Decidability for Sturmian Words

Hieronymi, Philipp; Ma, Dun; Oei, Reed; Schaeffer, Luke; Schulz, Christian; Shallit, Jeffrey

doi:10.4230/lipics.csl.2022.24

Cited by 6 publications

(5 citation statements)

References 9 publications

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“…We remark that it is unclear to us whether some of the results proved in this section could be proved automatically using the very recent tool Pecan developed in [36,26] Minimal complexity words can be seen as a generalization of Sturmian words to larger alphabets: if a word (containing all letters of A) has less than n+|A|−1 factors of length n for some n, then it is ultimately periodic. Otherwise it is aperiodic (a consequence of the Morse-Hedlund theorem).…”

Section: Another Description Of the Extended Boundary Sequencementioning

confidence: 97%

On Extended Boundary Sequences of Morphic and Sturmian Words

Rigo,

Stipulanti,

Whiteland

2024

Electron. J. Combin.

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Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for all pairs of factors of length $\ell$ separated by $n-\ell$ symbols. For the large class of addable abstract numeration systems $S$, we show that if an infinite word is $S$-automatic, then the same holds for its $\ell$-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). To precise the limits of this result, we discuss examples of non-addable numeration systems and $S$-automatic words for which the boundary sequence is nevertheless $S$-automatic and conversely, $S$-automatic words with a boundary sequence that is not $S$-automatic. In the second part of the paper, we study the $\ell$-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope. We also show that it is the image under a morphism of some other characteristic Sturmian word.

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Section: Another Description Of the Extended Boundary Sequencementioning

confidence: 97%

On Extended Boundary Sequences of Morphic and Sturmian Words

Rigo,

Stipulanti,

Whiteland

2024

Electron. J. Combin.

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“…As our data set, we used 39,837 BAs from the -repository [33] (used before by, e.g., [25,26,34]), which contains BAs from the following sources: (i) randomly generated BAs used in [49] (21,876 BAs), (ii) BAs obtained from LTL formulae from the literature and randomly generated LTL formulae [5] (3,442 BAs), (iii) BAs obtained from U A [11] (915 BAs), (iv) BAs obtained from the solver for first-order logic over Sturmian words P [28] (13,216 BAs), (v) BAs obtained from an S1S solver [23] (370 BAs), and (vi) BAs from LTL to SDBA translation [46] (18 BAs). From these BAs, 23,850 are deterministic, 6,147 are SDBAs (but not deterministic), 4,105 are elevator (but not SDBAs), and 5,735 are the rest.…”

Section: Experimental Evaluationmentioning

confidence: 99%

“…Nondeterministic Büchi automata (BAs) [8] are an elegant and conceptually simple framework to model infinite behaviors of systems and the properties they are expected to satisfy. BAs are widely used in many important verification tasks, such as termination analysis of programs [27], model checking [51], or as the underlying formal model of decision procedures for some logics (such as S1S [8] or a fragment of the first-order logic over Sturmian words [28]). Many of these applications require to perform complementation of BAs: For instance, in termination analysis of programs within U A [27], complementation is used to keep track of the set of paths whose termination still needs to be proved.…”

Section: Introductionmentioning

confidence: 99%

Modular Mix-and-Match Complementation of Büchi Automata (Technical Report)

Havlena¹,

Lengál²,

Li³

et al. 2023

Preprint

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Complementation of nondeterministic Büchi automata (BAs) is an important problem in automata theory with numerous applications in formal verification, such as termination analysis of programs, model checking, or in decision procedures of some logics. We build on ideas from a recent work on BA determinization by Li et al. and propose a new modular algorithm for BA complementation. Our algorithm allows to combine several BA complementation procedures together, with one procedure for a subset of the BA's strongly connected components (SCCs). In this way, one can exploit the structure of particular SCCs (such as when they are inherently weak or deterministic) and use more efficient specialized algorithms, regardless of the structure of the whole BA. We give a general framework into which partial complementation procedures can be plugged in, and its instantiation with several algorithms. The framework can, in general, produce a complement with an Emerson-Lei acceptance condition, which can often be more compact. Using the algorithm, we were able to establish an exponentially better new upper bound of O (4 𝑛 ) for complementation of the recently introduced class of elevator automata. We implemented the algorithm in a prototype and performed a comprehensive set of experiments on a large set of benchmarks, showing that our framework complements well the state of the art and that it can serve as a basis for future efficient BA complementation and inclusion checking algorithms.Here, we consider model checking w.r.t. a specification given in some more expressive logic, such as S1S [8], QPTL [47], or HyperLTL [12], rather than LTL [41], where negation is simple.

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“…The BAs of the Pecan [31] benchmarks encode sets of solutions of predicates, hence a logical implication between predicates reduces to a language inclusion problem between BAs. The benchmarks correspond to theorems of type ∀x, ∃y, P (x) =⇒ Q(y) about Sturmian words [21]. We collected 60 benchmarks from Pecan for which inclusion holds, where the BAs have alphabets of up to 256 symbols and have up to 21 395 states.…”

Section: Experimental Evaluationmentioning

confidence: 99%

FORQ-Based Language Inclusion Formal Testing

Doveri

Ganty

Mazzocchi

2022

Computer Aided Verification

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We propose a novel algorithm to decide the language inclusion between (nondeterministic) Büchi automata, a PSpace-complete problem. Our approach, like others before, leverage a notion of quasiorder to prune the search for a counterexample by discarding candidates which are subsumed by others for the quasiorder. Discarded candidates are guaranteed to not compromise the completeness of the algorithm. The novelty of our work lies in the quasiorder used to discard candidates. We introduce FORQs (family of right quasiorders) that we obtain by adapting the notion of family of right congruences put forward by Maler and Staiger in 1993. We define a FORQ-based inclusion algorithm which we prove correct and instantiate it for a specific FORQ, called the structural FORQ, induced by the Büchi automaton to the right of the inclusion sign. The resulting implementation, called Forklift, scales up better than the state-of-the-art on a variety of benchmarks including benchmarks from program verification and theorem proving for word combinatorics. Artifact:https://doi.org/10.5281/zenodo.6552870

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Decidability for Sturmian Words

Cited by 6 publications

References 9 publications

On Extended Boundary Sequences of Morphic and Sturmian Words

On Extended Boundary Sequences of Morphic and Sturmian Words

Modular Mix-and-Match Complementation of Büchi Automata (Technical Report)

FORQ-Based Language Inclusion Formal Testing

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