Alternation

Chandra, Ashok K.; Kozen, Dexter; Stockmeyer, Larry J.

doi:10.1145/322234.322243

Cited by 1,290 publications

(749 citation statements)

References 9 publications

Supporting

Mentioning

732

Contrasting

Unclassified

Order By: Relevance

“…2.12 there is a (k − 1)AExpSpace machine M s.t. L(M) is kExpTime-hard [4]. Using padding we can assume the space required by M on an input word of length n to be bounded by F k−2 (p(n)) for some polynomial p(n) > n. Thm.…”

Section: The Lower Boundmentioning

confidence: 99%

The Complexity of Model Checking Higher-Order Fixpoint Logic

Axelsson¹,

Lange²,

Somla³

2007

Logical Methods in Computer Science

View full text Add to dashboard Cite

Abstract. Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed λ-calculus and the modal µ-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal µ-calculus.This paper provides complexity results for its model checking problem. In particular, we consider those fragments of HFL that are built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment.The lower bound is established by a reduction from the word problem for alternating (k − 1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.

show abstract

Section: The Lower Boundmentioning

confidence: 99%

The Complexity of Model Checking Higher-Order Fixpoint Logic

Axelsson¹,

Lange²,

Somla³

2007

Logical Methods in Computer Science

View full text Add to dashboard Cite

show abstract

“…A word w is accepted by an ATM M if there exists a run tree of M on w whose all leaf nodes are accepting configurations (see [3] for details). The AND-OR graph of the polynomial space ATM (M, p) on the input word w ∈ Σ * is G(M, p) = S ∨ , S ∧ , s 0 , ⇒, R where…”

Section: Lower Boundsmentioning

confidence: 99%

“…The membership problem is to decide if a given word w is accepted by a given polynomial space ATM (M, p). This problem is known to be ExpTime-hard [3].…”

Section: Lower Boundsmentioning

confidence: 99%

Algorithms for Omega-Regular Games with Imperfect Information

Chatterjee¹,

Doyen²,

Henzinger³

et al. 2007

Logical Methods in Computer Science

View full text Add to dashboard Cite

Abstract. We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixed point is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Büchi objective. This set is of interest because in the absence of perfect information, randomized strategies are more powerful than deterministic ones. We show that our algorithms are optimal by proving matching lower bounds.

show abstract

“…An alternating TM M is a natural extension of the nondeterministic T M introduced in [45]. We observe that existential states of an alternating T M has the same meaning as states of nondeterministic T Ms. One has to choose one of the possible actions, and one accepts if at least one of these possibilities leads to acceptance.…”

Section: Parallel Computation Thesis and Alternationmentioning

confidence: 99%