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Deepak Kapur

Publications and source records attributed to Deepak Kapur.

16 recordsLinked to original sources
arXiv2026

Computing Certificates in Archimedean Univariate Saturated Quadratic Modules

A new symbolic algorithm to compute sums of squares multipliers (certificates) to witness the membership of non-negative univariate polynomials in a saturated univariate quadratic module is presented. Certificates are first computed in terms of natural generators introduced by Kuhlmann and Marshall for an Archimedean saturated quadratic module; natural generators can be easily read-off from a semialgebraic set. In the univariate case, an Archimedean quadratic module is also a preordering since it is closed under multiplication; certificates have different representations when a polynomial is viewed as a member in a quadratic module versus in a preordering An algorithm is given to compute certificates of natural generators in terms of the original generators; it uses a construction introduced by Kuhlmann, Marshall, and Schwartz known as the ``Basic Lemma'', which splits the non-negative factors of generators. To compute a quadratic module certificate, certificates of products of natural generators are computed using a detailed case analysis based on the types of natural generators. An implementation of the algorithms proposed in Maple is also discussed. The certificates obtained using this implementation are compared with those generated by RealCertify. We discuss examples where RealCertify is unable to find certificates while the proposed method is successful.

cs.SC↗
arXiv2025

Computing Certificates of Strictly Positive Polynomials in Archimedean Quadratic Modules

New results on computing certificates of strictly positive polynomials in Archimedean quadratic modules are presented. The results build upon (i) Averkov's method for generating a strictly positive polynomial for which a membership certificate can be more easily computed than the input polynomial whose certificate is being sought, and (ii) Lasserre's method for generating a certificate by successively approximating a nonnegative polynomial by sums of squares. First, a fully constructive method based on Averkov's result is given by providing details about the parameters; further, his result is extended to work on arbitrary subsets, in particular, the whole Euclidean space $\mathbb{R}^n$, producing globally strictly positive polynomials. Second, Lasserre's method is integrated with the extended Averkov construction to generate certificates. Third, the methods have been implemented and their effectiveness is illustrated. Examples are given on which the existing software package RealCertify appears to struggle, whereas the proposed method succeeds in generating certificates. Several situations are identified where an Archimedean polynomial does not have to be explicitly included in a set of generators of an Archimedean quadratic module. Unlike other approaches for addressing the problem of computing certificates, the methods/approach presented is easier to understand as well as implement.

math.AC↗
arXiv2023

Verifying Quantum Phase Estimation (QPE) using Prove-It

The general-purpose interactive theorem-proving assistant called Prove-It was used to verify the Quantum Phase Estimation (QPE) algorithm, specifically claims about its outcome probabilities. Prove-It is unique in its ability to express sophisticated mathematical statements, including statements about quantum circuits, integrated firmly within its formal theorem-proving framework. We demonstrate our ability to follow a textbook proof to produce a formally certified proof, highlighting useful automation features to fill in obvious steps and make formal proving nearly as straightforward as informal theorem proving. Finally, we make comparisons with formal theorem-proving in other systems where similar claims about QPE have been proven.

quant-ph↗
arXiv2022

General Interpolation and Strong Amalgamation for Contiguous Arrays

Interpolation is an essential tool in software verification, where first-order theories are used to constrain datatypes manipulated by programs. In this paper, we introduce the datatype theory of contiguous arrays with maxdiff, where arrays are completely defined in their allocation memory and for which maxdiff returns the max index where they differ. This theory is strictly more expressive than the array theories previously studied. By showing via an algebraic analysis that its models strongly amalgamate, we prove that this theory admits quantifier-free interpolants and, notably, that interpolation transfers to theory combinations. Finally, we provide an algorithm that significantly improves the ones for related array theories: it relies on a polysize reduction to general interpolation in linear arithmetics, thus avoiding impractical full terms instantiations and unbounded loops.

cs.LO↗
arXiv2021

Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties

Algorithms for computing congruence closure of ground equations over uninterpreted symbols and interpreted symbols satisfying associativity and commutativity (AC) properties are proposed. The algorithms are based on a framework for computing a congruence closure by abstracting nonflat terms by constants as proposed first in Kapur's congruence closure algorithm (RTA97). The framework is general, flexible, and has been extended also to develop congruence closure algorithms for the cases when associative-commutative function symbols can have additional properties including idempotency, nilpotency, identities, cancellativity and group properties as well as their various combinations. Algorithms are modular; their correctness and termination proofs are simple, exploiting modularity. Unlike earlier algorithms, the proposed algorithms neither rely on complex AC compatible well-founded orderings on nonvariable terms nor need to use the associative-commutative unification and extension rules in completion for generating canonical rewrite systems for congruence closures. They are particularly suited for integrating into the Satisfiability modulo Theories (SMT) solvers. A new way to view Groebner basis algorithm for polynomial ideals with integer coefficients as a combination of the congruence closures over the AC symbol * with the identity 1 and the congruence closure over an Abelian group with + is outlined.

cs.LO↗
arXiv2021

Interpolation and Amalgamation for Arrays with MaxDiff (Extended Version)

In this paper, the theory of McCarthy's extensional arrays enriched with a maxdiff operation (this operation returns the biggest index where two given arrays differ) is proposed. It is known from the literature that a diff operation is required for the theory of arrays in order to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom). Our maxdiff operation significantly increases the level of expressivity; however, obtaining interpolation results for the resulting theory becomes a surprisingly hard task. We obtain such results via a thorough semantic analysis of the models of the theory and of their amalgamation properties. The results are modular with respect to the index theory and it is shown how to convert them into concrete interpolation algorithms via a hierarchical approach.

cs.LO↗
arXiv2020

An Algorithm for Computing a Minimal Comprehensive Gröbner\, Basis of a Parametric Polynomial System

An algorithm to generate a minimal comprehensive Gröbner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gröbner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive Gröbner\, basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gröbner\, basis of the associated specialized polynomial ideal. The key idea used in ensuring minimality is that of a polynomial being essential with respect to a comprehensive Gröbner\, basis. The essentiality check is performed by determining whether a polynomial can be covered for various specializations by other polynomials in the associated branches in a comprehensive Gröbner\, system. The algorithm has been implemented and successfully tried on many examples from the literature.

cs.SC↗
arXiv2020

Uniform Interpolants in EUF: Algorithms using DAG-representations

The concept of uniform interpolant for a quantifier-free formula from a given formula with a list of symbols, while well-known in the logic literature, has been unknown to the formal methods and automated reasoning community for a long time. This concept is precisely defined. Two algorithms for computing quantifier-free uniform interpolants in the theory of equality over uninterpreted symbols (EUF) endowed with a list of symbols to be eliminated are proposed. The first algorithm is non-deterministic and generates a uniform interpolant expressed as a disjunction of conjunctions of literals, whereas the second algorithm gives a compact representation of a uniform interpolant as a conjunction of Horn clauses. Both algorithms exploit efficient dedicated DAG representations of terms. Correctness and completeness proofs are supplied, using arguments combining rewrite techniques with model theory.

cs.LO↗
arXiv2019

NIL: Learning Nonlinear Interpolants

Nonlinear interpolants have been shown useful for the verification of programs and hybrid systems in contexts of theorem proving, model checking, abstract interpretation, etc. The underlying synthesis problem, however, is challenging and existing methods have limitations on the form of formulae to be interpolated. We leverage classification techniques with space transformations and kernel tricks as established in the realm of machine learning, and present a counterexample-guided method named NIL for synthesizing polynomial interpolants, thereby yielding a unified framework tackling the interpolation problem for the general quantifier-free theory of nonlinear arithmetic, possibly involving transcendental functions. We prove the soundness of NIL and propose sufficient conditions under which NIL is guaranteed to converge, i.e., the derived sequence of candidate interpolants converges to an actual interpolant, and is complete, namely the algorithm terminates by producing an interpolant if there exists one. The applicability and effectiveness of our technique are demonstrated experimentally on a collection of representative benchmarks from the literature, where in particular, our method suffices to address more interpolation tasks, including those with perturbations in parameters, and in many cases synthesizes simpler interpolants compared with existing approaches.

cs.LO↗
arXiv2019

Using Dynamic Analysis to Generate Disjunctive Invariants

Program invariants are important for defect detection, program verification, and program repair. However, existing techniques have limited support for important classes of invariants such as disjunctions, which express the semantics of conditional statements. We propose a method for generating disjunctive invariants over numerical domains, which are inexpressible using classical convex polyhedra. Using dynamic analysis and reformulating the problem in non-standard "max-plus" and "min-plus" algebras, our method constructs hulls over program trace points. Critically, we introduce and infer a weak class of such invariants that balances expressive power against the computational cost of generating nonconvex shapes in high dimensions. Existing dynamic inference techniques often generate spurious invariants that fit some program traces but do not generalize. With the insight that generating dynamic invariants is easy, we propose to verify these invariants statically using k-inductive SMT theorem proving which allows us to validate invariants that are not classically inductive. Results on difficult kernels involving nonlinear arithmetic and abstract arrays suggest that this hybrid approach efficiently generates and proves correct program invariants.

cs.SE↗
arXiv2019

Connecting Program Synthesis and Reachability: Automatic Program Repair using Test-Input Generation

We prove that certain formulations of program synthesis and reachability are equivalent. Specifically, our constructive proof shows the reductions between the template-based synthesis problem, which generates a program in a pre-specified form, and the reachability problem, which decides the reachability of a program location. This establishes a link between the two research fields and allows for the transfer of techniques and results between them. To demonstrate the equivalence, we develop a program repair prototype using reachability tools. We transform a buggy program and its required specification into a specific program containing a location reachable only when the original program can be repaired, and then apply an off-the-shelf test-input generation tool on the transformed program to find test values to reach the desired location. Those test values correspond to repairs for the original programm. Preliminary results suggest that our approach compares favorably to other repair methods.

cs.PL↗
arXiv2016

Interpolation synthesis for quadratic polynomial inequalities and combination with EUF

An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities (both strict and nonstrict) is proposed. The algorithm is based on a key observation that quadratic polynomial inequalities can be linearized if they are concave. A generalization of Motzkin's transposition theorem is proved, which is used to generate an interpolant between two mutually contradictory conjunctions of polynomial inequalities, using semi-definite programming in time complexity $\mathcal{O}(n^3+nm))$, where $n$ is the number of variables and $m$ is the number of inequalities. Using the framework proposed by \cite{SSLMCS2008} for combining interpolants for a combination of quantifier-free theories which have their own interpolation algorithms, a combination algorithm is given for the combined theory of concave quadratic polynomial inequalities and the equality theory over uninterpreted functions symbols (\textit{EUF}). The proposed approach is applicable to all existing abstract domains like \emph{octagon}, \emph{polyhedra}, \emph{ellipsoid} and so on, therefore it can be used to improve the scalability of existing verification techniques for programs and hybrid systems. In addition, we also discuss how to extend our approach to formulas beyond concave quadratic polynomials using Gröbner basis.

cs.LO↗
arXiv2013

Synthesizing Switching Controllers for Hybrid Systems by Continuous Invariant Generation

We extend a template-based approach for synthesizing switching controllers for semi-algebraic hybrid systems, in which all expressions are polynomials. This is achieved by combining a QE (quantifier elimination)-based method for generating continuous invariants with a qualitative approach for predefining templates. Our synthesis method is relatively complete with regard to a given family of predefined templates. Using qualitative analysis, we discuss heuristics to reduce the numbers of parameters appearing in the templates. To avoid too much human interaction in choosing templates as well as the high computational complexity caused by QE, we further investigate applications of the SOS (sum-of-squares) relaxation approach and the template polyhedra approach in continuous invariant generation, which are both well supported by efficient numerical solvers.

eess.SY↗
arXiv2012

A "Hybrid" Approach for Synthesizing Optimal Controllers of Hybrid Systems: A Case Study of the Oil Pump Industrial Example

In this paper, we propose an approach to reduce the optimal controller synthesis problem of hybrid systems to quantifier elimination; furthermore, we also show how to combine quantifier elimination with numerical computation in order to make it more scalable but at the same time, keep arising errors due to discretization manageable and within bounds. A major advantage of our approach is not only that it avoids errors due to numerical computation, but it also gives a better optimal controller. In order to illustrate our approach, we use the real industrial example of an oil pump provided by the German company HYDAC within the European project Quasimodo as a case study throughout this paper, and show that our method improves (up to 7.5%) the results reported in [3] based on game theory and model checking.

eess.SY↗
arXiv2010

Unification modulo a partial theory of exponentiation

Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement protocols. In our earlier work, we gave many decidability as well as undecidability results for multiple equational theories, involving various properties of modular exponentiation. Here, we consider a partial subtheory focussing only on exponentiation and multiplication operators. Two main results are proved. The first result is positive, namely, that the unification problem for the above theory (in which no additional property is assumed of the multiplication operators) is decidable. The second result is negative: if we assume that the two multiplication operators belong to two different abelian groups, then the unification problem becomes undecidable.

cs.SC↗