Alternating and empty alternating auxiliary stack automata

AbstractWe consider variants of alternating auxiliary stack automata and characterize their computational power when the number of alternations is bounded by a constant or unlimited. In this way we get new characterizations of NP, the polynomial hierarchy, PSpace, and bounded query classes like co-DP=NL〈NP[1]〉 and Θ2P=PNP[O(logn)], in a uniform framework

Reachability in Higher-Order-Counters

Higher-order counter automata (\HOCS) can be either seen as a restriction of higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of \HOCS: those that can test whether the topmost counter value is zero and those which cannot. We show that control-state reachability for level $k$ \HOCS with $0$-test is complete for \mbox{$(k-2)$}-fold exponential space; leaving out the $0$-test leads to completeness for \mbox{$(k-2)$}-fold exponential time. Restricting \HOCS (without $0$-test) to level $2$, we prove that global (forward or backward) reachability analysis is \PTIME-complete. This enhances the known result for pushdown systems which are subsumed by level $2$ \HOCS without $0$-test. We transfer our results to the formal language setting. Assuming that \PTIME \subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet's constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201

Tree-size bounded alternation

AbstractThe size of an accepting computation tree of an alternating Turing machine (ATM) is introduced as a complexity measure. We present a number of applications of tree-size to the study of more traditional complexity classes. Tree-size on ATMs is shown to closely correspond to time on nondeterministic TMs and on nondeterministic auxiliary pushdown automata. One application of the later is a useful new characterization of the class of languages log-space-reducible to context-free languages. Surprising relationships with parallel-time complexity are also demonstrated. ATM computations using at most space S(n) and tree-size Z(n) (simultaneously) can be simulated in alternating space S(n) and time S(n) · log Z(n) (simultaneously). Several well-known simulations, e.g., Savitch's theorem, are special cases of this result. It also leads to improved parallel complexity bounds for many problems in terms of both time and number of “processors.” As one example we show that context-free language recognition in time O(log2 n) is possible on several parallel models. Further, this bound is achievable with only a polynomial number of processors, in contrast to all previously known sub-linear time CFL recognizers

Game Characterization of Probabilistic Bisimilarity, and Applications to 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). We first show a general characterization of probabilistic bisimilarity in terms of two-player games, which naturally reduces checking bisimilarity of probabilistic labelled transition systems to checking bisimilarity of standard (non-deterministic) labelled transition systems. This reduction can be easily implemented in the framework of pPDA, allowing to use known results for standard (non-probabilistic) PDA and their subclasses. A direct use of the reduction incurs an exponential increase of complexity, which does not matter in deriving decidability of bisimilarity for pPDA due to the non-elementary complexity of the problem. In the cases of probabilistic one-counter automata (pOCA), of probabilistic visibly pushdown automata (pvPDA), and of probabilistic basic process algebras (i.e., single-state pPDA) we show that an implicit use of the reduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and 2-EXPTIME upper bounds, respectively, like for the respective non-probabilistic versions. The bisimilarity problems for OCA and vPDA are known to have matching lower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively); we show that these lower bounds also hold for fully probabilistic versions that do not use non-determinism

Model Checking Probabilistic Pushdown Automata

We consider the model checking problem for probabilistic pushdown automata (pPDA) and properties expressible in various probabilistic logics. We start with properties that can be formulated as instances of a generalized random walk problem. We prove that both qualitative and quantitative model checking for this class of properties and pPDA is decidable. Then we show that model checking for the qualitative fragment of the logic PCTL and pPDA is also decidable. Moreover, we develop an error-tolerant model checking algorithm for PCTL and the subclass of stateless pPDA. Finally, we consider the class of omega-regular properties and show that both qualitative and quantitative model checking for pPDA is decidable

Unary Pushdown Automata and Straight-Line Programs

We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs---one for the prefix, one for the lasso---that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNP-hardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is $\Pi_2 \mathrm P$-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply $\Pi_2 \mathrm P$-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards

Visibly Linear Dynamic Logic

We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the $\omega$-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. The main technical contribution of this work is a translation of VLDL into $\omega$-visibly pushdown automata of exponential size via one-way alternating jumping automata. This translation yields exponential-time algorithms for satisfiability, validity, and model checking. We also show that visibly pushdown games with VLDL winning conditions are solvable in triply-exponential time. We prove all these problems to be complete for their respective complexity classes.Comment: 25 Page