Moses Ganardi

Constructing small tree grammars and small circuits for formulas

It is shown that every tree of size n over a fixed set of different ranked symbols can be produced by a so called straight-line linear context-free tree grammar of size O ( n log n ) , which can be used as a compressed representation of the input tree. Two algorithms for computing such tree grammar are presented: One working in logarithmic space and the other working in linear time. As a consequence, it is shown that every arithmetical formula of size n, in which only m n different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size O ( n log m log n ) and depth O ( log n ) . This refines a classical result of Brent from 1974, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O ( n ) . A grammar-based tree compressor based on BISECTION is developed.The output grammar has size O ( n / log n ) , where n is the size of the input tree.A logspace as well as a linear time implementation is presented.

Querying Regular Languages over Sliding Windows.

We study the space complexity of querying regular languages over data streams in the sliding window model. The algorithm has to answer at any point of time whether the content of the sliding window belongs to a fixed regular language. A trichotomy is shown: For every regular language the optimal space requirement is either in Theta(n), Theta(log(n)), or constant, where

Tree Compression Using String Grammars

n

A Universal Tree Balancing Theorem

is the size of the sliding window.

Parity Games of Bounded Tree- and Clique-Width

We study the compressed representation of a ranked tree by a straight-line program (SLP) for its preorder traversal string, and compare it with the previously studied representation by straight-line context-free tree grammars (also known as tree straight-line programs or TSLPs). Although SLPs may be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height of a tree, tree navigation, or evaluation of Boolean expressions. Other problems like pattern matching and evaluation of tree automata become intractable.

Sliding Window Algorithms for Regular Languages

We present a general framework for balancing expressions (terms) in the form of so-called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations, which insert terms/contexts into contexts. In Ref. [16], it was shown that one can compute for a given term of size n in logspace a tree straight-line program of depth O(log n) and size O(n/ log n). In the present article, it is shown that the conversion can be done in DLOGTIME-uniform TC0. This allows reducing the term evaluation problem over an arbitrary algebra A to the term evaluation problem over a derived two-sorted algebra F (A). Three applications are presented: (i) an alternative proof for a recent result by Krebs et al. [25] on the expression evaluation problem is given; (ii) it is shown that expressions for an arbitrary (possibly non-commutative) semiring can be transformed in DLOGTIME-uniform TC0 into equivalent circuits of logarithmic depth and size O(n/ log n); and, (iii) a corresponding result for regular expressions is shown.

Circuits and Expressions over Finite Semirings

In this paper it is shown that deciding the winner of a parity game is in LogCFL, if the underlying graph has bounded tree-width, and in LogDCFL, if the tree-width is at most 2. It is also proven that parity games of bounded clique-width can be solved in LogCFL via a log-space reduction to the bounded tree-width case, assuming that a k-expression for the parity game is part of the input.

Circuit Evaluation for Finite Semirings

This paper gives a survey on recent results for sliding window streaming algorithms for regular languages. Details can be found in the recent papers [18, 19].

On the Parallel Complexity of Bisimulation on Finite Systems.

The computational complexity of the circuit and expression evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or a multiplicative identity. The following dichotomy is shown: If a finite semiring is such that (i) the multiplicative semigroup is solvable and (ii) it does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 ≠ 0, then the circuit evaluation problem is in DET ⊆ NC2, and the expression evaluation problem for the semiring is in TC0. For all other finite semirings, the circuit evaluation problem is P-complete and the expression evaluation problem is NC1-complete. As an application, we determine the complexity of intersection non-emptiness problems for given context-free grammars (regular expressions) with a fixed regular language.

Automata Theory on Sliding Windows.

The circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or multiplicative identity. The following dichotomy is shown: If a finite semiring R (i) has a solvable multiplicative semigroup and (ii) does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 != 0, then its circuit evaluation problem is in the complexity class DET (which is contained in NC^2). In all other cases, the circuit evaluation problem is P-complete.