A Very Hard log-Space Counting Class

  title={A Very Hard log-Space Counting Class},
  author={Carme {\`A}lvarez and Birgit Jenner},
  journal={Theor. Comput. Sci.},

#NFA Admits an FPRAS: Efficient Enumeration, Counting, and Uniform Generation for Logspace Classes

This work solves the open problem of counting the number of strings of length n and obtains as a welcome corollary that every function in SpanL admits an FPRAS.

Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation

This work investigates the complexity of three fundamental algorithmic problems for two simple yet general complexity classes, based on logspace Turing machines, and proves constant delay enumeration, and both counting and uniform generation of solutions in polynomial time.

Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation

This work solves the open problem of whether the fundamental problem #NFA admits an FPRAS, and obtains as a welcome corollary that every function in SpanL admits a fully polynomial-time randomized approximation scheme (FPRAS).

Semantical Counting Circuits

This work considers semantically defined probabilistic complexity classes corresponding to AC0 and NC1 and proves that in the case of unbounded error, they are identical to their syntactical counterparts.

On the connection between interval size functions and path counting

This work provides inclusion and separation relations between TotP and interval size counting classes, and defines a new class of interval size functions which strictly contains FP and is strictly contained in TotP under reasonable complexity-theoretic assumptions, and shows that this new class contains hard counting problems.

On the power of unambiguity in log-space

It is shown that counting the number of s-t paths in graphs where the numberof s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class.

A Combinatorial Property of ♯L Assuming Nl = Ul and its Implications for Modl

—We first show that SLDAGSTCON which is the st - connectivity problem for simple layered directed acyclic graphs, where the vertex s is in the 1 st row and the vertex t is in the last row of the

Descriptive Complexity for counting complexity classes

This paper obtains a logic called Quantitative Second Order Logics (QSO), and shows how some of its fragments can be used to capture fundamental counting complexity classes such as FP, #P and FPSPACE, among others.

Unambiguous logarithmic space bounded computations

The NL versus UL question led to another important problem in complexity theory - space complexity of deciding if a graph has a perfect matching, and it is proved that perfect matching in bipartite bounded genus graphs is in SPL (a class which is a generalization of UL and not known to be comparable with NL).

On TC0, AC0, and Arithmetic Circuits

A characterization of TC0 in terms of #AC0, as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates, and a restatement of this characterization is that TC0 can be simulated by Constant-depth arithmetic circuits, with a single threshold gate.



Two Applications of Inductive Counting for Complementation Problems

It is shown that small numbers of “role switches” in two- person pebbling can be eliminated and a general result that shows closure under complementation of classes defined by semi-unbounded fan-in circuits is shown.

The complexity of optimization problems

The central result is that any FPSAT function decomposes into an OptP function followed by polynomial-time computation, and it quantifies "how much" NP-completeness is in a problem, i.e., the number of NP queries it takes to compute the function.

Space-Bounded Reducibility among Combinatorial Problems

  • N. Jones
  • Computer Science, Mathematics
    J. Comput. Syst. Sci.
  • 1975

On the Complexity of Ranking

Languages Simultaneously Complete for One-Way and Two-Way Log-Tape Automata

It is shown that for the one-way $\log n$-tape automata the nondeterministic model (1-NL) is computationally much more powerful than the deterministic model(1-L), that under one- way $\logn$-Tape reductions there exist natural complete languages for these automata and that the complete languages cannot be sparse.

The Complexity of Enumeration and Reliability Problems

  • L. Valiant
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1979
For a large number of natural counting problems for which there was no previous indication of intractability, that they belong to the class of computationally eqivalent counting problems that are at least as difficult as the NP-complete problems.

The polynomial-time hierarchy and sparse oracles

It is proved that the polynomial-time hierarchy collapses if and only if for every sparse set S, the hierarchy relative to S collapses and the question is answered if it is answered for any arbitrary sparse oracle set.

Space-bounded hierarchies and probabilistic computations

Two aspects of the power of space-bounded probabilistic Turing machines are studied, one of which raises interesting questions about space hierarchies, and the other demonstrates that any language in the log n space hierarchy can be recognized by an log n Space Turing machine with small error.

Computing the Counting Function of Context-Free Languages

It is shown that, if L is unambiguous context-free, then FL can be computed in NC2, hence admitting a fast parallel algorithm.

Functional Oracle Queries as a Measure of Parallel Time

By imposing appropriate bounds on the number of functional oracle queries made in this computation model, this work obtains new characterizations of the NC and AC hierarchies and solves open questions of Wilson.