32 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Unboundedness Problems for Machines with Reversal-Bounded Counters

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    We consider a general class of decision problems concerning formal languages, called (one-dimensional) unboundedness predicates, for automata that feature reversal-bounded counters (RBCA). We show that each problem in this class reduces-non-deterministically in polynomial time to the same problem for just nite automata. We also show an analogous reduction for automata that have access to both a push- down stack and reversal-bounded counters (PRBCA). This allows us to answer several open questions: For example, we settle the complexity of deciding whether a given (P)RBCA language L is bounded, meaning whether there exist words w1, . . . , wn with L ⊆ w1∗ · · · wn∗ . For PRBCA, even decidability was open. Our methods also show that there is no language of a (P)RBCA of intermediate growth. Part of our proof is likely of independent interest: We show that one can translate an RBCA into a machine with Z-counters in logarithmic space

    On the Commutative Equivalence of Algebraic Formal Series and Languages

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    The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated

    A guide to F-automatic sets

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    A self-contained introduction to the theory of F-automatic sets is given. Building on [Bell, Moosa, F-sets and finite automata, Journal de théorie des nombres de Bordeaux, 2019], contributions are made to both the foundations of this theory and to questions of a model-theoretic nature. Suppose Γ is an abelian group and F : Γ → Γ is an injective endomorphism. If (Γ, F) admits a "spanning set" then the notion of an F-automatic set can be defined. It is shown that this notion is independent of the spanning set chosen. A characterization of the existence of a spanning set is given in terms of certain functions on Γ, called "height functions". It is shown that if Γ is finitely generated then (Γ, F) admits a spanning set if and only if no eigenvalue of the matrix of F lies in the complex unit disk. A notion of sparsity among F-automatic sets, called F-sparsity, is studied. Outstanding questions from [Bell, Moosa, F-sets and finite automata, Journal de théorie des nombres de Bordeaux, 2019] are resolved, including independence from the spanning set chosen and closure under set summation. In addition, it is shown that sparsity can be characterized in terms of another natural class of functions introduced here, called "length functions". Model-theoretic tameness properties of F-automatic sets are studied. In the case where Γ is finitely generated, a combinatorial description is given of the stable F-sparse sets in terms of the F-sets introduced in [Moosa, Scanlon, F-structures and integral points on semiabelian varieties over finite fields, American Journal of Mathematics, 2004]. When Γ = ℤ, this description is extended to a characterization of the stable F-automatic sets. It is shown that if A ⊆ Γ is F-sparse then (Γ, +, A) is NIP. Automatic methods are used to show that the following structures have NIP theories: (ℤ, +, d^ℕ, ×↾d^ℕ) for d ≥ 2, (_p[t], +, t^ℕ, ×↾t^ℕ) for prime p ≥ 9, and (ℤ, +, <, d^ℕ) for d ≥ 2. (Here d^ℕ = {1, d, d^2, ...}, and likewise with t^ℕ.

    On Bounded Linear Codes and the Commutative Equivalence

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    The problem of the commutative equivalence of semigroups generated by semi-linear languages is studied. In particular conditions ensuring that the Kleene closure of a bounded semi-linear code is commutatively equivalent to a regular language are investigated

    Sparse Automatic Sets

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    The theory of automatic sets and sequences arises naturally in many different areas of mathematics, notably in the study of algebraic power series in positive characteristic, due to work of Christol, and in Derksen's classification of zero sets for sequences satisfying a linear recurrence over fields of positive characteristic. A fundamental dichotomy for automatic sets shows that they are either sparse, having counting functions that grow relatively slowly, or they are not sparse, in which case their counting functions grow reasonably fast. While this dichotomy has been known to hold for some time, there has not---to this point in time---been a systematic study of the algebraic and number theoretic properties of sparse automatic sets. This thesis rectifies this situation and gives multiple results dealing specifically with sparse automatic sets. In particular, we give a stronger version of a classical result of Cobham for automatic sets where one now specializes to sparse automatic sets; we then prove that a conjecture of Erdos and Turan holds for automatic sets, again using the theory of sparseness; finally, we give a refinement of a classical result of Christol where we consider algebraic power series whose support set is a sparse automatic set

    Relationships Between Bounded Languages, Counter Machines, Finite-Index Grammars, Ambiguity, and Commutative Equivalence

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    It is shown that for every language family that is a trio containing only semilinear languages, all bounded languages in it can be accepted by one-way deterministic reversal-bounded multicounter machines (DCM). This implies that for every semilinear trio (where these properties are effective), it is possible to decide containment, equivalence, and disjointness concerning its bounded languages. A condition is also provided for when the bounded languages in a semilinear trio coincide exactly with those accepted by DCM machines, and it is used to show that many grammar systems of finite index — such as finite-index matrix grammars (Mfin) and finite-index ET0L (ET0Lfin) — have identical bounded languages as DCM. Then connections between ambiguity, counting regularity, and commutative regularity are made, as many machines and grammars that are unambiguous can only generate/accept counting regular or com- mutatively regular languages. Thus, such a system that can generate/accept a non-counting regular or non-commutatively regular language implies the existence of inherently ambiguous languages over that system. In addition, it is shown that every language generated by an unambiguous Mfin has a rational char- acteristic series in commutative variables, and is counting regular. This result plus the connections are used to demonstrate that the grammar systems Mfin and ET0Lfin can generate inherently ambiguous languages (over their grammars), as do several machine models. It is also shown that all bounded languages generated by these two grammar systems (those in any semilinear trio) can be generated unambiguously within the systems. Finally, conditions on Mfin and ET0Lfin languages implying commutative regularity are obtained. In particular, it is shown that every finite-index ED0L language is commutatively regular

    A Burnside Approach to the Termination of Mohri’s Algorithm for Polynomially Ambiguous Min-Plus-Automata

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    We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm

    Aperiodic Weighted Automata and Weighted First-Order Logic

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    By fundamental results of Sch\"utzenberger, McNaughton and Papert from the 1970s, the classes of first-order definable and aperiodic languages coincide. Here, we extend this equivalence to a quantitative setting. For this, weighted automata form a general and widely studied model. We define a suitable notion of a weighted first-order logic. Then we show that this weighted first-order logic and aperiodic polynomially ambiguous weighted automata have the same expressive power. Moreover, we obtain such equivalence results for suitable weighted sublogics and finitely ambiguous or unambiguous aperiodic weighted automata. Our results hold for general weight structures, including all semirings, average computations of costs, bounded lattices, and others.Comment: An extended abstract of the paper appeared at MFCS'1
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