Synchronous Subsequentiality and Approximations to Undecidable Problems

We introduce the class of synchronous subsequential relations, a subclass of the synchronous relations which embodies some properties of subsequential relations. If we take relations of this class as forming the possible transitions of an infinite automaton, then most decision problems (apart from membership) still remain undecidable (as they are for synchronous and subsequential rational relations), but on the positive side, they can be approximated in a meaningful way we make precise in this paper. This might make the class useful for some applications, and might serve to establish an intermediate position in the trade-off between issues of expressivity and (un)decidability.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

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

Model Theoretic Complexity of Automatic Structures

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by \omega^\omega ; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal.Comment: 23 pages. Extended abstract appeared in Proceedings of TAMC '08, LNCS 4978 pp 514-52

Splicing Systems from Past to Future: Old and New Challenges

A splicing system is a formal model of a recombinant behaviour of sets of double stranded DNA molecules when acted on by restriction enzymes and ligase. In this survey we will concentrate on a specific behaviour of a type of splicing systems, introduced by P\u{a}un and subsequently developed by many researchers in both linear and circular case of splicing definition. In particular, we will present recent results on this topic and how they stimulate new challenging investigations.Comment: Appeared in: Discrete Mathematics and Computer Science. Papers in Memoriam Alexandru Mateescu (1952-2005). The Publishing House of the Romanian Academy, 2014. arXiv admin note: text overlap with arXiv:1112.4897 by other author