The reachability problem for vector addition systems with a stack is - PDF document

The reachability problem for vector addition systems with a stack is not elementary Ranko Lazi´ c DIMAP, Department of Computer Science, University of Warwick, UK Abstract Whereas computations of VAS Branching VAS. are words of vectors of natural numbers, BVAS are By adapting the iterative yardstick construction of a natural generalisation whose computations are Stockmeyer, we show that the reachability problem trees of such vectors. Although their reachabil- for vector addition systems with a stack does not ity problem has been shown inter-reducible with have elementary complexity. As a corollary, the the emptiness problem for multiple-valued linear same lower bound holds for the satisfiability prob- index grammars [15, 17], and with the provability lem for a two-variable first-order logic on trees in problem for multiplicative exponential linear logic which unbounded data may label only leaf nodes. [6], the decidability status remains an open ques- Whether the two problems are decidable remains tion. However, curiously, a lower bound that is an open question. two notches above adding alternation to Lipton’s result, namely 2 ExpSpace -hardness, was recently shown [10]. 1 Introduction Priority VAS. Equipping two counters (in Petri- Before presenting details of this small contribu- speak, places) with zero tests, of course, makes VAS tion to the on-going investigation of complexity- as powerful as Minsky machines and the reachabil- theoretic properties of vector addition systems, ity problem undecidable. It has turned out, though, their extensions and related logics on words and that the Mayr-Kosaraju-Lambert proof can be ex- trees with unbounded data, we provide a brief tended when only one counter may be tested for overview of most-closely related research. A dia- zero. In fact, Reinhardt has obtained a highly non- grammatic summary is in Figure 1, where boldface trivial proof of an even more general result: that and a thicker line indicate the new results. reachability is decidable for PVAS , where one may test whether all counters from any one of a series of VAS. Vector addition systems, or equivalently sets C 1 ⊆ C 2 ⊆ · · · C k are zero [16]. So far with Petri nets, are a fundamental and established model one zero-testable counter, Bonnet has succeeded of concurrency. They support an active and broad in greatly simplifying Reinhardt’s proof along the research community, with long-standing links to in- lines of Leroux [5]. dustry, where VAS are an important modelling for- Let us say that PVAS whose series of zero- malism and associated tools are extensively used. testable sets of counters have length k are of in- In spite of several decades of research, the com- dex k . putational complexity of the reachability problem for VAS remains one of the most well-known open questions in theoretical computer science. While Another natural extension of VAS is Stack VAS. decidability was still unknown, Lipton made initial to allow them to use a stack over a finite alphabet. progress on the problem by showing it ExpSpace - Equivalently to these systems, which we call SVAS hard [12], which is still the highest known lower and whose motivations include modelling software bound. A few years later, Mayr showed the prob- with integer variables and call-return procedures, lem decidable [13]. Although his proof was sub- one may consider intersections of VAS languages sequently substantially simplified by Kosaraju [8], and context-free languages. For an SVAS in that al- Lambert [9] and in a remarkable recent series of ternative presentation, let us say that it is of index articles by Leroux [11], it is still unknown even k if and only if the context-free language is of index whether there exists a primitive recursive algorithm k , i.e. there is a context-free grammar such that ev- for the problem. ery word in the language has a derivation whose ev- 1

ery step contains at most k non-terminal symbols. in the pre-traversal (“document order”). Atig and Ganty have recently shown that finite- On data words, Boja´ nczyk et al. [3] showed that index SVAS are essentially equivalent to PVAS: ev- the satisfiability problem for such a logic reduces in ery index- k SVAS can be simulated by an index- k doubly-exponential time to the reachability prob- PVAS, and every index- k PVAS can be simulated lem for VAS, and is therefore decidable. Moreover, by an index-( k + 1) SVAS [1]. Incidentally, that they exhibited a polynomial-time converse reduc- seems to be the only interesting known relationship tion, and so Lipton’s lower bound carries over to among BVAS, PVAS and SVAS. the logic. The reachability problem for finite-index SVAS On data trees, the picture is more complicated. is consequently decidable since it is decidable for Already without document order, Boja´ nczyk et al. PVAS. Although decidability for unrestricted SVAS [4] observed that the satisfiability problem is at remains an open question, we make some progress least as hard as the reachability problem for BVAS here in the opposite direction, obtaining that the (whose decidability is open), but obtained decid- ability by disallowing also the transitive naviga- problem is not elementary. That puts SVAS in tional predicates (“descendant” and “following sib- contrast to BVAS, for which decidability is also un- ling”). Another way of getting decidability was known but so far there is only an elementary lower found by Bj¨ orklund and Boja´ nczyk [2]: no restric- bound [10]. tions on the navigational predicates are required provided the depth of data trees is bounded. With The well-known coverability prob- Coverability. that assumption, they showed how to reduce satis- lem for VAS and their extensions corresponds to fiability to the reachability problem for PVAS. “control-state reachability”: it asks whether a given An alternative restriction on data trees suggests system can reach a configuration that is point- itself: that data labels be allowed only on leaf wise (i.e., for each counter) greater than or equal nodes. Although decidability of the full 2-variable to a given configuration. Lipton’s and Rackoff’s FO on such structures remains open, we show that classical results show that coverability for VAS is even without the “descendant” and “following sib- ExpSpace -complete [12, 14], and by building on ling” predicates, satisfiability is at least as hard as those works, Demri et al. have shown 2 ExpTime - the reachability problem for SVAS, and so is not completeness of the problem for BVAS [7]. elementary. Unfortunately, for PVAS and SVAS, there is no hope for such results, since for both classes of sys- tems, there are straightforward reductions of reach- 2 Lower bound ability to coverability. It is convenient for our purposes to formalise SVAS as programs which operate on non-negative coun- 2-variable FO on data words and data trees. ters and a finite-alphabet stack. More precisely, we Partly motivated by verification of concurrent sys- define them as finite sequences of commands which tems and by querying of XML databases, in recent may be labelled, where a command is one of: an years there has been extensive research in logics increment of a counter ( x := x + 1), a decrement on data words and data trees. In addition to let- of a counter ( x := x − 1), a push ( push a ), a pop ters from a finite alphabet as classically, the lat- ( pop a ), a non-deterministic jump to one of two la- ter structures have labels from an infinite domain, belled commands ( goto L or L ′ ), or termination which are called data and on which only certain op- ( halt ). Initially, all counters have value 0 and the erations are available. In fact, typically, the data stack is empty. Whenever a decrement of a counter can only be compared for equality, and that is the with value 0 or an erroneous pop is attempted, the only operation we consider here. program aborts. In every program, halt occurs Remarkably, there are several connections be- only as the last command. tween, on one hand, VAS and their extensions that The reachability problem can now be stated as we have introduced, and on the other hand, two- follows: given an SVAS, does it have a computation variable first-order logics on data words and data which reaches the halt command with all counters trees. For positions x and y of a data word, the being 0 and the stack being empty? logics have navigational predicates y = x + 1 and x < y , as well as equality of data labels x ∼ y . On Theorem 1 The reachability problem for SVAS is data trees, where variables range over nodes, navi- not elementary. gational predicates are either vertical (“child” and “descendant”), or horizontal (“next sibling” and The proof is by reducing from the (2 ⇑ n )- “following sibling”), or compare nodes for positions bounded halting problem for counter programs with 2