Oleg Golubitsky

A Study of Optimal 4-Bit Reversible Toffoli Circuits and Their Synthesis

Optimal synthesis of reversible functions is a nontrivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! ≈ 244 functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesizes all optimal implementations. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of all optimal 4-bit permutations, synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, and synthesis of existing benchmark functions; we compose a list of the hardest permutations to synthesize, and show distribution of optimal circuits. We further illustrate that our proposed approach may be extended to accommodate physical constraints via reporting LNN-optimal reversible circuits. Our results have important implications in the design and optimization of reversible and quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.

Synthesis of the optimal 4-bit reversible circuits

Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a complexity explosion (16! ≈244 functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present an algorithm, that synthesizes an optimal circuit for any 4-bit reversible specification. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions, and distribution of optimal circuits. Our results have important implications for the design and optimization of quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.

Distance-based classification of handwritten symbols

We study online classification of isolated handwritten symbols using distance measures on spaces of curves. We compare three distance-based measures on a vector space representation of curves to elastic matching and ensembles of SVM. We consider the Euclidean and Manhattan distances and the distance to the convex hull of nearest neighbors. We show experimentally that of all these methods the distance to the convex hull of nearest neighbors yields the best classification accuracy of about 97.5%. Any of the above distance measures can be used to find the nearest neighbors and prune totally irrelevant classes, but the Manhattan distance is preferable for this because it admits a very efficient implementation. We use the first few Legendre-Sobolev coefficients of the coordinate functions to represent the symbol curves in a finite-dimensional vector space and choose the optimal dimension and number of bits per coefficient by cross-validation. We discuss an implementation of the proposed classification scheme that will allow classification of a sample among hundreds of classes in a setting with strict time and storage limitations.

A bound for orders in differential Nullstellensatz

Abstract We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed in [A. Seidenberg, An elimination theory for differential algebra, Univ. of California Publ. in Math. III (2) (1956) 31–66] but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz [G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1) (1926) 736–788; E.W. Mayr, A.W. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (3) (1982) 305–329; W.D. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. 126 (3) (1987) 577–591; J. Kollar, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (4) (1988) 963–975; L. Caniglia, A. Galligo, J. Heintz, Some new effectivity bounds in computational geometry, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Rome, 1988, in: Lecture Notes in Comput. Sci., vol. 357, Springer, Berlin, 1989, pp. 131–151; N. Fitchas, A. Galligo, Nullstellensatz effectif et conjecture de Serre (theoreme de Quillen–Suslin) pour le calcul formel, Math. Nachr. 149 (1990) 231–253; T. Krick, L.M. Pardo, M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (3) (2001) 521–598; Z. Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (1) (2005) 1–17; T. Dube, A combinatorial proof of the effective Nullstellensatz, J. Symbolic Comput. 15 (3) (1993) 277–296]. This paper is dedicated to the memory of Eugeny Pankratiev, who was the advisor of the first three authors at Moscow State University.

Online Recognition of Multi-Stroke Symbols with Orthogonal Series

We propose an efficient method to recognize multi-stroke handwritten symbols. The method is based on computing the truncated Legendre-Sobolev expansions of the coordinate functions of the stroke curves and classifying them using linear support vector machines. Earlier work has demonstrated the efficiency and robustness of this approach in the case of single-stroke characters. Here we show that the method can be successfully applied to multi-stroke characters by joining the strokes and including the number of strokes in the feature vector or in the class labels. Our experiments yield an error rate of 11-20%, and in 99% of cases the correct class is among the top 4. The recognition process causes virtually no delay, because computation of Legendre-Sobolev expansions and SVM classification proceed on-line, as the strokes are written.

Toward affine recognition of handwritten mathematical characters

We address the problem of handwritten symbol classification in the presence of distortions modeled by affine transformations. We consider shear, rotation, scaling and translation, since these types of transformations occur most often in practice, and focus most on shear within this framework. We present a distance-based classification method, in which feature vectors are constructed from Legendre-Sobolev expansions of the coordinate functions and of the affine integral invariants of the curves given by the symbols ink strokes. We analyze different size normalization methods and conclude that integral invariants provide the most robust norm. Finally, we propose a new parameterization, a combination of arc length and time, insensitive to variations in curve tracing speed and affine distortion.

On the generalized Ritt problem as a computational problem

The Ritt problem asks if there is an algorithm that decides whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing whether a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of this equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.

Algebraic transformation of differential characteristic decompositions from one ranking to another

Canonical characteristic sets of characterizable differential ideals are studied. For the ordinary case, an estimate of orders of elements from the canonical characteristic set is proved. It is also shown how one can verify the equality of characterizable ideals without using canonical characteristic sets. DOI: 10.3103/S0027132208020095We propose an algorithm for transforming a characteristic decomposition of a radical differential ideal from one ranking into another. The algorithm is based on a new bound: we show that, in the ordinary case, for any ranking, the order of each element of the canonical characteristic set of a characterizable differential ideal is bounded by the order of the ideal. Applying this bound, the algorithm determines the number of times one needs to differentiate the given differential polynomials, so that a characteristic decomposition w.r.t. the target ranking could be computed by a purely algebraic algorithm (that is, without further differentiations). We also propose a factorization-free algorithm for computing the canonical characteristic set of a characterizable differential ideal represented as a radical ideal by a set of generators. This algorithm is not restricted to the ordinary case and is applicable for an arbitrary ranking.

A bound for the Rosenfeld–Gröbner algorithm

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.

Improved classification through runoff elections

We consider the problem of dealing with irrelevant votes when a multi-case classifier is built from an ensemble of binary classifiers. We show how run-off elections can be used to limit the effects of irrelevant votes and the occasional errors of binary classifiers, improving classification accuracy. We consider as a concrete classification problem the recognition of handwritten mathematical characters. A succinct representation of handwritten symbol curves can be obtained by computing truncated Legendre-Sobolev expansions of the coordinate functions. With this representation, symbol classes are well linearly separable in low dimension which yields fast classification algorithms based on linear support vector machines. A set of 280 different symbols was considered, which gave 1635 classes when different variants are labelled separately. With this number of classes, however, the effect of irrelevant classifiers becomes significant, often causing the correct class to be ranked lower. We introduce a general technique to correct this effect by replacing the conventional majority voting scheme with a runoff election scheme. We have found that such runoff elections further cut the top-1 mis-classification rate by about half.