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D. Vendraminetto - One of the best experts on this subject based on the ideXlab platform.
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Reducing interpolant circuit size by ad-hoc logic synthesis and SAT-based weakening
2016 Formal Methods in Computer-Aided Design (FMCAD), 2016Co-Authors: G. Cabodi, P. E. Camurati, M. Palena, P. Pasini, D. VendraminettoAbstract:We address the problem of reducing the size of Craig Interpolants used in SAT-based Model Checking. Craig Interpolants are AND-OR circuits, generated by post-processing refutation proofs of SAT solvers. Whereas it is well known that Interpolants are highly redundant, their compaction is typically tackled by reducing the proof graph and/or by exploiting standard logic synthesis techniques. Furthermore, strengthening and weakening have been studied as an option to control interpolant quality. In this paper we propose two interpolant compaction techniques: (1) A set of ad-hoc logic synthesis functions that, revisiting known logic synthesis approaches, specifically address speed and scalability. Though general and not restricted to Interpolants, these techniques target the main sources of redundancy in interpolant circuits. (2) An interpolant weakening technique, where the UNSAT core extracted from an additional SAT query is used to obtain a gate-level abstraction of the interpolant. The abstraction introduces fresh new variables at gate cuts that must be quantified out in order to obtain a valid interpolant. We show how to efficiently quantify them out, by working on an NNF representation of the circuit. The paper includes an experimental evaluation, showing the benefits of the proposed techniques, on a set of benchmark Interpolants arising from both hardware and software model checking problems.
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optimization techniques for craig interpolant compaction in unbounded model checking
Formal Methods, 2015Co-Authors: Gianpiero Cabodi, Carmelo Loiacono, D. VendraminettoAbstract:This paper addresses the problem of reducing the size of Craig Interpolants generated within inner steps of SAT-based Unbounded Model Checking. Craig Interpolants are obtained from refutation proofs of unsatisfiable SAT runs, in terms of and/or circuits of linear size, w.r.t. the proof. Existing techniques address proof reduction, whereas interpolant circuit compaction is typically considered as an implementation problem, tackled with standard logic synthesis techniques. We propose a three step interpolant computation process, specifically oriented to scalability, in which we: (1) exploit an existing technique to detect and remove redundancies in refutation proofs, (2) apply customized light weight combinational logic reductions (constant propagation, ODC-based simplifications, and BDD-based sweeping) directly on the proof graph data structure, (3) introduce an ad-hoc combinational reduction procedure for large interpolant circuits, based on the incrementality and divide-and-conquer paradigms. The main contributions of our approach are represented by the overall approach, the proposed combinational reduction technique, and a detailed experimental evaluation of the interpolant generation process, on a set of benchmarks from the Hardware Model Checking Competitions 2013 and 2014.
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optimization techniques for craig interpolant compaction in unbounded model checking
Design Automation and Test in Europe, 2013Co-Authors: Gianpiero Cabodi, Carmelo Loiacono, D. VendraminettoAbstract:This paper addresses the problem of reducing the size of Craig Interpolants generated within inner steps of SAT-based Unbounded Model Checking. Craig Interpolants are obtained from refutation proofs of unsatisfiable SAT runs, in terms of and/or circuits of linear size, w.r.t. the proof. Existing techniques address proof reduction, whereas interpolant compaction is typically considered as an implementation problem, tackled using standard logic synthesis techniques. We propose an integrated three step process, in which we: (1) exploit an existing technique to detect and remove redundancies in refutation proofs, (2) apply combinational logic reductions (constant propagation, ODC-based simplifications, and BDD-based sweeping) directly on the proof graph data structure, (3) eventually apply ad hoc combinational logic synthesis steps on interpolant circuits. The overall procedure is novel (as well as parts of the above listed steps), and represents an advance w.r.t. the state-of-the art. The paper includes an experimental evaluation, showing the benefits of the proposed technique, on a set of benchmarks from the Hardware Model Checking Competition 2011.
Gianpiero Cabodi - One of the best experts on this subject based on the ideXlab platform.
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optimization techniques for craig interpolant compaction in unbounded model checking
Formal Methods, 2015Co-Authors: Gianpiero Cabodi, Carmelo Loiacono, D. VendraminettoAbstract:This paper addresses the problem of reducing the size of Craig Interpolants generated within inner steps of SAT-based Unbounded Model Checking. Craig Interpolants are obtained from refutation proofs of unsatisfiable SAT runs, in terms of and/or circuits of linear size, w.r.t. the proof. Existing techniques address proof reduction, whereas interpolant circuit compaction is typically considered as an implementation problem, tackled with standard logic synthesis techniques. We propose a three step interpolant computation process, specifically oriented to scalability, in which we: (1) exploit an existing technique to detect and remove redundancies in refutation proofs, (2) apply customized light weight combinational logic reductions (constant propagation, ODC-based simplifications, and BDD-based sweeping) directly on the proof graph data structure, (3) introduce an ad-hoc combinational reduction procedure for large interpolant circuits, based on the incrementality and divide-and-conquer paradigms. The main contributions of our approach are represented by the overall approach, the proposed combinational reduction technique, and a detailed experimental evaluation of the interpolant generation process, on a set of benchmarks from the Hardware Model Checking Competitions 2013 and 2014.
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optimization techniques for craig interpolant compaction in unbounded model checking
Design Automation and Test in Europe, 2013Co-Authors: Gianpiero Cabodi, Carmelo Loiacono, D. VendraminettoAbstract:This paper addresses the problem of reducing the size of Craig Interpolants generated within inner steps of SAT-based Unbounded Model Checking. Craig Interpolants are obtained from refutation proofs of unsatisfiable SAT runs, in terms of and/or circuits of linear size, w.r.t. the proof. Existing techniques address proof reduction, whereas interpolant compaction is typically considered as an implementation problem, tackled using standard logic synthesis techniques. We propose an integrated three step process, in which we: (1) exploit an existing technique to detect and remove redundancies in refutation proofs, (2) apply combinational logic reductions (constant propagation, ODC-based simplifications, and BDD-based sweeping) directly on the proof graph data structure, (3) eventually apply ad hoc combinational logic synthesis steps on interpolant circuits. The overall procedure is novel (as well as parts of the above listed steps), and represents an advance w.r.t. the state-of-the art. The paper includes an experimental evaluation, showing the benefits of the proposed technique, on a set of benchmarks from the Hardware Model Checking Competition 2011.
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ICCAD - Stepping forward with Interpolants in unbounded model checking
Proceedings of the 2006 IEEE ACM international conference on Computer-aided design - ICCAD '06, 2006Co-Authors: Gianpiero Cabodi, Marco Murciano, Sergio Nocco, Stefano QuerAbstract:This paper addresses SAT-based Unbounded Model Checking based on Craig Interpolants. This recently introduced methodology is often able to outperform BDDs and other SAT-based techniques on large verification instances. Based on refutation proofs generated by SAT solvers, Interpolants provide compact circuit representations of state sets, and abstract away several details non relevant for proofs. We propose three main contributions, aimed at controlling interpolant size and traversal depth. First of all, we introduce interpolant-based dynamic abstraction to reduce the support of the computed interpolant. Second, we propose new advances in interpolant compaction by redundancy removal. Both techniques rely on an effective application of the incremental SAT paradigm. Finally, we also introduce interpolant computation exploiting circuit quantification, instead of SAT refutation proofs. Experimental results are specifically oriented to prove properties, rather than disproving them (bug hunting). They show how the methodology is able to extend the applicability of interpolant based Model Checking to larger and deeper verification instances.
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Stepping Forward with Interpolants in Unbounded Model Checking
2006 IEEE ACM International Conference on Computer Aided Design, 2006Co-Authors: Gianpiero Cabodi, Marco Murciano, Sergio Nocco, Stefano QuerAbstract:This paper addresses SAT-based unbounded model checking based on Craig Interpolants. This recently introduced methodology is often able to outperform BDDs and other SAT-based techniques on large verification instances. Based on refutation proofs generated by SAT solvers, Interpolants provide compact circuit representations of state sets, and abstract away several details non relevant for proofs. We propose three main contributions, aimed at controlling interpolant size and traversal depth. First of all, we introduce interpolant-based dynamic abstraction to reduce the support of the computed interpolant. Second, we propose new advances in interpolant compaction by redundancy removal. Both techniques rely on an effective application of the incremental SAT paradigm. Finally, we also introduce interpolant computation exploiting circuit quantification, instead of SAT refutation proofs. Experimental results are specifically oriented to prove properties, rather than disproving them (bug hunting). They show how the methodology is able to extend the applicability of interpolant based Model Checking to larger and deeper verification instances
Anders Lindquist - One of the best experts on this subject based on the ideXlab platform.
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On Degree-Constrained Analytic Interpolation With Interpolation Points Close to the Boundary
IEEE Transactions on Automatic Control, 2009Co-Authors: Johan Karlsson, Anders LindquistAbstract:In the recent article , a theory for complexity-constrained interpolation of contractive functions is developed. In particular, it is shown that any such interpolant may be obtained as the unique minimizer of a (convex) weighted entropy gain. In this technical note we study this optimization problem in detail and describe how the minimizer depends on weight selection and on interpolation conditions. We first show that, if, for a sequence of Interpolants, the values of the entropy gain of the Interpolants converge to the optimum, then the Interpolants converge in H 2 , but not in H infin . This result is then used to describe the asymptotic behavior of the interpolant as an interpolation point approaches the boundary of the domain of analyticity. For loop shaping to specifications in control design, it might at first seem natural to place strategically additional interpolation points close to the boundary. However, our results indicate that such a strategy will have little effect on the shape. Another consequence of our results relates to model reduction based on minimum-entropy principles, where one should avoid placing interpolation points too close to the boundary.
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Stability-Preserving Rational Approximation Subject to Interpolation Constraints
IEEE Transactions on Automatic Control, 2008Co-Authors: Johan Karlsson, Anders LindquistAbstract:A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic Interpolants with an a priori bound, has been developed in recent years. In this paper, we consider the limit case when this bound is removed, and only stable Interpolants with a prescribed maximum degree are sought. This leads to weighted H 2 minimization, where the Interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded analytic Interpolants.
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Weight selection for gap robustness with degree-constrained controllers
2008 47th IEEE Conference on Decision and Control, 2008Co-Authors: Johan Karlsson, Tryphon Georgiou, Anders LindquistAbstract:In modern robust control, control synthesis may be cast as an interpolation problem where the interpolant relates to robustness and performance criteria. In particular, robustness in the gap fits into this framework and the magnitude of the corresponding interpolant dictate the robustness to perturbations of the plant as a function of frequency. In this paper we consider the correspondence between weighted entropy functionals and minimizing Interpolants in order to find appropriate Interpolants for e.g. control synthesis. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. These results are used in order to systematize feedback control synthesis to obtain frequency dependent robustness bounds with a constraint on the controller degree.
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Stable rational approximation in the context of interpolation and convex optimization
2007 46th IEEE Conference on Decision and Control, 2007Co-Authors: Johan Karlsson, Anders LindquistAbstract:A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational Interpolants with an a priori bound, has been developed in recent years. In this paper we consider the limit case when this bound is removed, and only stable Interpolants with a prescribed maximum degree are sought. This leads to weighted H 2 minimization, where the Interpolants are parameterized by the weights. The inverse problem of determining the weight and the interpolation points given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded Interpolants.
Rida T. Farouki - One of the best experts on this subject based on the ideXlab platform.
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Algorithm 952: PHquintic: A library of basic functions for the construction and analysis of planar quintic pythagorean-hodograph curves
ACM Transactions on Mathematical Software, 2015Co-Authors: B. Dong, Rida T. FaroukiAbstract:© 2015 ACM. The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as Interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C2 PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique "good" interpolant among them.
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identification of spatial ph quintic hermite Interpolants with near optimal shape measures
Computer Aided Geometric Design, 2008Co-Authors: Rida T. Farouki, Carlotta Giannelli, Carla Manni, Alessandra SestiniAbstract:The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic Interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the ''ordinary'' cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the Interpolants depends on only one of the parameters, and that four (general) helical PH quintic Interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of Interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how ''close'' the PH quintic Interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting Interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the ''ordinary'' cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.
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geometric hermite interpolation by spatial pythagorean hodograph cubics
Advances in Computational Mathematics, 2005Co-Authors: Francesca Pelosi, Rida T. Farouki, Carla Manni, Alessandra SestiniAbstract:It is shown that, depending upon the orientation of the end tangents t0,t1 relative to the end point displacement vector Δp=p1−p0, the problem of G1 Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two Interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p0,. . .,pn in R3, compatible with a G1 piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.
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construction and shape analysis of ph hermite Interpolants
Computer Aided Geometric Design, 2001Co-Authors: Hwan Pyo Moon, Rida T. Farouki, Hyeong In ChoiAbstract:Abstract In general, the problem of interpolating given first-order Hermite data (end points and derivatives) by quintic Pythagorean-hodograph (PH) curves has four distinct formal solutions. Ordinarily, only one of these Interpolants is of acceptable shape. Previous interpolation algorithms have relied on explicitly constructing all four solutions, and invoking a suitable measure of shape—e.g., the absolute rotation index or elastic bending energy—to select the “good” interpolant. We introduce here a new means to differentiate among the solutions, namely, the winding number of the closed loop formed by a union of the hodographs of the PH quintic and of the unique “ordinary” cubic interpolant. We also show that, for “reasonable” Hermite data, the good PH quintic can be directly constructed with certainty, obviating the need to compute and compare all four solutions. Finally, we present an algorithm based on the subdivision, degree elevation, and convex hull properties of the Bernstein form, that gives rapidly convergent curvature bounds for PH curves, using only rational arithmetic operations on their coefficients.
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construction of c 2 pythagorean hodograph interpolating splines by the homotopy method
Advances in Computational Mathematics, 1996Co-Authors: Gudrun Albrecht, Rida T. FaroukiAbstract:The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing a C 2 PH quintic "'spline" that interpolates a given sequence of points P0, Pt,.-., Pu and end-derivatives d o and du to be reduced to solving a "tridiagonal" system of N quadratic equations in N complex unknowns. The system can also be easily modified to incorporate PH-spline end conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 ~+~ distinct Interpolants for each of several different data sets. We observe empirically that all but one of these Interpolants exhibits undesirable "looping" behavior (which may be quantified in terms of the elastic bending energy, i.e., the integral of the square of the curvature with respect to arc length). The remaining "good" interpolant, however, is invariably a fairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding "ordinary" C 2 cubic spline. Moreover, the PH spline has the advantage that its offsets are rational curves and its arc length is a polynomial function of the curve parameter.
Stefano Quer - One of the best experts on this subject based on the ideXlab platform.
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ICCAD - Stepping forward with Interpolants in unbounded model checking
Proceedings of the 2006 IEEE ACM international conference on Computer-aided design - ICCAD '06, 2006Co-Authors: Gianpiero Cabodi, Marco Murciano, Sergio Nocco, Stefano QuerAbstract:This paper addresses SAT-based Unbounded Model Checking based on Craig Interpolants. This recently introduced methodology is often able to outperform BDDs and other SAT-based techniques on large verification instances. Based on refutation proofs generated by SAT solvers, Interpolants provide compact circuit representations of state sets, and abstract away several details non relevant for proofs. We propose three main contributions, aimed at controlling interpolant size and traversal depth. First of all, we introduce interpolant-based dynamic abstraction to reduce the support of the computed interpolant. Second, we propose new advances in interpolant compaction by redundancy removal. Both techniques rely on an effective application of the incremental SAT paradigm. Finally, we also introduce interpolant computation exploiting circuit quantification, instead of SAT refutation proofs. Experimental results are specifically oriented to prove properties, rather than disproving them (bug hunting). They show how the methodology is able to extend the applicability of interpolant based Model Checking to larger and deeper verification instances.
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Stepping Forward with Interpolants in Unbounded Model Checking
2006 IEEE ACM International Conference on Computer Aided Design, 2006Co-Authors: Gianpiero Cabodi, Marco Murciano, Sergio Nocco, Stefano QuerAbstract:This paper addresses SAT-based unbounded model checking based on Craig Interpolants. This recently introduced methodology is often able to outperform BDDs and other SAT-based techniques on large verification instances. Based on refutation proofs generated by SAT solvers, Interpolants provide compact circuit representations of state sets, and abstract away several details non relevant for proofs. We propose three main contributions, aimed at controlling interpolant size and traversal depth. First of all, we introduce interpolant-based dynamic abstraction to reduce the support of the computed interpolant. Second, we propose new advances in interpolant compaction by redundancy removal. Both techniques rely on an effective application of the incremental SAT paradigm. Finally, we also introduce interpolant computation exploiting circuit quantification, instead of SAT refutation proofs. Experimental results are specifically oriented to prove properties, rather than disproving them (bug hunting). They show how the methodology is able to extend the applicability of interpolant based Model Checking to larger and deeper verification instances
