arXiv subjects

Supratik Chakraborty

Publications and source records attributed to Supratik Chakraborty.

At least 37 records · Page 2Linked to original sources

arXiv2021

Diffy: Inductive Reasoning of Array Programs using Difference Invariants

We present a novel verification technique to prove interesting properties of a class of array programs with a symbolic parameter N denoting the size of arrays. The technique relies on constructing two slightly different versions of the same program. It infers difference relations between the corresponding variables at key control points of the joint control-flow graph of the two program versions. The desired post-condition is then proved by inducting on the program parameter $N$, wherein the difference invariants are crucially used in the inductive step. This contrasts with classical techniques that rely on finding potentially complex loop invaraints for each loop in the program. Our synergistic combination of inductive reasoning and finding simple difference invariants helps prove properties of programs that cannot be proved even by the winner of Arrays sub-category from SV-COMP 2021. We have implemented a prototype tool called diffy to demonstrate these ideas. We present results comparing the performance of diffy with that of state-of-the-art tools.

cs.PL↗

arXiv2021

A Normal Form Characterization for Efficient Boolean Skolem Function Synthesis

Boolean Skolem function synthesis concerns synthesizing outputs as Boolean functions of inputs such that a relational specification between inputs and outputs is satisfied. This problem, also known as Boolean functional synthesis, has several applications, including design of safe controllers for autonomous systems, certified QBF solving, cryptanalysis etc. Recently, complexity theoretic hardness results have been shown for the problem, although several algorithms proposed in the literature are known to work well in practice. This dichotomy between theoretical hardness and practical efficacy has motivated the research into normal forms or representations of input specifications that permit efficient synthesis, thus explaining perhaps the efficacy of these algorithms. In this paper we go one step beyond this and ask if there exists a normal form representation that can in fact precisely characterize "efficient" synthesis. We present a normal form called SAUNF that precisely characterizes tractable synthesis in the following sense: a specification is polynomial time synthesizable iff it can be compiled to SAUNF in polynomial time. Additionally, a specification admits a polynomial-sized functional solution iff there exists a semantically equivalent polynomial-sized SAUNF representation. SAUNF is exponentially more succinct than well-established normal forms like BDDs and DNNFs, used in the context of AI problems, and strictly subsumes other more recently proposed forms like SynNNF. It enjoys compositional properties that are similar to those of DNNF. Thus, SAUNF provides the right trade-off in knowledge representation for Boolean functional synthesis.

cs.LO↗

arXiv2020

On Uniformly Sampling Traces of a Transition System (Extended Version)

A key problem in constrained random verification (CRV) concerns generation of input stimuli that result in good coverage of the system's runs in targeted corners of its behavior space. Existing CRV solutions however provide no formal guarantees on the distribution of the system's runs. In this paper, we take a first step towards solving this problem. We present an algorithm based on Algebraic Decision Diagrams for sampling bounded traces (i.e. sequences of states) of a sequential circuit with provable uniformity (or bias) guarantees, while satisfying given constraints. We have implemented our algorithm in a tool called TraceSampler. Extensive experiments show that TraceSampler outperforms alternative approaches that provide similar uniformity guarantees.

cs.LO↗

arXiv2020

Verifying Array Manipulating Programs with Full-Program Induction

We present a full-program induction technique for proving (a sub-class of) quantified as well as quantifier-free properties of programs manipulating arrays of parametric size N. Instead of inducting over individual loops, our technique inducts over the entire program (possibly containing multiple loops) directly via the program parameter N. Significantly, this does not require generation or use of loop-specific invariants. We have developed a prototype tool Vajra to assess the efficacy of our technique. We demonstrate the performance of Vajra vis-a-vis several state-of-the-art tools on a set of array manipulating benchmarks.

cs.SE↗

arXiv2019

Knowledge Compilation for Boolean Functional Synthesis

Given a Boolean formula F(X,Y), where X is a vector of outputs and Y is a vector of inputs, the Boolean functional synthesis problem requires us to compute a Skolem function vector G(Y)for X such that F(G(Y),Y) holds whenever \exists X F(X,Y) holds. In this paper, we investigate the relation between the representation of the specification F(X,Y) and the complexity of synthesis. We introduce a new normal form for Boolean formulas, called SynNNF, that guarantees polynomial-time synthesis and also polynomial-time existential quantification for some order of quantification of variables. We show that several normal forms studied in the knowledge compilation literature are subsumed by SynNNF, although SynNNFcan be super-polynomially more succinct than them. Motivated by these results, we propose an algorithm to convert a specification in CNF to SynNNF, with the intent of solving the Boolean functional synthesis problem. Experiments with a prototype implementation show that this approach solves several benchmarks beyond the reach of state-of-the-art tools.

cs.LO↗

arXiv2019

On Symbolic Approaches for Computing the Matrix Permanent

Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The permanent is #P-Complete; hence it is unlikely that a polynomial-time algorithm exists for the problem. Researchers have therefore focused on finding tractable subclasses of matrices for permanent computation. One such subclass that has received much attention is that of sparse matrices i.e. matrices with few entries set to 1, the rest being 0. For this subclass, improved theoretical upper bounds and practically efficient algorithms have been developed. In this paper, we ask whether it is possible to go beyond sparse matrices in our quest for developing scalable techniques for the permanent, and answer this question affirmatively. Our key insight is to represent permanent computation symbolically using Algebraic Decision Diagrams (ADDs). ADD-based techniques naturally use dynamic programming, and hence avoid redundant computation through memoization. This permits exploiting the hidden structure in a large class of matrices that have so far remained beyond the reach of permanent computation techniques. The availability of sophisticated libraries implementing ADDs also makes the task of engineering practical solutions relatively straightforward. While a complete characterization of matrices admitting a compact ADD representation remains open, we provide strong experimental evidence of the effectiveness of our approach for computing the permanent, not just for sparse matrices, but also for dense matrices and for matrices with "similar" rows.

cs.DS↗

arXiv2018

Functional Synthesis via Input-Output Separation

Boolean functional synthesis is the process of constructing a Boolean function from a Boolean specification that relates input and output variables. Despite significant recent developments in synthesis algorithms, Boolean functional synthesis remains a challenging problem even when state-of-the-art methods are used for decomposing the specification. In this work we bring a fresh decomposition approach, orthogonal to existing methods, that explores the decomposition of the specification into separate input and output components. We make use of an input-output decomposition of a given specification described as a CNF formula, by alternatingly analyzing the separate input and output components. We exploit well-defined properties of these components to ultimately synthesize a solution for the entire specification. We first provide a theoretical result that, for input components with specific structures, synthesis for CNF formulas via this framework can be performed more efficiently than in the general case. We then show by experimental evaluations that our algorithm performs well also in practice on instances which are challenging for existing state-of-the-art tools, serving as a good complement to modern synthesis techniques.

cs.LO↗

arXiv2018

What's hard about Boolean Functional Synthesis

Given a relational specification between Boolean inputs and outputs, the goal of Boolean functional synthesis is to synthesize each output as a function of the inputs such that the specification is met. In this paper, we first show that unless some hard conjectures in complexity theory are falsified, Boolean functional synthesis must necessarily generate exponential-sized Skolem functions, thereby requiring exponential time, in the worst-case. Given this inherent hardness, what does one do to solve the problem? We present a two-phase algorithm for Boolean functional synthesis, where the first phase is efficient both in terms of time and sizes of synthesized functions, and solves an overwhelming majority of benchmarks. To explain this surprisingly good performance, we provide a sufficient condition under which the first phase must produce exact correct answers. When this condition fails, the second phase builds upon the result of the first phase, possibly requiring exponential time and generating exponential-sized functions in the worst-case. Detailed experimental evaluation shows our algorithm to perform better than state-of-the-art techniques for a majority of benchmarks.

cs.LO↗

arXiv2018

Demand-driven Alias Analysis : Formalizing Bidirectional Analyses for Soundness and Precision

A demand-driven approach to program analysis have been viewed as efficient algorithms to compute only the information required to serve a target demand. In contrast, an exhaustive approach computes all information in anticipation of it being used later. However, for a given set of demands, demand-driven methods are believed to compute the same information that would be computed by the corresponding exhaustive methods. We investigate the precision and bidirectional nature of demand-driven methods and show that: (a) demand-driven methods can be formalized inherently as bidirectional data flow analysis because the demands are propagated against the control flow and the information to satisfy the demands is propagated along the control flow. We extend the formalization of the Meet Over Paths solution to bidirectional flows. This formalization helps us to prove the soundness and precision of our analysis, and (b) since a demand-driven method computes only the required information to meet a demand, it should be able to reduce the imprecision caused by data abstractions and hence should be more precise than an exhaustive method. We show that while this is indeed the case with Java, for C/C++, the precision critically hinges on how indirect assignments are handled. We use this insight and propose a demand-driven alias analysis that is more precise than an exhaustive analysis for C/C++ too. We have chosen devirtualization as an application. Our measurements show that our method is not only more efficient but more precise than the existing demand-driven method, as well as the corresponding exhaustive method.

cs.PL↗

arXiv2017

Verifying Array Manipulating Programs by Tiling

Formally verifying properties of programs that manipulate arrays in loops is computationally challenging. In this paper, we focus on a useful class of such programs, and present a novel property-driven verification method that first infers array access patterns in loops using simple heuristics, and then uses this information to compositionally prove universally quantified assertions about arrays. Specifically, we identify tiles of array accesses patterns in a loop, and use the tiling information to reduce the problem of checking a quantified assertion at the end of a loop to an inductive argument that checks only a slice of the assertion for a single iteration of the loop body. We show that this method can be extended to programs with sequentially composed loops and nested loops as well. We have implemented our method in a tool called Tiler. Initial experiments show that Tiler outperforms several state-of-the-art tools on a suite of interesting benchmarks.

cs.SE↗

arXiv2017

On Petri Nets with Hierarchical Special Arcs

We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy on places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that the termination problem is as hard as deciding positivity for linear recurrent sequences -- a long-standing open problem.

cs.LO↗

arXiv2017

Towards Parallel Boolean Functional Synthesis

Given a relational specification R(X, Y), where X and Y are sequences of input and output variables, we wish to synthesize each output as a function of the inputs such that the specification holds. This is called the Boolean functional synthesis problem and has applications in several areas. In this paper, we present the first parallel approach for solving this problem, using compositional and CEGAR-style reasoning as key building blocks. We show by means of extensive experiments that our approach outperforms existing tools on a large class of benchmarks.

cs.LO↗

arXiv2016

Matching Multiplications in Bit-Vector Formulas

Bit-vector formulas arising from hardware verification problems often contain word-level arithmetic operations. Empirical evidence shows that state-of-the-art SMT solvers are not very efficient at reasoning about bit-vector formulas with multiplication. This is particularly true when multiplication operators are decomposed and represented in alternative ways in the formula.We present a pre-processing heuristic that identifies certain types of decomposed multipliers, and adds special assertions to the input formula encoding the equivalence of sub-terms to word-level multiplication. The pre-processed formulas are then solved using an SMT solver. Our experiments with three SMT solvers show that our heuristic allows several formulas to be solved quickly, while the same formulas time out without the pre-processing step.

cs.LO↗

arXiv2016

Approximate Probabilistic Inference via Word-Level Counting

Hashing-based model counting has emerged as a promising approach for large-scale probabilistic inference on graphical models. A key component of these techniques is the use of xor-based 2-universal hash functions that operate over Boolean domains. Many counting problems arising in probabilistic inference are, however, naturally encoded over finite discrete domains. Techniques based on bit-level (or Boolean) hash functions require these problems to be propositionalized, making it impossible to leverage the remarkable progress made in SMT (Satisfiability Modulo Theory) solvers that can reason directly over words (or bit-vectors). In this work, we present the first approximate model counter that uses word-level hashing functions, and can directly leverage the power of sophisticated SMT solvers. Empirical evaluation over an extensive suite of benchmarks demonstrates the promise of the approach.

cs.AI↗

arXiv2015

Constrained Sampling and Counting: Universal Hashing Meets SAT Solving

Constrained sampling and counting are two fundamental problems in artificial intelligence with a diverse range of applications, spanning probabilistic reasoning and planning to constrained-random verification. While the theory of these problems was thoroughly investigated in the 1980s, prior work either did not scale to industrial size instances or gave up correctness guarantees to achieve scalability. Recently, we proposed a novel approach that combines universal hashing and SAT solving and scales to formulas with hundreds of thousands of variables without giving up correctness guarantees. This paper provides an overview of the key ingredients of the approach and discusses challenges that need to be overcome to handle larger real-world instances.

cs.AI↗

arXiv2015

Skolem Functions for Factored Formulas

Given a propositional formula F(x,y), a Skolem function for x is a function Ψ(y), such that substituting Ψ(y) for x in F gives a formula semantically equivalent to \exists F. Automatically generating Skolem functions is of significant interest in several applications including certified QBF solving, finding strategies of players in games, synthesising circuits and bit-vector programs from specifications, disjunctive decomposition of sequential circuits etc. In many such applications, F is given as a conjunction of factors, each of which depends on a small subset of variables. Existing algorithms for Skolem function generation ignore any such factored form and treat F as a monolithic function. This presents scalability hurdles in medium to large problem instances. In this paper, we argue that exploiting the factored form of F can give significant performance improvements in practice when computing Skolem functions. We present a new CEGAR style algorithm for generating Skolem functions from factored propositional formulas. In contrast to earlier work, our algorithm neither requires a proof of QBF satisfiability nor uses composition of monolithic conjunctions of factors. We show experimentally that our algorithm generates smaller Skolem functions and outperforms state-of-the-art approaches on several large benchmarks.

cs.LO↗

arXiv2015

Word-level Symbolic Trajectory Evaluation

Symbolic trajectory evaluation (STE) is a model checking technique that has been successfully used to verify industrial designs. Existing implementations of STE, however, reason at the level of bits, allowing signals to take values in {0, 1, X}. This limits the amount of abstraction that can be achieved, and presents inherent limitations to scaling. The main contribution of this paper is to show how much more abstract lattices can be derived automatically from RTL descriptions, and how a model checker for the general theory of STE instantiated with such abstract lattices can be implemented in practice. This gives us the first practical word-level STE engine, called STEWord. Experiments on a set of designs similar to those used in industry show that STEWord scales better than word-level BMC and also bit-level STE.

cs.LO↗

arXiv2014

Distribution-Aware Sampling and Weighted Model Counting for SAT

Given a CNF formula and a weight for each assignment of values to variables, two natural problems are weighted model counting and distribution-aware sampling of satisfying assignments. Both problems have a wide variety of important applications. Due to the inherent complexity of the exact versions of the problems, interest has focused on solving them approximately. Prior work in this area scaled only to small problems in practice, or failed to provide strong theoretical guarantees, or employed a computationally-expensive maximum a posteriori probability (MAP) oracle that assumes prior knowledge of a factored representation of the weight distribution. We present a novel approach that works with a black-box oracle for weights of assignments and requires only an {\NP}-oracle (in practice, a SAT-solver) to solve both the counting and sampling problems. Our approach works under mild assumptions on the distribution of weights of satisfying assignments, provides strong theoretical guarantees, and scales to problems involving several thousand variables. We also show that the assumptions can be significantly relaxed while improving computational efficiency if a factored representation of the weights is known.

cs.AI↗