Journal of the ACM (JACM)

Jančar, Petr; Srba, Jivří

doi: 10.1145/1326554.1326559pmid: N/A

Stirling 1996, 1998 proved the decidability of bisimilarity on so-called normed pushdown processes. This result was substantially extended by Sénizergues 1998, 2005 who showed the decidability of bisimilarity for regular (or equational) graphs of finite out-degree; this essentially coincides with weak bisimilarity of processes generated by (unnormed) pushdown automata where the ε -transitions can only deterministically pop the stack. The question of decidability of bisimilarity for the more general class of so called Type -1 systems, which is equivalent to weak bisimilarity on unrestricted ε -popping pushdown processes, was left open. This was repeatedly indicated by both Stirling and Sénizergues. Here we answer the question negatively, that is, we show the undecidability of bisimilarity on Type -1 systems, even in the normed case. We achieve the result by applying a technique we call Defender's Forcing, referring to the bisimulation games. The idea is simple, yet powerful. We demonstrate its versatility by deriving further results in a uniform way. First, we classify several versions of the undecidable problems for prefix rewrite systems (or pushdown automata) as Π 0 1 -complete or Σ 1 1 -complete. Second, we solve the decidability question for weak bisimilarity on PA (Process Algebra) processes, showing that the problem is undecidable and even Σ 1 1 -complete. Third, we show Σ 1 1 -completeness of weak bisimilarity for so-called parallel pushdown (or multiset) automata, a subclass of (labeled, place/transition) Petri nets.

Halpern, Joseph Y.; Weissman, Vicky

doi: 10.1145/1326554.1326558pmid: N/A

XrML is becoming a popular language in industry for writing software licenses. The semantics for XrML is implicitly given by an algorithm that determines if a permission follows from a set of licenses. We focus on a fragment of the language and use it to highlight some problematic aspects of the algorithm. We then correct the problems, introduce formal semantics, and show that our semantics captures the (corrected) algorithm. Next, we consider the complexity of determining if a permission is implied by a set of XrML licenses. We prove that the general problem is undecidable, but it is polynomial-time computable for an expressive fragment of the language. We extend XrML to capture a wider range of licenses by adding negation to the language. Finally, we discuss the key differences between XrML and MPEG-21, an international standard based on XrML.

Flesca, S.; Furfaro, F.; Masciari, E.

doi: 10.1145/1326554.1326556pmid: N/A

XPath expressions define navigational queries on XML data and are issued on XML documents to select sets of element nodes. Due to the wide use of XPath, which is embedded into several languages for querying and manipulating XML data, the problem of efficiently answering XPath queries has received increasing attention from the research community. As the efficiency of computing the answer of an XPath query depends on its size, replacing XPath expressions with equivalent ones having the smallest size is a crucial issue in this direction. This article investigates the minimization problem for a wide fragment of XPath (namely X P ✶ ), where the use of the most common operators (child, descendant, wildcard and branching) is allowed with some syntactic restrictions. The examined fragment consists of expressions which have not been specifically studied in the relational setting before: neither are they mere conjunctive queries (as the combination of “//” and “*” enables an implicit form of disjunction to be expressed) nor do they coincide with disjunctive ones (as the latter are more expressive). Three main contributions are provided. The “global minimality” property is shown to hold: the minimization of a given XPath expression can be accomplished by removing pieces of the expression, without having to re-formulate it (as for “general” disjunctive queries). Then, the complexity of the minimization problem is characterized, showing that it is the same as the containment problem. Finally, specific forms of XPath expressions are identified, which can be minimized in polynomial time.

Yekhanin, Sergey

doi: 10.1145/1326554.1326555pmid: N/A

A q -query Locally Decodable Code (LDC) encodes an n -bit message x as an N -bit codeword C ( x ), such that one can probabilistically recover any bit x i of the message by querying only q bits of the codeword C ( x ), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2 t − 1, we design three query LDCs of length N = exp( O ( n 1/ t )), for every n . Based on the largest known Mersenne prime, this translates to a length of less than exp( O ( n 10 − 7 )) compared to exp( O ( n 1/2 )) in the previous constructions. It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp( n O (1/log log n ) ) for infinitely many n . We also obtain analogous improvements for Private Information Retrieval (PIR) schemes. We give 3-server PIR schemes with communication complexity of O ( n 10 − 7 ) to access an n -bit database, compared to the previous best scheme with complexity O ( n 1/5.25 ). Assuming again that there are infinitely many Mersenne primes, we get 3-server PIR schemes of communication complexity n O (1/log log n ) ) for infinitely many n . Previous families of LDCs and PIR schemes were based on the properties of low-degree multivariate polynomials over finite fields. Our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set.

Dinitz, Yefim; Moran, Shlomo; Rajsbaum, Sergio

doi: 10.1145/1326554.1326557pmid: N/A

We consider a failure-free, asynchronous message passing network with n links, where the processors are arranged on a ring or a chain. The processors are identically programmed but have distinct identities, taken from {0, 1,… , M − 1}. We investigate the communication costs of three well studied tasks: Consensus, Leader, and MaxF (finding the maximum identity). We show that in chain and ring topologies, the message complexities of all three tasks are the same. Hence, we study a finer measure of complexity: the number of transmitted bits required to solve a task T , denoted BitC ( T ). We prove several new lower bounds (and some simple upper bounds) that imply the following results: For the two processors case, BitC (Consensus) = 2 and BitC (Leader) = BitC (MaxF) = 2log 2 M ± O (1), where the gap between the lower and upper bounds is almost always 1. For a chain, BitC (Consensus) = Θ( n ), BitC (Leader) = Θ( n + log M ), and BitC (MaxF) = Θ( n log M ). For the ring topology, we prove the lower bound of Ω( n log M ) for Leader, and (hence) MaxF. We consider also a chain where the intermediate processors have no identities. We prove that BitC (Leader) = Θ( n log M ), which is equal to n times the bit complexity of the problem for two processors. For the specific case when the chain length is even, we prove that BitC (Leader) = Θ( n ), for both above settings. In addition, we show that for any algorithm solving MaxF, there exists an input, for which every execution has the bit complexity Ω( n log M ) (this is not the case for Leader). In our proofs, we use both methods of distributed computing and of communication complexity theory, establishing new links between the two areas.