Computer Science

A classical result by Floyd ("On the non-existence of a phrase structure grammar for ALGOL 60", 1962) states that the complete syntax of any sensible programming language cannot be described by the ordinary kind of formal grammars (Chomsky's ``context-free''). This paper uses grammars extended with conjunction and negation operators, known as conjunctive grammars and Boolean grammars, to describe the set of well-formed programs in a simple typeless procedural programming language. A complete Boolean grammar, which defines such concepts as declaration of variables and functions before their use, is constructed and explained. Using the Generalized LR parsing algorithm for Boolean grammars, a program can then be parsed in time O( n 4 ) in its length, while another known algorithm allows subcubic-time parsing. Next, it is shown how to transform this grammar to an unambiguous conjunctive grammar, with square-time parsing. This becomes apparently the first specification of the syntax of a programming language entirely by a computationally feasible formal grammar.

Splicing systems are generative mechanisms introduced by Tom Head in 1987 to model the biological process of DNA recombination. The computational engine of a splicing system is the "splicing operation", a cut-and-paste binary string operation defined by a set of "splicing rules" r=( α 1 , α 2 ; α 3 , α 4 ) where α 1 , α 2 , α 3 , α 4 are words over an alphabet Σ . For two strings x= x 1 α 1 α 2 x 2 and y= y 1 α 3 α 4 y 2 , applying the splicing rule r produces the string z= x 1 α 1 α 4 y 2 . In this paper we focus on a particular type of splicing systems, called (i,j) semi-simple splicing systems, i=1,2 and j=3,4 , wherein all splicing rules have the property that the two strings in positions i and j are singleton letters, while the other two strings are empty. The language generated by such a system consists of the set of words that are obtained starting from an initial set called "axiom set", by iteratively applying the splicing rules to strings in the axiom set as well as to intermediately produced strings. We consider semi-simple splicing systems where the axiom set is a regular language, and investigate the descriptional complexity of such systems in terms of the size of the minimal deterministic finite automata that recognize the languages they generate.

We investigate certain word-construction games with variable turn orders. In these games, Alice and Bob take turns on choosing consecutive letters of a word of fixed length, with Alice winning if the result lies in a predetermined target language. The turn orders that result in a win for Alice form a binary language that is regular whenever the target language is, and we prove some upper and lower bounds for its state complexity based on that of the target language.

In this paper, by developing appropriate methods, we for the first time obtain characterization of four fundamental notions of detectability for general labeled weighted automata over monoids (denoted by A M for short), where the four notions are strong (periodic) detectability (SD and SPD) and weak (periodic) detectability (WD and WPD). Firstly, we formulate the notions of concurrent composition, observer, and detector for A M . Secondly, we use the concurrent composition to give an equivalent condition for SD, use the detector to give an equivalent condition for SPD, and use the observer to give equivalent conditions for WD and WPD, all for general A M without any assumption. Thirdly, we prove that for a labeled weighted automaton over monoid ( Q k ,+) (denoted by A Q k ), its concurrent composition, observer, and detector can be computed in NP, 2 -EXPTIME, and 2 -EXPTIME, respectively, by developing novel connections between A Q k and the NP-complete exact path length problem (proved by [Nykänen and Ukkonen, 2002]) and a subclass of Presburger arithmetic. As a result, we prove that for A Q k , SD can be verified in coNP, while SPD, WD, and WPD can be verified in 2 -EXPTIME. Particularly, for A Q k in which from every state, a distinct state can be reached through some unobservable, instantaneous path, its detector can be computed in NP, and SPD can be verified in coNP. Finally, we prove that the problems of verifying SD and SPD of deterministic A N over monoid (N,+) are both NP-hard. The original methods developed in this paper will provide foundations for characterizing other fundamental properties (e.g., diagnosability and opacity) in A M .

The deterministic membership problem for timed automata asks whether the timed language recognised by a nondeterministic timed automaton can be recognised by a deterministic timed automaton. We show that the problem is decidable when the input automaton is a one-clock nondeterministic timed automaton without epsilon transitions and the number of clocks of the deterministic timed automaton is fixed. We show that the problem in all the other cases is undecidable, i.e., when either 1) the input nondeterministic timed automaton has two clocks or more, or 2) it uses epsilon transitions, or 3) the number of clocks of the output deterministic automaton is not fixed.

Determinization of Büchi automata is a long-known difficult problem and after the seminal result of Safra, who developed the first asymptotically optimal construction from Büchi into Rabin automata, much work went into improving, simplifying or avoiding Safra's construction. A different, less known determinization construction was derived by Muller and Schupp and appears to be unrelated to Safra's construction on the first sight. In this paper we propose a new meta-construction from nondeterministic Büchi to deterministic parity automata which strictly subsumes both the construction of Safra and the construction of Muller and Schupp. It is based on a correspondence between structures that are encoded in the macrostates of the determinization procedures - Safra trees on one hand, and levels of the split-tree, which underlies the Muller and Schupp construction, on the other. Our construction allows for combining the mentioned constructions and opens up new directions for the development of heuristics.

Let exp k,α denote a tetration function defined as follows: exp 1,α = 2 α and exp k+1,α = 2 exp k,α , where k,α are positive integers. Let Δ n denote an alphabet with n letters. If L⊆ Δ ∗ n is an infinite language such that for each u∈L there is v∈L with |u|<|v|≤ exp k,α |u| then we call L a language with the \emph{growth bounded by} (k,α) -tetration. Given two infinite languages L 1 , L 2 ∈ Δ ∗ n , we say that L 1 \emph{dissects} L 2 if | L 1 ∩ L 2 |=∞ and |( Δ ∗ n ∖ L 1 )∩ L 2 |=∞ . Given a context free language L , let κ(L) denote the size of the smallest context free grammar G that generates L . We define the size of a grammar to be the total number of symbols on the right sides of all production rules. Given positive integers n,k with k≥2 , we show that there are context free languages L 1 , L 2 ,…, L 3k−3 ⊆ Δ ∗ n with κ( L i )≤40k such that if α is a positive integer and L⊆ Δ ∗ n is an infinite language with the growth bounded by (k,α) -tetration then there is a regular language M such that M∩( ⋂ 3k−3 i=1 L i ) dissects L and the minimal deterministic finite automaton accepting M has at most k+α+3 states.

The process of decomposing a complex system into simpler subsystems has been of interest to computer scientists over many decades, for instance, for the field of distributed computing. In this paper, motivated by the desire to distribute the process of active automata learning onto multiple subsystems, we study the equivalence between a system and the total behaviour of its decomposition which comprises subsystems with communication between them. We show synchronously- and asynchronously-communicating decompositions that maintain branching bisimilarity, and we prove that there is no decomposition operator that maintains divergence-preserving branching bisimilarity over all LTSs.

We study the dynamic membership problem for regular languages: fix a language L, read a word w, build in time O(|w|) a data structure indicating if w is in L, and maintain this structure efficiently under letter substitutions on w. We consider this problem on the unit cost RAM model with logarithmic word length, where the problem always has a solution in O(log |w| / log log |w|) per operation. We show that the problem is in O(log log |w|) for languages in an algebraically-defined, decidable class QSG, and that it is in O(1) for another such class QLZG. We show that languages not in QSG admit a reduction from the prefix problem for a cyclic group, so that they require {\Omega}(log |w| / log log |w|) operations in the worst case; and that QSG languages not in QLZG admit a reduction from the prefix problem for the multiplicative monoid U 1 = {0, 1}, which we conjecture cannot be maintained in O(1). This yields a conditional trichotomy. We also investigate intermediate cases between O(1) and O(log log |w|). Our results are shown via the dynamic word problem for monoids and semigroups, for which we also give a classification. We thus solve open problems of the paper of Skovbjerg Frandsen, Miltersen, and Skyum [30] on the dynamic word problem, and additionally cover regular languages.

We give a new proof of the result of Comon and Jurski that the binary reachability relation of a timed automaton is definable in linear arithmetic.