Dynamic Complexity of Formal Languages

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.Comment: Contains the material presenten at STACS 2009, extendes with proofs and examples which were omitted due lack of spac

Reasoning about XML with temporal logics and automata

We show that problems arising in static analysis of XML specifications and transformations can be dealt with using techniques similar to those developed for static analysis of programs. Many properties of interest in the XML context are related to navigation, and can be formulated in temporal logics for trees. We choose a logic that admits a simple single-exponential translation into unranked tree automata, in the spirit of the classical LTL-to-Büchi automata translation. Automata arising from this translation have a number of additional properties; in particular, they are convenient for reasoning about unary node-selecting queries, which are important in the XML context. We give two applications of such reasoning: one deals with a classical XML problem of reasoning about navigation in the presence of schemas, and the other relates to verifying security properties of XML views

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Two-Way Visibly Pushdown Automata and Transducers

Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three following models are equivalent: deterministic two-way transducers, monadic second-order (MSO) transducers, and deterministic one-way automata equipped with a finite number of registers. Nested words are words with a nesting structure, allowing to model unranked trees as their depth-first-search linearisations. In this paper, we consider transformations from nested words to words, allowing in particular to produce unranked trees if output words have a nesting structure. The model of visibly pushdown transducers allows to describe such transformations, and we propose a simple deterministic extension of this model with two-way moves that has the following properties: i) it is a simple computational model, that naturally has a good evaluation complexity; ii) it is expressive: it subsumes nested word-to-word MSO transducers, and the exact expressiveness of MSO transducers is recovered using a simple syntactic restriction; iii) it has good algorithmic/closure properties: the model is closed under composition with a unambiguous one-way letter-to-letter transducer which gives closure under regular look-around, and has a decidable equivalence problem

Monadic Datalog Containment on Trees

We show that the query containment problem for monadic datalog on finite unranked labeled trees can be solved in 2-fold exponential time when (a) considering unordered trees using the axes child and descendant, and when (b) considering ordered trees using the axes firstchild, nextsibling, child, and descendant. When omitting the descendant-axis, we obtain that in both cases the problem is EXPTIME-complete.Comment: This article is the full version of an article published in the proccedings of the 8th Alberto Mendelzon Workshop (AMW 2014

One-Tape Turing Machine Variants and Language Recognition

We present two restricted versions of one-tape Turing machines. Both characterize the class of context-free languages. In the first version, proposed by Hibbard in 1967 and called limited automata, each tape cell can be rewritten only in the first $d$ visits, for a fixed constant $d\geq 2$. Furthermore, for $d=2$ deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages. Further restricting the possible operations, we consider strongly limited automata. These models still characterize context-free languages. However, the deterministic version is less powerful than the deterministic version of limited automata. In fact, there exist deterministic context-free languages that are not accepted by any deterministic strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of the September 2015 issue of SIGACT New

Detecting Decidable Classes of Finitely Ground Logic Programs with Function Symbols

In this article, we propose a new technique for checking whether the bottom-up evaluation of logic programs with function symbols terminates. The technique is based on the definition of mappings from arguments to strings of function symbols, representing possible values which could be taken by arguments during the bottom-up evaluation. Starting from mappings, we identify mapping-restricted arguments, a subset of limited arguments, namely arguments that take values from finite domains. Mapping-restricted programs, consisting of rules whose arguments are all mapping restricted, are terminating under the bottom-up computation, as all of its arguments take values from finite domains. We show that mappings can be computed by transforming the original program into a unary logic program: this allows us to establish decidability of checking if a program is mapping restricted. We study the complexity of the presented approach and compare it to other techniques known in the literature. We also introduce an extension of the proposed approach that is able to recognize a wider class of logic programs. The presented technique provides a significant improvement, as it can detect terminating programs not identified by other criteria proposed so far. Furthermore, it can be combined with other techniques to further enlarge the class of programs recognized as terminating under the bottom-up evaluation. </jats:p