arXiv subjects

Shankara Narayanan Krishna

Publications and source records attributed to Shankara Narayanan Krishna.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2017

Making Metric Temporal Logic Rational

We study an extension of $\mtl$ in pointwise time with rational expression guarded modality $\reg_I(\re)$ where $\re$ is a rational expression over subformulae. We study the decidability and expressiveness of this extension ($\mtl$+$φ\ureg_{I, \re} φ$+$\reg_{I,\re}φ$), called $\regmtl$, as well as its fragment $\sfmtl$ where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that $\regmtl$ has decidable satisfiability by giving an equisatisfiable reduction to $\mtl$. We also identify a subclass $\mitl+\ureg$ of $\regmtl$ for which our equi-satisfiable reduction gives rise to formulae of $\mitl$, yielding elementary decidability. As our second main result, we show a tight automaton-logic connection between $\sfmtl$ and partially ordered (or very weak) 1-clock alternating timed automata.

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arXiv2016

On Nonlinear Prices in Timed Automata

Priced timed automata provide a natural model for quantitative analysis of real-time systems and have been successfully applied in various scheduling and planning problems. The optimal reachability problem for linearly-priced timed automata is known to be PSPACE-complete. In this paper we investigate priced timed automata with more general prices and show that in the most general setting the optimal reachability problem is undecidable. We adapt and implement the construction of Audemard, Cimatti, Kornilowicz, and Sebastiani for non-linear priced timed automata using state-of-the-art theorem prover Z3 and present some preliminary results.

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arXiv2016

Stochastic Timed Games Revisited

Stochastic timed games (STGs), introduced by Bouyer and Forejt, naturally generalize both continuous-time Markov chains and timed automata by providing a partition of the locations between those controlled by two players (Player Box and Player Diamond) with competing objectives and those governed by stochastic laws. Depending on the number of players---$2$, $1$, or $0$---subclasses of stochastic timed games are often classified as $2\frac{1}{2}$-player, $1\frac{1}{2}$-player, and $\frac{1}{2}$-player games where the $\frac{1}{2}$ symbolizes the presence of the stochastic "nature" player. For STGs with reachability objectives it is known that $1\frac{1}{2}$-player one-clock STGs are decidable for qualitative objectives, and that $2\frac{1}{2}$-player three-clock STGs are undecidable for quantitative reachability objectives. This paper further refines the gap in this decidability spectrum. We show that quantitative reachability objectives are already undecidable for $1\frac{1}{2}$ player four-clock STGs, and even under the time-bounded restriction for $2\frac{1}{2}$-player five-clock STGs. We also obtain a class of $1\frac{1}{2}$, $2\frac{1}{2}$ player STGs for which the quantitative reachability problem is decidable.

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arXiv2016

FO-definable transformations of infinite strings

The theory of regular and aperiodic transformations of finite strings has recently received a lot of interest. These classes can be equivalently defined using logic (Monadic second-order logic and first-order logic), two-way machines (regular two-way and aperiodic two-way transducers), and one-way register machines (regular streaming string and aperiodic streaming string transducers). These classes are known to be closed under operations such as sequential composition and regular (star-free) choice; and problems such as functional equivalence and type checking, are decidable for these classes. On the other hand, for infinite strings these results are only known for $ω$-regular transformations: Alur, Filiot, and Trivedi studied transformations of infinite strings and introduced an extension of streaming string transducers over $ω$-strings and showed that they capture monadic second-order definable transformations for infinite strings. In this paper we extend their work to recover connection for infinite strings among first-order logic definable transformations, aperiodic two-way transducers, and aperiodic streaming string transducers.

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arXiv2015

Metric Temporal Logic with Counting

Ability to count number of occurrences of events within a specified time interval is very useful in specification of resource bounded real time computation. In this paper, we study an extension of Metric Temporal Logic ($\mathsf{MTL}$) with two different counting modalities called $\mathsf{C}$ and $\mathsf{UT}$ (until with threshold), which enhance the expressive power of $\mathsf{MTL}$ in orthogonal fashion. We confine ourselves only to the future fragment of $\mathsf{MTL}$ interpreted in a pointwise manner over finite timed words. We provide a comprehensive study of the expressive power of logic $\mathsf{CTMTL}$ and its fragments using the technique of EF games extended with suitable counting moves. Finally, as our main result, we establish the decidability of $\mathsf{CTMTL}$ by giving an equisatisfiable reduction from $\mathsf{CTMTL}$ to $\mathsf{MTL}$. The reduction provides one more example of the use of temporal projections with oversampling introduced earlier for proving decidability. Our reduction also implies that $\mathsf{MITL}$ extended with $\mathsf{C}$ and $\mathsf{UT}$ modalities is elementarily decidable.

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arXiv2015

Revisiting Robustness in Priced Timed Games

Priced timed games are optimal-cost reachability games played between two players---the controller and the environment---by moving a token along the edges of infinite graphs of configurations of priced timed automata. The goal of the controller is to reach a given set of target locations as cheaply as possible, while the goal of the environment is the opposite. Priced timed games are known to be undecidable for timed automata with $3$ or more clocks, while they are known to be decidable for automata with $1$ clock. In an attempt to recover decidability for priced timed games Bouyer, Markey, and Sankur studied robust priced timed games where the environment has the power to slightly perturb delays proposed by the controller. Unfortunately, however, they showed that the natural problem of deciding the existence of optimal limit-strategy---optimal strategy of the controller where the perturbations tend to vanish in the limit---is undecidable with $10$ or more clocks. In this paper we revisit this problem and improve our understanding of the decidability of these games. We show that the limit-strategy problem is already undecidable for a subclass of robust priced timed games with $5$ or more clocks. On a positive side, we show the decidability of the existence of almost optimal strategies for the same subclass of one-clock robust priced timed games by adapting a classical construction by Bouyer at al. for one-clock priced timed games.

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arXiv2015

Hybrid Automata for Formal Modeling and Verification of Cyber-Physical Systems

The presence of a tight integration between the discrete control (the "cyber") and the analog environment (the "physical")---via sensors and actuators over wired or wireless communication networks---is the defining feature of cyber-physical systems. Hence, the functional correctness of a cyber- physical system is crucially dependent not only on the dynamics of the analog physical environment, but also on the decisions taken by the discrete control that alter the dynamics of the environment. The framework of Hybrid automata---introduced by Alur, Courcoubetis, Henzinger, and Ho---provides a formal modeling and specification environment to analyze the interaction between the discrete and continuous parts of a cyber-physical system. Hybrid automata can be considered as generalizations of finite state automata augmented with a finite set of real-valued variables whose dynamics in each state is governed by a system of ordinary differential equations. Moreover, the discrete transitions of hybrid automata are guarded by constraints over the values of these real-valued variables, and enable discontinuous jumps in the evolution of these variables. Considering the richness of the dynamics in a hybrid automaton, it is perhaps not surprising that the fundamental verification questions, like reachability and schedulability, for the general model are undecidable. In this article we present a review of hybrid automata as modeling and verification framework for cyber-physical systems, and survey some of the key results related to practical verification questions related to hybrid automata.

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arXiv2014

Bounded-Rate Multi-Mode Systems Based Motion Planning

Bounded-rate multi-mode systems are hybrid systems that can switch among a finite set of modes. Its dynamics is specified by a finite number of real-valued variables with mode-dependent rates that can vary within given bounded sets. Given an arbitrary piecewise linear trajectory, we study the problem of following the trajectory with arbitrary precision, using motion primitives given as bounded-rate multi-mode systems. We give an algorithm to solve the problem and show that the problem is co-NP complete. We further prove that the problem can be solved in polynomial time for multi-mode systems with fixed dimension. We study the problem with dwell-time requirement and show the decidability of the problem under certain positivity restriction on the rate vectors. Finally, we show that introducing structure to the multi-mode systems leads to undecidability, even when using only a single clock variable.

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arXiv2014

Improved Undecidability Results for Reachability Games on Recursive Timed Automata

We study reachability games on recursive timed automata (RTA) that generalize Alur-Dill timed automata with recursive procedure invocation mechanism similar to recursive state machines. It is known that deciding the winner in reachability games on RTA is undecidable for automata with two or more clocks, while the problem is decidable for automata with only one clock. Ouaknine and Worrell recently proposed a time-bounded theory of real-time verification by claiming that restriction to bounded-time recovers decidability for several key decision problem related to real-time verification. We revisited games on recursive timed automata with time-bounded restriction in the hope of recovering decidability. However, we found that the problem still remains undecidable for recursive timed automata with three or more clocks. Using similar proof techniques we characterize a decidability frontier for a generalization of RTA to recursive stopwatch automata.

cs.FL↗

arXiv2014

On The Reachability Problem for Recursive Hybrid Automata with One and Two Players

Motivated by the success of bounded model checking framework for finite state machines, Ouaknine and Worrell proposed a time-bounded theory of real-time verification by claiming that restriction to bounded-time recovers decidability for several key decision problem related to real-time verification. In support of this theory, the list of undecidable problems recently shown decidable under time-bounded restriction is rather impressive: language inclusion for timed automata, emptiness problem for alternating timed automata, and emptiness problem for rectangular hybrid automata. The objective of our study was to recover decidability for general recursive timed automata---and perhaps for recursive hybrid automata---under time-bounded restriction in order to provide an appealing verification framework for powerful modeling environments such as Stateflow/Simulink. Unfortunately, however, we answer this question in negative by showing that time-bounded reachability problem stays undecidable for recursive timed automata with five or more clocks. While the bad news continues even when one considers well-behaved subclasses of recursive hybrid automata, we recover decidability by considering recursive hybrid automata with bounded context using a pass-by-reference mechanism, or by restricting the number of variables to two, with rates in $\{0,1\}$.

cs.LO↗

arXiv2014

First-order definable string transformations

The connection between languages defined by computational models and logic for languages is well-studied. Monadic second-order logic and finite automata are shown to closely correspond to each-other for the languages of strings, trees, and partial-orders. Similar connections are shown for first-order logic and finite automata with certain aperiodicity restriction. Courcelle in 1994 proposed a way to use logic to define functions over structures where the output structure is defined using logical formulas interpreted over the input structure. Engelfriet and Hoogeboom discovered the corresponding "automata connection" by showing that two-way generalised sequential machines capture the class of monadic-second order definable transformations. Alur and Cerny further refined the result by proposing a one-way deterministic transducer model with string variables---called the streaming string transducers---to capture the same class of transformations. In this paper we establish a transducer-logic correspondence for Courcelle's first-order definable string transformations. We propose a new notion of transition monoid for streaming string transducers that involves structural properties of both underlying input automata and variable dependencies. By putting an aperiodicity restriction on the transition monoids, we define a class of streaming string transducers that captures exactly the class of first-order definable transformations.

cs.LO↗

arXiv2014

Weak Singular Hybrid Automata

The framework of Hybrid automata, introduced by Alur, Courcourbetis, Henzinger, and Ho, provides a formal modeling and analysis environment to analyze the interaction between the discrete and the continuous parts of cyber-physical systems. Hybrid automata can be considered as generalizations of finite state automata augmented with a finite set of real-valued variables whose dynamics in each state is governed by a system of ordinary differential equations. Moreover, the discrete transitions of hybrid automata are guarded by constraints over the values of these real-valued variables, and enable discontinuous jumps in the evolution of these variables. Singular hybrid automata are a subclass of hybrid automata where dynamics is specified by state-dependent constant vectors. Henzinger, Kopke, Puri, and Varaiya showed that for even very restricted subclasses of singular hybrid automata, the fundamental verification questions, like reachability and schedulability, are undecidable. In this paper we present \emph{weak singular hybrid automata} (WSHA), a previously unexplored subclass of singular hybrid automata, and show the decidability (and the exact complexity) of various verification questions for this class including reachability (NP-Complete) and LTL model-checking (PSPACE-Complete). We further show that extending WSHA with a single unrestricted clock or extending WSHA with unrestricted variable updates lead to undecidability of reachability problem.

cs.FL↗

arXiv2014

Partially Punctual Metric Temporal Logic is Decidable

Metric Temporal Logic $\mathsf{MTL}[\until_I,\since_I]$ is one of the most studied real time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints $I$. Henzinger et al., in their seminal paper showed that the non-punctual fragment of $\mathsf{MTL}$ called $\mathsf{MITL}$ is decidable. In this paper, we sharpen this decidability result by showing that the partially punctual fragment of $\mathsf{MTL}$ (denoted $\mathsf{PMTL}$) is decidable over strictly monotonic finite point wise time. In this fragment, we allow either punctual future modalities, or punctual past modalities, but never both together. We give two satisfiability preserving reductions from $\mathsf{PMTL}$ to the decidable logic $\mathsf{MTL}[\until_I]$. The first reduction uses simple projections, while the second reduction uses a novel technique of temporal projections with oversampling. We study the trade-off between the two reductions: while the second reduction allows the introduction of extra action points in the underlying model, the equisatisfiable $\mathsf{MTL}[\until_I]$ formula obtained is exponentially succinct than the one obtained via the first reduction, where no oversampling of the underlying model is needed. We also show that $\mathsf{PMTL}$ is strictly more expressive than the fragments $\mathsf{MTL}[\until_I,\since]$ and $\mathsf{MTL}[\until,\since_I]$.

cs.LO↗

arXiv2013

On the Decidability and Complexity of Some Fragments of Metric Temporal Logic

Metric Temporal Logic, $\mtlfull$ is amongst the most studied real-time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints $I$. \oomit{The classical results of Alur and Henzinger showed that $\mtlfull$ is undecidable where as $\mitl$ which uses only non-singular intervals $NS$ is decidable. In a surprizing result, Ouaknine and Worrell showed that the satisfiability of $\mtl$ is decidable over finite pointwise models, albeit with NPR decision complexity, whereas it remains undecidable for infinite pointwise models or for continuous time.} In this paper, we sharpen the decidability results by showing that the satisfiability of $\mtlsns$ (where $NS$ denotes non-singular intervals) is also decidable over finite pointwise strictly monotonic time. We give a satisfiability preserving reduction from the logic $\mtlsns$ to decidable logic $\mtl$ of Ouaknine and Worrell using the technique of temporal projections. We also investigate the decidability of unary fragment $\mtlfullunary$ (a question posed by A. Rabinovich) and show that $\mtlfut$ over continuous time as well as $\mtlfullunary$ over finite pointwise time are both undecidable. Moreover, $\mathsf{MTL}^{pw}[\fut_I]$ over finite pointwise models already has NPR lower bound for satisfiability checking. We also compare the expressive powers of some of these fragments using the technique of EF games for $\mathsf{MTL}$.

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arXiv2013

A Unifying Approach to Decide Relations for Timed Automata and their Game Characterization

In this paper we present a unifying approach for deciding various bisimulations, simulation equivalences and preorders between two timed automata states. We propose a zone based method for deciding these relations in which we eliminate an explicit product construction of the region graphs or the zone graphs as in the classical methods. Our method is also generic and can be used to decide several timed relations. We also present a game characterization for these timed relations and show that the game hierarchy reflects the hierarchy of the timed relations. One can obtain an infinite game hierarchy and thus the game characterization further indicates the possibility of defining new timed relations which have not been studied yet. The game characterization also helps us to come up with a formula which encodes the separation between two states that are not timed bisimilar. Such distinguishing formulae can also be generated for many relations other than timed bisimilarity.

cs.FL↗

arXiv2012

Compositional Verification of Evolving Software Product Lines

This paper presents a novel approach to the design verification of Software Product Lines(SPL). The proposed approach assumes that the requirements and designs are modeled as finite state machines with variability information. The variability information at the requirement and design levels are expressed differently and at different levels of abstraction. Also the proposed approach supports verification of SPL in which new features and variability may be added incrementally. Given the design and requirements of an SPL, the proposed design verification method ensures that every product at the design level behaviorally conforms to a product at the requirement level. The conformance procedure is compositional in the sense that the verification of an entire SPL consisting of multiple features is reduced to the verification of the individual features. The method has been implemented and demonstrated in a prototype tool SPLEnD (SPL Engine for Design Verification) on a couple of fairly large case studies.

cs.SE↗

arXiv2012

Game Characterizations of Timed Relations for Timed Automata Processes

In this work, we design the game semantics for timed equivalences and preorders of timed processes. The timed games corresponding to the various timed relations form a hierarchy. These games are similar to Stirling's bisimulation games. If it is the case that the existence of a winning strategy for the defender in a game ${\cal G}_1$ implies that there exists a winning strategy for the defender in another game ${\cal G}_2$, then the relation that corresponds to ${\cal G}_1$ is stronger than the relation corresponding to ${\cal G}_2$. The game hierarchy also throws light into several timed relations that are not considered in this paper.

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