arXiv subjects

S. Akshay

Publications and source records attributed to S. Akshay.

33 records · Page 2Linked to original sources

arXiv2021

Synthesizing Pareto-Optimal Interpretations for Black-Box Models

We present a new multi-objective optimization approach for synthesizing interpretations that "explain" the behavior of black-box machine learning models. Constructing human-understandable interpretations for black-box models often requires balancing conflicting objectives. A simple interpretation may be easier to understand for humans while being less precise in its predictions vis-a-vis a complex interpretation. Existing methods for synthesizing interpretations use a single objective function and are often optimized for a single class of interpretations. In contrast, we provide a more general and multi-objective synthesis framework that allows users to choose (1) the class of syntactic templates from which an interpretation should be synthesized, and (2) quantitative measures on both the correctness and explainability of an interpretation. For a given black-box, our approach yields a set of Pareto-optimal interpretations with respect to the correctness and explainability measures. We show that the underlying multi-objective optimization problem can be solved via a reduction to quantitative constraint solving, such as weighted maximum satisfiability. To demonstrate the benefits of our approach, we have applied it to synthesize interpretations for black-box neural-network classifiers. Our experiments show that there often exists a rich and varied set of choices for interpretations that are missed by existing approaches.

cs.LG↗

arXiv2021

Fast zone-based algorithms for reachability in pushdown timed automata

Given the versatility of timed automata a huge body of work has evolved that considers extensions of timed automata. One extension that has received a lot of interest is timed automata with a, possibly unbounded, stack, also called the pushdown timed automata (PDTA) model. While different algorithms have been given for reachability in different variants of this model, most of these results are purely theoretical and do not give rise to efficient implementations. One main reason for this is that none of these algorithms (and the implementations that exist) use the so-called zone-based abstraction, but rely either on the region-abstraction or other approaches, which are significantly harder to implement. In this paper, we show that a naive extension of the zone based reachability algorithm for the control state reachability problem of timed automata is not sound in the presence of a stack. To understand this better we give an inductive rule based view of the zone reachability algorithm for timed automata. This alternate view allows us to analyze and adapt the rules to also work for pushdown timed automata. We obtain the first zone-based algorithm for PDTA which is terminating, sound and complete. We implement our algorithm in the tool TChecker and perform experiments to show its efficacy, thus leading the way for more practical approaches to the verification of pushdown timed systems.

cs.FL↗

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

Revisiting Underapproximate Reachability for Multipushdown Systems

Boolean programs with multiple recursive threads can be captured as pushdown automata with multiple stacks. This model is Turing complete, and hence, one is often interested in analyzing a restricted class that still captures useful behaviors. In this paper, we propose a new class of bounded under approximations for multi-pushdown systems, which subsumes most existing classes. We develop an efficient algorithm for solving the under-approximate reachability problem, which is based on efficient fix-point computations. We implement it in our tool BHIM and illustrate its applicability by generating a set of relevant benchmarks and examining its performance. As an additional takeaway, BHIM solves the binary reachability problem in pushdown automata. To show the versatility of our approach, we then extend our algorithm to the timed setting and provide the first implementation that can handle timed multi-pushdown automata with closed guards.

cs.FL↗

arXiv2020

Sparse Hashing for Scalable Approximate Model Counting: Theory and Practice

Given a CNF formula F on n variables, the problem of model counting or #SAT is to compute the number of satisfying assignments of F . Model counting is a fundamental but hard problem in computer science with varied applications. Recent years have witnessed a surge of effort towards developing efficient algorithmic techniques that combine the classical 2-universal hashing with the remarkable progress in SAT solving over the past decade. These techniques augment the CNF formula F with random XOR constraints and invoke an NP oracle repeatedly on the resultant CNF-XOR formulas. In practice, calls to the NP oracle calls are replaced a SAT solver whose runtime performance is adversely affected by size of XOR constraints. The standard construction of 2-universal hash functions chooses every variable with probability p = 1/2 leading to XOR constraints of size n/2 in expectation. Consequently, the challenge is to design sparse hash functions where variables can be chosen with smaller probability and lead to smaller sized XOR constraints. In this paper, we address this challenge from theoretical and practical perspectives. First, we formalize a relaxation of universal hashing, called concentrated hashing and establish a novel and beautiful connection between concentration measures of these hash functions and isoperimetric inequalities on boolean hypercubes. This allows us to obtain (log m) tight bounds on variance and dispersion index and show that p = O( log(m)/m ) suffices for design of sparse hash functions from {0, 1}^n to {0, 1}^m. We then use sparse hash functions belonging to this concentrated hash family to develop new approximate counting algorithms. A comprehensive experimental evaluation of our algorithm on 1893 benchmarks demonstrates that usage of sparse hash functions can lead to significant speedups.

cs.DS↗

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

Timed Systems through the Lens of Logic

In this paper, we analyze timed systems with data structures, using a rich interplay of logic and properties of graphs. We start by describing behaviors of timed systems using graphs with timing constraints. Such a graph is called realizable if we can assign time-stamps to nodes or events so that they are consistent with the timing constraints. The logical definability of several graph properties has been a challenging problem, and we show, using a highly non-trivial argument, that the realizability property for collections of graphs with strict timing constraints is logically definable in a class of propositional dynamic logic (EQ-ICPDL), which is strictly contained in MSO. Using this result, we propose a novel, algorithmically efficient and uniform proof technique for the analysis of timed systems enriched with auxiliary data structures, like stacks and queues. Our technique unravels new results (for emptiness checking as well as model checking) for timed systems with richer features than considered so far, while also recovering existing results.

cs.LO↗

arXiv2019

Continuous Reachability for Unordered Data Petri nets is in PTime

Unordered data Petri nets (UDPN) are an extension of classical Petri nets with tokens that carry data from an infinite domain and where transitions may check equality and disequality of tokens. UDPN are well-structured, so the coverability and termination problems are decidable, but with higher complexity than for Petri nets. On the other hand, the problem of reachability for UDPN is surprisingly complex, and its decidability status remains open. In this paper, we consider the continuous reachability problem for UDPN, which can be seen as an over-approximation of the reachability problem. Our main result is a characterization of continuous reachability for UDPN and polynomial time algorithm for solving it. This is a consequence of a combinatorial argument, which shows that if continuous reachability holds then there exists a run using only polynomially many data values.

cs.FL↗

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

Analyzing Timed Systems Using Tree Automata

Timed systems, such as timed automata, are usually analyzed using their operational semantics on timed words. The classical region abstraction for timed automata reduces them to (untimed) finite state automata with the same time-abstract properties, such as state reachability. We propose a new technique to analyze such timed systems using finite tree automata instead of finite word automata. The main idea is to consider timed behaviors as graphs with matching edges capturing timing constraints. When a family of graphs has bounded tree-width, they can be interpreted in trees and MSO-definable properties of such graphs can be checked using tree automata. The technique is quite general and applies to many timed systems. In this paper, as an example, we develop the technique on timed pushdown systems, which have recently received considerable attention. Further, we also demonstrate how we can use it on timed automata and timed multi-stack pushdown systems (with boundedness restrictions).

cs.FL↗

arXiv2018

Distribution-based objectives for Markov Decision Processes

We consider distribution-based objectives for Markov Decision Processes (MDP). This class of objectives gives rise to an interesting trade-off between full and partial information. As in full observation, the strategy in the MDP can depend on the state of the system, but similar to partial information, the strategy needs to account for all the states at the same time. In this paper, we focus on two safety problems that arise naturally in this context, namely, existential and universal safety. Given an MDP A and a closed and convex polytope H of probability distributions over the states of A, the existential safety problem asks whether there exists some distribution d in H and a strategy of A, such that starting from d and repeatedly applying this strategy keeps the distribution forever in H. The universal safety problem asks whether for all distributions in H, there exists such a strategy of A which keeps the distribution forever in H. We prove that both problems are decidable, with tight complexity bounds: we show that existential safety is PTIME-complete, while universal safety is co-NP-complete. Further, we compare these results with existential and universal safety problems for Rabin's probabilistic finite-state automata (PFA), the subclass of Partially Observable MDPs which have zero observation. Compared to MDPs, strategies of PFAs are not state-dependent. In sharp contrast to the PTIME result, we show that existential safety for PFAs is undecidable, with H having closed and open boundaries. On the other hand, it turns out that the universal safety for PFAs is decidable in EXPTIME, with a co-NP lower bound. Finally, we show that an alternate representation of the input polytope allows us to improve the complexity of universal safety for MDPs and PFAs.

cs.LO↗

arXiv2017

Towards an Efficient Tree Automata based technique for Timed Systems

The focus of this paper is the analysis of real-time systems with recursion, through the development of good theoretical techniques which are implementable. Time is modeled using clock variables, and recursion using stacks. Our technique consists of modeling the behaviours of the timed system as graphs, and interpreting these graphs on tree terms by showing a bound on their tree-width. We then build a tree automaton that accepts exactly those tree terms that describe realizable runs of the timed system. The emptiness of the timed system thus boils down to emptiness of a finite tree automaton that accepts these tree terms. This approach helps us in obtaining an optimal complexity, not just in theory (as done in earlier work), but also in going towards an efficient implementation of our technique. To do this, we make several improvements in the theory and exploit these to build a first prototype tool that can analyze timed systems with recursion.

cs.FL↗

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↗

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↗