A Unified Method for Placing Problems in Polylogarithmic Depth

In this work we consider the term evaluation problem which is, given a term over some algebra and a valid input to the term, computing the value of the term on that input. In contrast to previous methods we allow the algebra to be completely general and consider the problem of obtaining an efficient upper bound for this problem. Many variants of the problems where the algebra is well behaved have been studied. For example, the problem over the Boolean semiring or over the semiring (N,+,*). We extend this line of work. Our efficient term evaluation algorithm then serves as a tool for obtaining polylogarithmic depth upper bounds for various well-studied problems. To demonstrate the utility of our result we show new bounds and reprove known results for a large spectrum of problems. In particular, the applications of the algorithm we consider include (but are not restricted to) arithmetic formula evaluation, word problems for tree and visibly pushdown automata, and various problems related to bounded tree-width and clique-width graphs

Sums of Palindromes: an Approach via Automata

Recently, Cilleruelo, Luca, & Baxter proved, for all bases b >= 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome. However, the cases b = 2, 3, 4 were left unresolved. We prove, using a decision procedure based on automata, that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem

Synchronizing automata over nested words

We extend the concept of a synchronizing word from deterministic finite-state automata (DFA) to nested word automata (NWA): A well-matched nested word is called synchronizing if it resets the control state of any configuration, i. e., takes the NWA from all control states to a single control state. We show that although the shortest synchronizing word for an NWA, if it exists, can be (at most) exponential in the size of the NWA, the existence of such a word can still be decided in polynomial time. As our main contribution, we show that deciding the existence of a short synchronizing word (of at most given length) becomes PSPACE-complete (as opposed to NP-complete for DFA). The upper bound makes a connection to pebble games and Strahler numbers, and the lower bound goes via small-cost synchronizing words for DFA, an intermediate problem that we also show PSPACE-complete. We also characterize the complexity of a number of related problems, using the observation that the intersection nonemptiness problem for NWA is EXP-complete

Synchronization of Deterministic Visibly Push-Down Automata

We generalize the concept of synchronizing words for finite automata, which map all states of the automata to the same state, to deterministic visibly push-down automata. Here, a synchronizing word w does not only map all states to the same state but also fulfills some conditions on the stack content of each run after reading w. We consider three types of these stack constraints: after reading w, the stack (1) is empty in each run, (2) contains the same sequence of stack symbols in each run, or (3) contains an arbitrary sequence which is independent of the other runs. We show that in contrast to general deterministic push-down automata, it is decidable for deterministic visibly push-down automata whether there exists a synchronizing word with each of these stack constraints, more precisely, the problems are in EXPTIME. Under the constraint (1), the problem is even in P. For the sub-classes of deterministic very visibly push-down automata, the problem is in P for all three types of constraints. We further study variants of the synchronization problem where the number of turns in the stack height behavior caused by a synchronizing word is restricted, as well as the problem of synchronizing a variant of a sequential transducer, which shows some visibly behavior, by a word that synchronizes the states and produces the same output on all runs

Synchronizing Deterministic Push-Down Automata Can Be Really Hard

The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata. There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference

Sofic-Dyck shifts

We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift