A Note on Nested String Replacements

We investigate the number of nested string replacements required to reduce a string of identical characters to one character

A Non-Oblivious Reduction of Counting Ones to Multiplication

An algorithm counting the number of ones in a binary word is presented running in time $O(\log\log b)$ where $b$ is the number of ones. The operations available include bit-wise logical operations and multiplication

Some Remarks on Real-Time Turing Machines

The power of real-time Turing machines using sublinear space is investigated. In contrast to a claim appearing in the literature, such machines can accept non-regular languages, even if working in deterministic mode. While maintaining a standard binary counter appears to be impossible in real-time, we present a guess and check approach that yields a binary representation of the input length. Based on this technique, we show that unary encodings of languages accepted in exponential time can be recognized by nondeterministic real-time Turing machines

A SWAR Approach to Counting Ones

We investigate the complexity of algorithms counting ones in different sets of operations. With addition and logical operations (but no shift) $O(\log^2(n))$ steps suffice to count ones. Parity can be computed with complexity $O(\log(n))$, which is the same bound as for methods using shift-operations. If multiplication is available, a solution of time complexity $O(\log^*(n))$ is possible improving the known bound $O(\log\log(n))$

Counting Ones Without Broadword Operations

A lower time bound $\Omega(\min(\nu(x), n-\nu(x))$ for counting the number of ones in a binary input word $x$ of length $n$ is presented, where $\nu(x)$ is the number of ones. The operations available are increment, decrement, bit-wise logical operations, and assignment. The only constant available is zero. An almost matching upper bound is also obtained

On Practical Regular Expressions

We report on simulation, hierarchy, and decidability results for Practical Regular Expressions (PRE), which may include back references in addition to the standard operations union, concatenation, and star. The following results are obtained: PRE can be simulated by the classical model of nondeterministic finite automata with sensing one-way heads. The number of heads depends on the number of different variables in the expressions. A space bound O(n log m) for matching a text of length m with a PRE with n variables based on the previous simulation. This improves the bound O(nm) from (C\^ampeanu and Santean 2009). PRE cannot be simulated by deterministic finite automata with at most three sensing one-way heads or deterministic finite automata with any number of non-sensing one-way heads. PRE with a bounded number of occurrences of variables in any match can be simulated by nondeterministic finite automata with one-way heads. There is a tight hierarchy of PRE with a growing number of non-nested variables over a fixed alphabet. A previously known hierarchy was based on nested variables and growing alphabets (Larsen 1998). Matching of PRE without star over a single-letter alphabet is NP-complete. This strengthens the corresponding result for expressions over larger alphabets and with star (Aho 1990). Inequivalence of PRE without closure operators is Sigma^P_2-complete. The decidability of universality of PRE over a single letter alphabet is linked to the existence of Fermat Primes. Greibach's Theorem applies to languages characterized by PRE

The Complexity of Some Combinatorial Puzzles

We show that the decision versions of the puzzles Knossos and The Hour-Glass are complete for NP

An NL-Complete Puzzle

We investigate the complexity of a puzzle that turns out to be NL-complete

Efficient Computation by Three Counter Machines

We show that multiplication can be done in polynomial time on a three counter machine that receives its input as the contents of two counters. The technique is generalized to functions of two variables computable by deterministic Turing machines in linear space

Some Remarks on Lower Bounds for Queue Machines (Preliminary Report)

We first give an improved lower bound for the deterministic online simulation of tapes or pushdown stores by queues. Then we inspect some proofs in a classical work on queue machines in the area of Formal Languages and outline why a main argument in the proofs is incomplete. Based on descriptional complexity, we show the intuition behind the argument to be correct