is reviewed. The method is found to be quite restrictive because often the Dixon matrix is singular, and the Dixon resultant vanished identically yielding no information about solutions for many algebraic and geometry problems. We extend Dixons method for the case when the Dixon matrix is singular, but satisfies a condition. An efficient algorithm is developed based on the proposed extension for extracting conditions for the existence of affine solutions of a finite set of polynomials. Using this algorithm, numerous geometric and algebraic identities are derived for examples which appear intractable with other techniques of triangulation such as the successive resultant method, the Gro¨bner basis method, Macaulay resultants and Characteristic set method. Experimental results suggest that the resultant of a set of polynomials which are symmetric in the variables is relatively easier to compute using the extended Dixons method.

On sufficient-completeness and related properties of term rewriting systems

SummaryThe decidability of the sufficient completeness property of equational specifications satisfying certain conditions is shown. In addition, the decidability of the related concept of quasi-reducibility of a term with respect to a set of rules is proved. Other results about irreducible ground terms of a term rewriting system also follow from a key technical lemma used in these decidability proofs; this technical lemma states that there is a finite bound on the substitutions of ground terms that need to be considered in order to check for a given term, whether the result obtained by any substitution of ground terms into the term is irreducible. These results are first shown for untyped systems and are subsequently extended to typed systems.

NP-completeness of the set unification and matching problems

The set-unification and set-matching problems, which are very restricted cases of the associative-commutative-idempotent unification and matching problems, respectively, are shown to be NP-complete. The NP-completeness of the subsumption check in first-order resolution follows from these results. It is also shown that commutative-idempotent matching and associative-idempotent matching are NP-hard, thus implying that the idempotency of a function does not help in reducing the complexity of matching and unification problems.

Generating all polynomial invariants in simple loops

This paper presents a method for automatically generating all polynomial invariants in simple loops. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Based on this connection, a fixpoint procedure using operations on ideals and Grobner basis constructions is proposed for finding all polynomial invariants. Most importantly, it is proved that the procedure terminates in at most m+1 iterations, where m is the number of program variables. The proof relies on showing that the irreducible components of the varieties associated with the ideals generated by the procedure either remain the same or increase their dimension at every iteration of the fixpoint procedure. This yields a correct and complete algorithm for inferring conjunctions of polynomial equalities as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover non-trivial invariants for several examples to illustrate the power of the technique.

Using Gröbner bases to reason about geometry problems

The use of Grobner basis computation for reasoning about geometry problems is demonstrated. Two kinds of geometry problems are considered: (i) Given a finite set of geometry relations expressed as polynomial equations, in conjunction with a finite set of subsidiary conditions stated as negations of polynomial equations to rule out certain degenerate eases, check whether another geometry relation expressed as a polynomial equation and given as a conclusion, holds. (ii) Given a finite set of geometry relations expressed as polynomial equations, find a finite set of subsidiary conditions, if any, stated as negations of polynomial equations which rule out certain values of variables, such that another geometry relation expressed as a polynomial equation and given as a conclusion, holds under these conditions. Using a refutational approach for theorem proving, both kinds of problems are converted into reasoning about a finite set of polynomial equations. The first problem is shown to be equivalent to checking whether a set of polynomial equations does not have a solution; this can be decided by computing a Grobner basis of these polynomials and checking whether I is included in such a basis. In addition, it is shown that the second problem can also be solved by computing a Grobner basis and appropriately picking polynomials from it. A number of geometry problems of both kinds have been solved using this approach.

Geometry theorem proving using Hilbert's Nullstellensatz

The theory of elementary algebra and elementary geometry was shown to be decidable by Tarski using a quantifier elimination technique in the 1930’s [26]. Subsquently, Tarski’s decision algorithm was improved by others notably among them Seidenberg [25], Monk [23], and Collins [12], and recently by Ben-Or et al [4]. These methods are algebraic and are based on translating geometry statements into first-order formulae using the operations 0, 1, -1, +, *, 2, = of an ordered field with variables rangmg over real numbers. Among these decision procedures, Collins’s method based on cylinderical algebraic decomposition technique is, to our knowledge, the only decision procedure implemented so far; see [2, 31 for details.

Interpolation for data structures

Interpolation based automatic abstraction is a powerful and robust technique for the automated analysis of hardware and software systems. Its use has however been limited to control-dominated applications because of a lack of algorithms for computing interpolants for data structures used in software programs. We present efficient procedures to construct interpolants for the theories of arrays, sets, and multisets using the reduction approach for obtaining decision procedures for complex data structures. The approach taken is that of reducing the theories of such data structures to the theories of equality and linear arithmetic for which efficient interpolating decision procedures exist. This enables interpolation based techniques to be applied to proving properties of programs that manipulate these data structures.