arXiv subjects

James Worrell

Publications and source records attributed to James Worrell.

104 records · Page 6Linked to original sources

arXiv2013

Controlling the Depth, Size, and Number of Subtrees for Two-variable Logic on Trees

Verification of properties of first order logic with two variables FO2 has been investigated in a number of contexts. Over arbitrary structures it is known to be decidable with NEXPTIME complexity, with finitely satisfiable formulas having exponential-sized models. Over word structures, where FO2 is known to have the same expressiveness as unary temporal logic, the same properties hold. Over finite labelled ordered trees FO2 is also of interest: it is known to have the same expressiveness as navigational XPath, a common query language for XML documents. Prior work on XPath and FO2 gives a 2EXPTIME bound for satisfiability of FO2. In this work we give the first in-depth look at the complexity of FO2 on trees, and on the size and depth of models. We show that the doubly-exponential bound is not tight, and neither do the NEXPTIME-completeness results from the word case carry over: the exact complexity varies depending on the vocabulary used, the presence or absence of a schema, and the encoding used for labels. Our results depend on an analysis of subformula types in models of FO2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to finite satisfiability for FO2 sentences over trees.

cs.LO↗

arXiv2013

Two Variable vs. Linear Temporal Logic in Model Checking and Games

Model checking linear-time properties expressed in first-order logic has non-elementary complexity, and thus various restricted logical languages are employed. In this paper we consider two such restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive but FO2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO2; we also consider the stutter-free fragment of FO2, obtained by omitting the successor relation, and the expressively equivalent fragment of UTL, obtained by omitting the next and previous connectives. We give three logic-to-automata translations which can be used to give upper bounds for FO2 and UTL and various sublogics. We apply these to get new bounds for both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple these with matching lower-bound arguments. Next, we look at combining FO2 verification techniques with those for LTL. We present here a language that subsumes both FO2 and LTL, and inherits the model checking properties of both languages. Our results give both a unified approach to understanding the behaviour of FO2 and LTL, along with a nearly comprehensive picture of the complexity of verification for these logics and their sublogics.

cs.LO↗

arXiv2013

Revisiting the Equivalence Problem for Finite Multitape Automata

The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhumäki in the early 1990s. Their proof of decidability yields a co_NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumaki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding language equivalence of deterministic multitape automata and, more generally, multiplicity equivalence of nondeterministic multitape automata. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ.

cs.FL↗

arXiv2013

On the Complexity of Equivalence and Minimisation for Q-weighted Automata

This paper is concerned with the computational complexity of equivalence and minimisation for automata with transition weights in the field Q of rational numbers. We use polynomial identity testing and the Isolation Lemma to obtain complexity bounds, focussing on the class NC of problems within P solvable in polylogarithmic parallel time. For finite Q-weighted automata, we give a randomised NC procedure that either outputs that two automata are equivalent or returns a word on which they differ. We also give an NC procedure for deciding whether a given automaton is minimal, as well as a randomised NC procedure that minimises an automaton. We consider probabilistic automata with rewards, similar to Markov Decision Processes. For these automata we consider two notions of equivalence: expectation equivalence and distribution equivalence. The former requires that two automata have the same expected reward on each input word, while the latter requires that each input word induce the same distribution on rewards in each automaton. For both notions we give algorithms for deciding equivalence by reduction to equivalence of Q-weighted automata. Finally we show that the equivalence problem for Q-weighted visibly pushdown automata is logspace equivalent to the polynomial identity testing problem.

cs.FL↗

arXiv2013

Expressive Completeness of Metric Temporal Logic

Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. A seminal result of Hirshfeld and Rabinovich shows that over the reals, first-order logic with binary order relation < and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<,+1). In this paper we show the surprising result that if we allow unary functions +q, (q rational), in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamp's theorem on the expressive completeness of LTL for FO(<) to the quantitative setting. The proof of this result involves a generalisation of Gabbay's notion of separation.

cs.LO↗

arXiv2013

When is Metric Temporal Logic Expressively Complete?

A seminal result of Kamp is that over the reals Linear Temporal Logic (LTL) has the same expressive power as first-order logic with binary order relation < and monadic predicates. A key question is whether there exists an analogue of Kamp's theorem for Metric Temporal Logic (MTL) -- a generalization of LTL in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich gave a negative answer, showing that first-order logic with binary order relation < and unary function +1 is strictly more expressive than MTL with integer constants. However, a recent result of Hunter, Ouaknine and Worrell shows that with rational timing constants, MTL has the same expressive power as first-order logic, giving a positive answer. In this paper we generalize these results by giving a precise characterization of those sets of constants for which MTL and first-order logic have the same expressive power. We also show that full first-order expressiveness can be recovered with the addition of counting modalities, strongly supporting the assertion of Hirshfeld and Rabinovich that Q2MLO is one of the most expressive decidable fragments of FO(<,+1).

cs.LO↗

arXiv2012

Time-bounded Reachability for Hybrid Automata: Complexity and Fixpoints

In this paper, we study thetime-bounded reachability problem for rectangular hybrid automata with non-negative rates (RHA+). This problem was recently shown to be decidable [Brihaye et al, ICALP11] (even though the unbounded reachability problem for even very simple classes of hybrid automata is well-known to be undecidable). However, [Brihaye et al, ICALP11] does not provide a precise characterisation of the complexity of the time-bounded reachability problem. The contribution of the present paper is threefold. First, we provide a new NExpTime algorithm to solve the timed-bounded reachability problem on RHA+. This algorithm improves on the one of [Brihaye et al, ICALP11] by at least one exponential. Second, we show that this new algorithm is optimal, by establishing a matching lower bound: time-bounded reachability for RHA+ is therefore NExpTime-complete. Third, we extend these results in a practical direction, by showing that we can effectively compute fixpoints that characterise the sets of states that are reachable (resp. co-reachable) within T time units from a given starting state.

cs.LO↗

arXiv2012

Bisimilarity of Probabilistic Pushdown Automata

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). Our first contribution is a general construction that reduces checking bisimilarity of probabilistic transition systems to checking bisimilarity of non-deterministic transition systems. This construction directly yields decidability of bisimilarity for pPDA, as well as an elementary upper bound for the bisimilarity problem on the subclass of probabilistic basic process algebras, i.e., single-state pPDA. We further show that, with careful analysis, the general reduction can be used to prove an EXPTIME upper bound for bisimilarity of probabilistic visibly pushdown automata. Here we also provide a matching lower bound, establishing EXPTIME-completeness. Finally we prove that deciding bisimilarity of probabilistic one-counter automata, another subclass of pPDA, is PSPACE-complete. Here we use a more specialised argument to obtain optimal complexity bounds.

cs.FL↗

arXiv2012

On the Complexity of the Equivalence Problem for Probabilistic Automata

Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we show how to check for equivalence two probabilistic automata with (cumulative) rewards. Our algorithm runs in deterministic polynomial time, if the number of reward counters is fixed. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the Arithmetic Circuit Identity Testing problem, which is to decide whether a polynomial represented by an arithmetic circuit is identically zero.

cs.FL↗

arXiv2011

On Reachability for Hybrid Automata over Bounded Time

This paper investigates the time-bounded version of the reachability problem for hybrid automata. This problem asks whether a given hybrid automaton can reach a given target location within T time units, where T is a constant rational value. We show that, in contrast to the classical (unbounded) reachability problem, the timed-bounded version is decidable for rectangular hybrid automata provided only non-negative rates are allowed. This class of systems is of practical interest and subsumes, among others, the class of stopwatch automata. We also show that the problem becomes undecidable if either diagonal constraints or both negative and positive rates are allowed.

cs.LO↗

arXiv2011

On Stabilization in Herman's Algorithm

Herman's algorithm is a synchronous randomized protocol for achieving self-stabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of the initial configuration. It is straightforward that the algorithm achieves stabilization almost surely from any initial configuration, and it is known that the worst-case expected time to stabilization (with respect to the initial configuration) is Theta(N^2). Our first contribution is to give an upper bound of 0.64 N^2 on the expected stabilization time, improving on previous upper bounds and reducing the gap with the best existing lower bound. We also introduce an asynchronous version of the protocol, showing a similar O(N^2) convergence bound in this case. Assuming that errors arise from the corruption of some number k of bits, where k is fixed independently of the size of the ring, we show that the expected time to stabilization is O(N). This reveals a hitherto unknown and highly desirable property of Herman's algorithm: it recovers quickly from bounded errors. We also show that if the initial configuration arises by resetting each bit independently and uniformly at random, then stabilization is significantly faster than in the worst case.

cs.DS↗

arXiv2008

Approximating a Behavioural Pseudometric without Discount for Probabilistic Systems

Desharnais, Gupta, Jagadeesan and Panangaden introduced a family of behavioural pseudometrics for probabilistic transition systems. These pseudometrics are a quantitative analogue of probabilistic bisimilarity. Distance zero captures probabilistic bisimilarity. Each pseudometric has a discount factor, a real number in the interval (0, 1]. The smaller the discount factor, the more the future is discounted. If the discount factor is one, then the future is not discounted at all. Desharnais et al. showed that the behavioural distances can be calculated up to any desired degree of accuracy if the discount factor is smaller than one. In this paper, we show that the distances can also be approximated if the future is not discounted. A key ingredient of our algorithm is Tarski's decision procedure for the first order theory over real closed fields. By exploiting the Kantorovich-Rubinstein duality theorem we can restrict to the existential fragment for which more efficient decision procedures exist.

cs.LO↗

arXiv2008

On Termination for Faulty Channel Machines

A channel machine consists of a finite controller together with several fifo channels; the controller can read messages from the head of a channel and write messages to the tail of a channel. In this paper, we focus on channel machines with insertion errors, i.e., machines in whose channels messages can spontaneously appear. Such devices have been previously introduced in the study of Metric Temporal Logic. We consider the termination problem: are all the computations of a given insertion channel machine finite? We show that this problem has non-elementary, yet primitive recursive complexity.

cs.IT↗

arXiv2007

On the decidability and complexity of Metric Temporal Logic over finite words

Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL--which includes invariance and time-bounded response properties--is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature.

cs.LO↗