Separation of complexity classes in Koiran's weak model

AbstractWe continue the study of complexity classes over the weak model introduced by P. Koiran. In particular we provide several separations of complexity classes, the most remarkable being the strict inclusion of P in NP. Other separations concern classes defined by weak polynomial time over parallel or alternating machines as well as over nondeterministic machines whose guesses are required to be 0 or 1

Alternating on-line turing machines with only universal states and small space bounds

AbstractLet L[AONTM(L(n))] be the class of sets accepted by L(n) space bounded alternating on-line Turing machines, and L[UONTM(L(n))] be the class of sets accepted by L(n) space bounded alternating on-line Turing machines with only universal states. This note first shows that, for any L(n) such that L(n) ⩾ log log n and limn → ∞[L(n)/log n] = 0, (i) L[UONTM(L(n))] ⊋ L[AONTM(L(n))], (ii) L[UONTM(L(n))] is not closed under complementation, and (iii) L[UONTM(L(n))] is properly contained in the class of sets accepted by L(n) space bounded alternating Turing machines with only universal states. We then show that there exists an infinite hierarchy among L[UONTM(L(n))]'s with log log n ⩽ L(n) ⩽ log n

Processing Succinct Matrices and Vectors

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of CSR 201

The First-Order Theory of Ground Tree Rewrite Graphs

We prove that the complexity of the uniform first-order theory of ground tree rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower bound, we show that there is some fixed ground tree rewrite graph whose first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a non-elementary first-order theory.Comment: accepted for Logical Methods in Computer Scienc

Model-Checking Problems as a Basis for Parameterized Intractability

Most parameterized complexity classes are defined in terms of a parameterized version of the Boolean satisfiability problem (the so-called weighted satisfiability problem). For example, Downey and Fellow's W-hierarchy is of this form. But there are also classes, for example, the A-hierarchy, that are more naturally characterised in terms of model-checking problems for certain fragments of first-order logic. Downey, Fellows, and Regan were the first to establish a connection between the two formalisms by giving a characterisation of the W-hierarchy in terms of first-order model-checking problems. We improve their result and then prove a similar correspondence between weighted satisfiability and model-checking problems for the A-hierarchy and the W^*-hierarchy. Thus we obtain very uniform characterisations of many of the most important parameterized complexity classes in both formalisms. Our results can be used to give new, simple proofs of some of the core results of structural parameterized complexity theory.Comment: Changes in since v2: Metadata update

On space-bounded synchronized alternating Turing machines

AbstractWe continue the study of the computational power of synchronized alternating Turing machines (SATM) introduced in (Hromkovič 1986, Slobodová 1987, 1988a, b) to allow communication via synchronization among processes of alternating Turing machines. We are interested in comparing the four main classes of space-bounded synchronized alternating Turing machines obtained by adding or removing off-line capability and nondeterminism (1SUTM(S(n)), SUTM(S(n)), 1SATM(S(n)), and SATM(S(n)) against one another and against other variants of alternating Turing machines. Denoting the class of languages accepted by machines in C by L(C), we show as our main results that L(1SUTM(S(n))) ⊂ L(SUTM(S(n))) ⊂ L(1SATM(S(n)))= L(SATM(S(n))) for all space-bounded functions S(n)ϵo(n), and L(1SUTM(S(n)))= L(SUTM(S(n))) ⊂ L(1SATM(S(n)))=L(SATM(S(n))) for S(n)) ⩾ n. Furthermore, we show that for log log(n) ⩽ S(n)ϵo(log(n)), L(1SUTM(S(n))) is incomparable to L[1] ATM(S(n))). L(UTM(S(n))), L(1MUTM(S(n))), and L(MUTM(S(n))), where MATMs are alternating Turing machines with modified acceptance proposed in (Inoue 1989); in contrast, we show that these relationships become proper inclusions when log(n) ⩽ S(n)ϵo(n).For deterministic synchronized alternating finite automata with at most k processes (1DSA(k)FA and DSA(k)FA) we establish a tight hierarchy on the number of processes for the one-way case, namely, L(1DSA(n)FA) ⊂ L(1DSA(n+1)FA) for all n > 0, and show that L(1DFA(2)) − ∪k=1∞L(DSA(k)FA) ≠ ∅, where DFA(k) denotes deterministic k-head finite automata. Finally we investigate closure properties under Boolean operations for some of these classes of languages

Fixed-parameter tractability, definability, and model checking

In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.Comment: To appear in SIAM Journal on Computin

Finite Automata with Generalized Acceptance Criteria

We examine the power of nondeterministic finite automata with acceptance of an input word defined by a leaf language, i.e., a condition on the sequence of leaves in the automaton's computation tree. We study leaf languages either taken from one of the classes of the Chomsky hierarchy, or taken from a time- or space-bounded complexity class. We contrast the obtained results with those known for leaf languages for Turing machines and Boolean circuits