A note on the space complexity of some decision problems for finite automata

Jiang, Tao; Ravikumar, Bala

doi:10.1016/s0020-0190(05)80006-7

Cited by 52 publications

(34 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Our construction can, for example, be used in hardware-based monitors to reduce the time needed to evaluate a block of path positions from linear to just logarithmic. The LTL path checking problem is closely related to the membership problems for the various types of regular expressions: the membership problem is in NL for regular expressions [14], in logCFL for semi-extended regular expressions [20], and P-complete for star-free regular expressions and extended regular expressions [19]. Of particular interest is the comparison to the star-free regular expressions, since they have the same expressive power as LTL on finite paths [16].…”

Section: Discussionmentioning

confidence: 99%

LTL Path Checking Is Efficiently Parallelizable

Kuhtz

Finkbeiner

2009

Automata, Languages and Programming

View full text Add to dashboard Cite

Abstract. We present an AC 1 (logDCFL) algorithm for checking LTL formulas over finite paths, thus establishing that the problem can be efficiently parallelized. Our construction provides a foundation for the parallelization of various applications in monitoring, testing, and verification.Linear-time temporal logic (LTL) is the standard specification language to describe properties of reactive computation paths. The problem of checking whether a given finite path satisfies an LTL formula plays a key role in monitoring and runtime verification [12,10,6,1,4], where individual paths are checked either online, during the execution of the system, or offline, for example based on an error report. Similarly, path checking occurs in testing [2] and in several static verification techniques, notably in Monte-Carlo-based probabilistic verification, where large numbers of randomly generated sample paths are analyzed [22].Somewhat surprisingly, given the widespread use of LTL, the complexity of the path checking problem is still open [18]. The established upper bound is P: The algorithms in the literature traverse the path sequentially (cf. [10,18,12]); by going backwards from the end of the path, one can ensure that, in each step, the value of each subformula is updated in constant time, which results in bilinear running time. The only known lower bound is NC 1 [8], the complexity of evaluating Boolean expressions. The large gap between the bounds is especially unsatisfying in light of the recent trend to implement path checking algorithms in hardware, which is inherently parallel. For example, the IEEE standard temporal logic PSL [13], an extension of LTL, has become part of the hardware description language VHDL, and several tools [6,4] are available to synthesize hardware-based monitors from assertions written in PSL. Can we improve over the sequential approach by evaluating entire blocks of path positions in parallel?In this paper, we show that LTL path checking can indeed be parallelized efficiently. Our approach is inspired by work in the related area of evaluating monotone Boolean circuits [11,9,15,3,17,5]. Rather than sequentially traversing the path, we consider the circuit that results from unrolling the formula over the path. Figure 1 shows such a circuit for the formula ((a U b) U (c U d)) U e and a path of length 5.

show abstract

Section: Discussionmentioning

confidence: 99%

LTL Path Checking Is Efficiently Parallelizable

Kuhtz

Finkbeiner

2009

Automata, Languages and Programming

View full text Add to dashboard Cite

show abstract

“…The different classes of regular expressions considered here have been well studied; in particular, RE(∩) and its membership [21,23,29] and equivalence and emptiness [10,28,30] problems. Also the classes RE(#) [22,27] and RE(&) [12,24] have received interest.…”

Section: Re(#)mentioning

confidence: 99%

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

Gelade

Lecture Notes in Computer Science

View full text Add to dashboard Cite

a b s t r a c tIn this paper, we study the succinctness of regular expressions (REs) extended with interleaving, intersection and counting operators. We show that in a translation from REs with interleaving to standard regular expressions a double exponential size increase cannot be avoided. We also consider the complexity of translations to finite automata. We give a tight exponential lower bound on the translation of REs with intersection to NFAs, and, for each of the three classes of REs, we show that in a translation to a DFA a double exponential size increase cannot be avoided. Together with known results, this gives a complete picture of the complexity of translating REs extended with interleaving, intersection or counting into (standard) regular expressions, NFAs, and DFAs.

show abstract

“…For regular expressions with intersection (RE ∩ ) (or semi-extended), several computational complexity decision problems, such as membership, equivalence and emptiness, were studied by various authors. Petersen [21] has shown that the membership problem is LOGCFL-complete, while for standard regular expressions (RE) it is NL-complete [19]. Fürer [14] has proved that inequivalence and non-empty complement are EXPSPACE-complete, which contrasts with the PSPACE-completeness of these problems for RE.…”

Section: Introductionmentioning

confidence: 99%

On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection

Bastos

Broda

Machiavelo

et al. 2016

Descriptional Complexity of Formal Systems

View full text Add to dashboard Cite

Abstract. Extended regular expressions (with complement and intersection) are used in many applications due to their succinctness. In particular, regular expressions extended with intersection only (also called semi-extended) can already be exponentially smaller than standard regular expressions or equivalent nondeterministic finite automata (NFA). For practical purposes it is important to study the average behaviour of conversions between these models. In this paper, we focus on the conversion of regular expressions with intersection to nondeterministic finite automata, using partial derivatives and the notion of support. First, we give a tight upper bound of 2 O(n) for the worst-case number of states of the resulting partial derivative automaton, where n is the size of the expression. Using the framework of analytic combinatorics, we then establish an upper bound of (1.056+o(1)) n for its asymptotic average-state complexity, which is significantly smaller than the one for the worst case.

show abstract

A note on the space complexity of some decision problems for finite automata

Cited by 52 publications

References 9 publications

LTL Path Checking Is Efficiently Parallelizable

LTL Path Checking Is Efficiently Parallelizable

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection

Contact Info

Product

Resources

About