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  • Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
    arXiv.cs.FL Pub Date : 2020-01-15
    Lars Jaffke; Mateus de Oliveira Oliveira; Hans Raj Tiwary

    It can be shown that each permutation group $G \sqsubseteq S_n$ can be embedded, in a well defined sense, in a connected graph with $O(n+|G|)$ vertices. Some groups, however, require much fewer vertices. For instance, $S_n$ itself can be embedded in the $n$-clique $K_n$, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group $G \sqsubseteq S_n$ can be upper bounded by three structural parameters of connected graphs embedding $G$: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group $G \sqsubseteq S_n$ that can be embedded into a connected graph with $m$ vertices, treewidth k, and maximum degree $\Delta$, can also be generated by a context-free grammar of size $2^{O(k\Delta\log\Delta)}\cdot m^{O(k)}$. By combining our upper bound with a connection between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity $2^{O(k \Delta\log \Delta)}\cdot m^{O(k)}$. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated $2^{\Omega(n)}$ lower bound on the grammar complexity of the symmetric group $S_n$ we have that connected graphs of treewidth $o(n/\log n)$ and maximum degree $o(n/\log n)$ embedding subgroups of $S_n$ of index $2^{cn}$ for some small constant $c$ must have $n^{\omega(1)}$ vertices. This lower bound can be improved to exponential on graphs of treewidth $n^{\varepsilon}$ for $\varepsilon<1$ and maximum degree $o(n/\log n)$.

    更新日期:2020-01-17
  • A Categorical Framework for Learning Generalised Tree Automata
    arXiv.cs.FL Pub Date : 2020-01-16
    Gerco van Heerdt; Tobias Kappé; Jurriaan Rot; Matteo Sammartino; Alexandra Silva

    Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify these using category theory in order to ease correctness proofs and guide the design of new algorithms. In this paper, we extend CALF to cover learning of algebraic structures that may not have a coalgebraic presentation. Furthermore, we provide a detailed algorithmic account of an abstract version of the popular L* algorithm, which was missing from CALF. We instantiate the abstract theory to a large class of Set functors, by which we recover for the first time practical tree automata learning algorithms from an abstract framework and at the same time obtain new algorithms to learn algebras of quotiented polynomial functors.

    更新日期:2020-01-17
  • Learning Concise Models from Long Execution Traces
    arXiv.cs.FL Pub Date : 2020-01-15
    Natasha Yogananda Jeppu; Tom Melham; Daniel Kroening; John O'Leary

    Abstract models of system-level behaviour have applications in design exploration, analysis, testing and verification. We describe a new algorithm for automatically extracting useful models, as automata, from execution traces of a HW/SW system driven by software exercising a use-case of interest. Our algorithm leverages modern program synthesis techniques to generate predicates on automaton edges, succinctly describing system behaviour. It employs trace segmentation to tackle complexity for long traces. We learn concise models capturing transaction-level, system-wide behaviour--experimentally demonstrating the approach using traces from a variety of sources, including the x86 QEMU virtual platform and the Real-Time Linux kernel.

    更新日期:2020-01-16
  • How Does Adiabatic Quantum Computation Fit into Quantum Automata Theory?
    arXiv.cs.FL Pub Date : 2020-01-15
    Tomoyuki Yamakami

    Quantum computation has emerged as a powerful computational medium of our time, having demonstrated the efficiency in solving the integer factoring and searching a database faster than any currently known classical computer algorithm. Adiabatic evolution of quantum systems have been studied as a potential means that physically realizes quantum computation. Up to now, all the research on adiabatic quantum systems has dealt with polynomial time-bounded computation and little attention has been paid to, for example, adiabatic quantum systems consuming only constant memory space. Such quantum systems can be modeled in a form similar to quantum finite automata. This exposition dares to ask a bold question of how to make adiabatic quantum computation fit into the rapidly progressing framework of quantum automata theory. As our answer to this eminent but profound question, we first lay out a fundamental platform for adiabatic evolutionary quantum systems (AEQSs) with limited computational resources and then establish how to construct AEQSs using quantum finite automata. We also explore fundamental structural properties of decision problems (or equivalently, languages) solved quickly by such AEQSs.

    更新日期:2020-01-16
  • Orbit Expandability of Automaton Semigroups and Groups
    arXiv.cs.FL Pub Date : 2018-12-18
    Daniele D'Angeli; Emanuele Rodaro; Jan Philipp Wächter

    We introduce the notion of expandability in the context of automaton semigroups and groups: a word is k-expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k. This definition is motivated by the question which {\omega}-words admit infinite orbits: for such a word, every prefix is expandable. In this paper, we show that, on input of a word u, an automaton T and a number k, it is decidable to check whether u is k-expandable with respect to the action of T. In fact, this can be done in exponential nondeterministic space. From this nondeterministic algorithm, we obtain a bound on the length of a potential orbit-increasing suffix x. Moreover, we investigate the situation if the automaton is invertible and generates a group. In this case, we give an algebraic characterization for the expandability of a word based on its shifted stabilizer. We also give a more efficient algorithm to decide expandability of a word in the case of automaton groups, which allows us to improve the upper bound on the maximal orbit-increasing suffix length. Then, we investigate the situation for reversible (and complete) automata and obtain that every word is expandable with respect to these automata. Finally, we give a lower bound example for the length of an orbit-increasing suffix.

    更新日期:2020-01-16
  • Nonstandard Cayley automatic representations of fundamental groups
    arXiv.cs.FL Pub Date : 2020-01-14
    Dmitry Berdinsky; Prohrak Kruengthomya

    We construct a new family of Cayley automatic representations of semidirect products $\mathbb{Z}^n \rtimes_A \mathbb{Z}$ for which none of the projections of the normal subgroup $\mathbb{Z}^n$ onto each of its cyclic components is finite automaton recognizable. For $n=2$ we describe a family of matrices from $\mathrm{GL}(2,\mathbb{Z})$ corresponding to these representations. We are motivated by a problem of characterization of all possible Cayley automatic representations of these groups.

    更新日期:2020-01-15
  • An Informal Method
    arXiv.cs.FL Pub Date : 2016-08-04
    Victor Yodaiken

    A method for specifying the behavior and architecture of discrete state systems such as digital electronic devices and software. The method draws on state machine theory, automata products, and recursive functions and is ordinary working mathematics, not involving formal methods or any foundational or meta-mathematical techniques. Systems in which there are levels of components that may operate in parallel or concurrently are specified in terms of function composition. Illustrative examples include real-time systems, distributed consensus, a Java producer/consumer solution, and digital circuits.

    更新日期:2020-01-15
  • Approximating probabilistic models as weighted finite automata
    arXiv.cs.FL Pub Date : 2019-05-21
    Ananda Theertha Suresh; Brian Roark; Michael Riley; Vlad Schogol

    Weighted finite automata (WFA) are often used to represent probabilistic models, such as $n$-gram language models, since they are efficient for recognition tasks in time and space. The probabilistic source to be represented as a WFA, however, may come in many forms. Given a generic probabilistic model over sequences, we propose an algorithm to approximate it as a weighted finite automaton such that the Kullback-Leiber divergence between the source model and the WFA target model is minimized. The proposed algorithm involves a counting step and a difference of convex optimization step, both of which can be performed efficiently. We demonstrate the usefulness of our approach on various tasks, including distilling $n$-gram models from neural models, building compact language models, and building open-vocabulary character models. The algorithms used for these experiments are available in an open-source software library.

    更新日期:2020-01-15
  • Games Where You Can Play Optimally with Finite Memory
    arXiv.cs.FL Pub Date : 2020-01-12
    Patricia Bouyer; Stéphane Le Roux; Youssouf Oualhadj; Mickael Randour; Pierre Vandenhove

    For decades, two-player (antagonistic) games on graphs have been a framework of choice for many important problems in theoretical computer science. A notorious one is controller synthesis, which can be rephrased through the game-theoretic metaphor as the quest for a winning strategy of the system in a game against its antagonistic environment. Depending on the specification, optimal strategies might be simple or quite complex, for example having to use (possibly infinite) memory. Hence, research strives to understand which settings allow for simple strategies. In 2005, Gimbert and Zielonka provided a complete characterization of preference relations (a formal framework to model specifications and game objectives) that admit memoryless optimal strategies for both players. In the last fifteen years however, practical applications have driven the community toward games with complex or multiple objectives, where memory --- finite or infinite --- is almost always required. Despite much effort, the exact frontiers of the class of preference relations that admit finite-memory optimal strategies still elude us. In this work, we establish a complete characterization of preference relations that admit optimal strategies using arena-independent finite memory, generalizing the work of Gimbert and Zielonka to the finite-memory case. We also prove an equivalent to their celebrated corollary of utmost practical interest: if both players have optimal (arena-independent-)finite-memory strategies in all one-player games, then it is also the case in all two-player games. Finally, we pinpoint the boundaries of our results with regard to the literature: our work completely covers the case of arena-independent memory (e.g., multiple parity objectives, lower- and upper-bounded energy objectives), and paves the way to the arena-dependent case (e.g., multiple lower-bounded energy objectives).

    更新日期:2020-01-14
  • An extended quantum process algebra (eQPAlg) approach for distributed quantum systems
    arXiv.cs.FL Pub Date : 2020-01-06
    Salman Haider; Dr. Syed Asad Raza Kazmi

    In this work, we have expounded the communication procedure of quantum systems by means of process algebra. The main objective of our research effort is to formally represent the communication between distributed quantum systems. In this new proposed communication model we have ameliorated the existing rules of Lalire's quantum process algebra QPAlg. We have brought some important modification in QPAlg by introducing the concept of formally specifying the Quantum teleportation protocol. We have further introduced the formal description of protocol by using programs that best explains its working and satisfies the specification. Examples have been provided to describe the working of the improved algebra that formally explain the sending and receiving of both classical as well as quantum data, keeping in mind the principal features of quantum mechanics.

    更新日期:2020-01-14
  • Reachability in fixed dimension vector addition systems with states
    arXiv.cs.FL Pub Date : 2020-01-13
    Wojciech Czerwiński; Sławomir Lasota; Ranko Lazić; Jérôme Leroux; Filip Mazowiecki

    The reachability problem is a central decision problem for formal verification based on vector addition systems with states (VASS), which are equivalent to Petri nets and form one of the most studied and applied models of concurrency. Reachability for VASS is also inter-reducible with a plethora of problems from a number of areas of computer science. In spite of recent progress, the complexity of the reachability problem remains unsettled, and it is closely related to the lengths of shortest VASS runs that witness reachability. We consider VASS of fixed dimension, and obtain three main results. For the first two, we assume that the integers in the input are given in unary, and that the control graph of the given VASS is flat (i.e., without nested cycles). We obtain a family of VASS in dimension~$3$ whose shortest reachability witnessing runs are exponential, and we show that the reachability problem is \np-hard in dimension~$7$. These results resolve negatively questions that had been posed by the works of Blondin et al.\ in LICS 2015 and Englert et al.\ in LICS 2016, and contribute a first construction that distinguishes $3$-dimensional flat VASS from $2$-dimensional VASS. Our third result, by means of a novel family of products of integer fractions, shows that $4$-dimensional VASS can have doubly exponentially long shortest reachability witnessing runs. The smallest dimension for which this was previously known is~$14$.

    更新日期:2020-01-14
  • A Universal Attractor Decomposition Algorithm for Parity Games
    arXiv.cs.FL Pub Date : 2020-01-13
    Marcin Jurdziński; Rémi Morvan

    An attractor decomposition meta-algorithm for solving parity games is given that generalizes the classic McNaughton-Zielonka algorithm and its recent quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and Wojtczak (2019). The central concepts studied and exploited are attractor decompositions of dominia in parity games and the ordered trees that describe the inductive structure of attractor decompositions. The main technical results include the embeddable decomposition theorem and the dominion separation theorem that together help establish a precise structural condition for the correctness of the universal algorithm: it suffices that the two ordered trees given to the algorithm as inputs embed the trees of some attractor decompositions of the largest dominia for each of the two players, respectively. The universal algorithm yields McNaughton-Zielonka, Parys's, and Lehtinen-Schewe-Wojtczak algorithms as special cases when suitable universal trees are given to it as inputs. The main technical results provide a unified proof of correctness and deep structural insights into those algorithms. A symbolic implementation of the universal algorithm is also given that improves the symbolic space complexity of solving parity games in quasi-polynomial time from $O(d \lg n)$---achieved by Chatterjee, Dvo\v{r}\'{a}k, Henzinger, and Svozil (2018)---down to $O(\lg d)$, where $n$ is the number of vertices and $d$ is the number of distinct priorities in a parity game. This not only exponentially improves the dependence on $d$, but it also entirely removes the dependence on $n$.

    更新日期:2020-01-14
  • Decisiveness of Stochastic Systems and its Application to Hybrid Models
    arXiv.cs.FL Pub Date : 2020-01-13
    Patricia Bouyer; Thomas Brihaye; Mickael Randour; Cédric Rivière; Pierre Vandenhove

    In [ABM07], Abdulla et al. introduced the concept of decisiveness, an interesting tool for lifting good properties of finite Markov chains to denumerable ones. Later, this concept was extended to more general stochastic transition systems (STSs), allowing the design of various verification algorithms for large classes of (infinite) STSs. We further improve the understanding and utility of decisiveness in two ways. First, we provide a general criterion for proving decisiveness of general STSs. This criterion, which is very natural but whose proof is rather technical, (strictly) generalizes all known criteria from the literature. Second, we focus on stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We establish the decisiveness of a large class of SHSs and, under a few classical hypotheses from mathematical logic, we show how to decide reachability problems in this class, even though they are undecidable for general SHSs. This provides a decidable stochastic extension of o-minimal hybrid systems. [ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007. Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).

    更新日期:2020-01-14
  • Good-for-games $ω$-Pushdown Automata
    arXiv.cs.FL Pub Date : 2020-01-13
    Karoliina Lehtinen; Martin Zimmermann

    We introduce good-for-games $\omega$-pushdown automata ($\omega$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the run constructed thus far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results show that $\omega$-GFG-PDA are more expressive than deterministic $\omega$-pushdown automata and that solving infinite games with winning conditions specified by $\omega$-GFG-PDA is EXPTIME-complete, i.e., we have identified a new class of $\omega$-contextfree winning conditions for which solving games is decidable. This means in particular that the universality problem is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $\omega$-GFG-PDA and decidability of good-for-gameness of $\omega$-pushdown automata and languages.

    更新日期:2020-01-14
  • One-Clock Priced Timed Games are PSPACE-hard
    arXiv.cs.FL Pub Date : 2020-01-13
    John Fearnley; Rasmus Ibsen-Jensen; Rahul Savani

    The main result of this paper is that computing the value of a one-clock priced timed game (OCPTG) is PSPACE-hard. Along the way, we provide a family of OCPTGs that have an exponential number of event points. Both results hold even in very restricted classes of games such as DAGs with treewidth three. Finally, we provide a number of positive results, including polynomial-time algorithms for even more restricted classes of OCPTGs such as trees.

    更新日期:2020-01-14
  • Synchronization under Dynamic Constraints
    arXiv.cs.FL Pub Date : 2019-10-04
    Petra Wolf

    Imagine an assembly line where a box with a lid and liquid in it enters in some unknown orientation. The box should leave the line with the open lid facing upwards with the liquid still in it. To save costs there are no complex sensors or image recognition software available on the assembly line, so a reset sequence needs to be computed. But how can the dependencies of the deforming impact of a transformation of the box, such as 'do not tilt the box over when the lid is open' or 'open the lid again each time it gets closed' be modeled? We present three attempts to model constraints of these kinds on the order in which the states of an automaton are transitioned by a synchronizing word. The first two concepts relate the last visits of states and form constraints on which states still need to be reached, whereas the third concept concerns the first visits of states and forms constraints on which states might still be reached. We examine the computational complexity of different variants of the problem, whether an automaton can be synchronized with a word that respects the constraints defined in the respective concept, and obtain nearly a full classification. While most of the problems are PSPACE-complete we also observe NP-complete variants and variants solvable in polynomial time. We will also observe a drop of the complexity if we track the orders of states on several paths simultaneously instead of tracking the set of active states. Further, we give upper bounds on the length of a synchronizing word depending on the size of the input relation and show that the Cerny conjecture holds for partial weakly acyclic automata.

    更新日期:2020-01-14
  • Finite models and the theory of concatenation
    arXiv.cs.FL Pub Date : 2019-12-12
    Dominik D. Freydenberger; Liat Peterfreund

    We propose FC, a logic on words that combines the previous approaches of finite-model theory and the theory of concatenation, and that has immediate applications in information extraction and database theory in the form of document spanners. Like the theory of concatenation, FC is built around word equations; in contrast to it, its semantics are defined to only allow finite models, by limiting the universe to a word and all its subwords. As a consequence of this, FC has many of the desirable properties of FO[<], while being far more expressive. Most noteworthy among these desirable properties are sufficient criteria for efficient model checking and capturing various complexity classes by extending the logic with appropriate closure or iteration operators. These results allows us to obtain new insights into and techniques for the expressive power and efficient evaluation of document spanners. In fact, FC provides us with a general framework for reasoning about words that has potential applications far beyond document spanners.

    更新日期:2020-01-14
  • Equivalence of Linear Tree Transducers with Output in the Free Group
    arXiv.cs.FL Pub Date : 2020-01-10
    Raphaela Löbel; Michael Luttenberger; Helmut Seidl

    We show that equivalence of deterministic linear tree transducers can be decided in polynomial time when their outputs are interpreted over the free group. Due to the cancellation properties offered by the free group, the required constructions are not only more general, but also simpler than the corresponding constructions for proving equivalence of deterministic linear tree-to-word transducers.

    更新日期:2020-01-13
  • Monitoring Event Frequencies
    arXiv.cs.FL Pub Date : 2019-10-14
    Thomas Ferrère; Thomas A. Henzinger; Bernhard Kragl

    The monitoring of event frequencies can be used to recognize behavioral anomalies, to identify trends, and to deduce or discard hypotheses about the underlying system. For example, the performance of a web server may be monitored based on the ratio of the total count of requests from the least and most active clients. Exact frequency monitoring, however, can be prohibitively expensive; in the above example it would require as many counters as there are clients. In this paper, we propose the efficient probabilistic monitoring of common frequency properties, including the mode (i.e., the most common event) and the median of an event sequence. We define a logic to express composite frequency properties as a combination of atomic frequency properties. Our main contribution is an algorithm that, under suitable probabilistic assumptions, can be used to monitor these important frequency properties with four counters, independent of the number of different events. Our algorithm samples longer and longer subwords of an infinite event sequence. We prove the almost-sure convergence of our algorithm by generalizing ergodic theory from increasing-length prefixes to increasing-length subwords of an infinite sequence. A similar algorithm could be used to learn a connected Markov chain of a given structure from observing its outputs, to arbitrary precision, for a given confidence.

    更新日期:2020-01-13
  • The repetition threshold for binary rich words
    arXiv.cs.FL Pub Date : 2019-08-08
    James D. Currie; Lucas Mol; Narad Rampersad

    A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article, we give a structure theorem for infinite binary rich words that avoid $14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.

    更新日期:2020-01-10
  • Incremental Monoidal Grammars
    arXiv.cs.FL Pub Date : 2020-01-02
    Dan Shiebler; Alexis Toumi; Mehrnoosh Sadrzadeh

    In this work we define formal grammars in terms of free monoidal categories, along with a functor from the category of formal grammars to the category of automata. Generalising from the Booleans to arbitrary semirings, we extend our construction to weighted formal grammars and weighted automata. This allows us to link the categorical viewpoint on natural language to the standard machine learning notion of probabilistic language model.

    更新日期:2020-01-09
  • VC-dimensions of nondeterministic finite automata for words of equal length
    arXiv.cs.FL Pub Date : 2020-01-07
    Bjørn Kjos-Hanssen; Clyde James Felix; Sun Young Kim; Ethan Lamb; Davin Takahashi

    Ishigami and Tani studied VC-dimensions of deterministic finite automata. We obtain analogous results for the nondeterministic case by extending a result of Champarnaud and Pin, who proved that the maximal deterministic state complexity of a set of binary words of length $n$ is \[ \sum_{i=0}^n\min(2^i,2^{2^{n-i}}-1). \] We show that for the nondeterministic case, if we fully restrict attention to words of length $n$, then we at most need the strictly increasing initial terms in this sum.

    更新日期:2020-01-09
  • An Incompressibility Theorem for Automatic Complexity
    arXiv.cs.FL Pub Date : 2019-08-26
    Bjørn Kjos-Hanssen

    Shallit and Wang showed that the automatic complexity $A(x)\ge n/13$ for almost all $x\in\{0,1\}^n$. They also stated that Holger Petersen had informed them that the constant 13 can be reduced to 7. Here we show that it can be reduced to $2+\epsilon$ for any $\epsilon>0$.

    更新日期:2020-01-09
  • A duality theoretic view on limits of finite structures
    arXiv.cs.FL Pub Date : 2019-07-09
    Mai Gehrke; Tomáš Jakl; Luca Reggio

    A systematic theory of structural limits for finite models has been developed by Nesetril and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises --- via Stone-Priestley duality and the notion of types from model theory --- by enriching the expressive power of first-order logic with certain ``probabilistic operators''. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively.

    更新日期:2020-01-08
  • Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear and additive cellular automata
    arXiv.cs.FL Pub Date : 2019-07-19
    Alberto Dennunzio; Enrico Formenti; Darij Grinberg; Luciano Margara

    Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this paper (Thm.~\ref{thm.finpowmat.main}) states that the set $\left\{ A^0,A^1,A^2,\ldots\right\}$ is finite if and only if the set $\left\{ B^0,B^1,B^2,\ldots\right\}$ is finite. We apply this result to Cellular Automata (CA). Indeed, it gives a complete and easy-to-check characterization of sensitivity to initial conditions and equicontinuity for linear CA over the alphabet $\mathbb{K}^n$ for $\mathbb{K} = \mathbb{Z}/m\Z$ (Thm.~\ref{froblca}), \ie, CA in which the local rule is defined by $n\times n$-matrices with elements in $\mathbb{Z}/m\Z$. To prove our main result, we derive an integrality criterion for matrices (Thm\ref{thm.finpowmat.char-int} and Prop.\ref{prop.finpowmat.char-int-conv}) that is likely of independent interest. Namely, let $\mathbb{K}$ be any commutative ring (not necessarily finite), and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Consider any $n \times n$-matrix $A$ over $\mathbb{L}$. Then, $A \in \mathbb{L}^{n \times n}$ is integral over $\mathbb{K}$ (that is, there exists a monic polynomial $f \in \mathbb{K}\left[t\right]$ satisfying $f\left(A\right) = 0$) if and only if all coefficients of the characteristic polynomial of $A$ are integral over $\mathbb{K}$. The proof of this fact relies on a strategic use of exterior powers (a trick pioneered by Gert Almkvist). Furthermore, we extend the decidability result concerning sensitivity and equicontinuity to the wider class of additive CA over a finite abelian group. For such CA, we also prove the decidability of injectivity, surjectivity, topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to the latter.

    更新日期:2020-01-08
  • Hardware/Software Co-verification Using Path-based Symbolic Execution
    arXiv.cs.FL Pub Date : 2020-01-05
    Rajdeep Mukherjee; Saurabh Joshi; John O'Leary; Daniel Kroening; Tom Melham

    Conventional tools for formal hardware/software co-verification use bounded model checking techniques to construct a single monolithic propositional formula. Formulas generated in this way are extremely complex and contain a great deal of irrelevant logic, hence are difficult to solve even by the state-of-the-art Satis ability (SAT) solvers. In a typical hardware/software co-design the firmware only exercises a fraction of the hardware state-space, and we can use this observation to generate simpler and more concise formulas. In this paper, we present a novel verification algorithm for hardware/software co-designs that identify partitions of the firmware and the hardware logic pertaining to the feasible execution paths by means of path-based symbolic simulation with custom path-pruning, property-guided slicing and incremental SAT solving. We have implemented this approach in our tool COVERIF. We have experimentally compared COVERIF with HW-CBMC, a monolithic BMC based co-verification tool, and observed an average speed-up of 5X over HW-CBMC for proving safety properties as well as detecting critical co-design bugs in an open-source Universal Asynchronous Receiver Transmitter design and a large SoC design.

    更新日期:2020-01-07
  • New tools for state complexity
    arXiv.cs.FL Pub Date : 2018-07-02
    Pascal Caron; Edwin Hamel-De le court; Jean-Gabriel Luque; Bruno Patrou

    A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. We revisit some language transformation algorithms in terms of modifier and monster. These new theoretical concepts allow one to find easily some state complexities. We illustrate this by retrieving the state complexity of the Star of Intersection and the one of the Square root operation.

    更新日期:2020-01-07
  • Words With Few Palindromes, Revisited
    arXiv.cs.FL Pub Date : 2019-11-27
    Lukas Fleischer; Jeffrey Shallit

    In 2013, Fici and Zamboni proved a number of theorems about finite and infinite words having only a small number of factors that are palindromes. In this paper we rederive some of their results, and obtain some new ones, by a different method based on finite automata.

    更新日期:2020-01-07
  • Representing Unordered Data Using Multiset Automata and Complex Numbers
    arXiv.cs.FL Pub Date : 2020-01-02
    Justin DeBenedetto; David Chiang

    Unordered, variable-sized inputs arise in many settings across multiple fields. The ability for set- and multiset- oriented neural networks to handle this type of input has been the focus of much work in recent years. We propose to represent multisets using complex-weighted multiset automata and show how the multiset representations of certain existing neural architectures can be viewed as special cases of ours. Namely, (1) we provide a new theoretical and intuitive justification for the Transformer model's representation of positions using sinusoidal functions, and (2) we extend the DeepSets model to use complex numbers, enabling it to outperform the existing model on an extension of one of their tasks.

    更新日期:2020-01-06
  • Abelian periods of factors of Sturmian words
    arXiv.cs.FL Pub Date : 2019-05-15
    Jarkko Peltomäki

    We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle $\alpha$ with continued fraction expansion $[0; a_1, a_2, \ldots]$ is either $tq_k$ with $1 \leq t \leq a_{k+1}$ (a multiple of a denominator $q_k$ of a convergent of $\alpha$) or $q_{k,\ell}$ (a denominator $q_{k,\ell}$ of a semiconvergent of $\alpha$). This result generalizes a result of Fici et. al stating that the abelian period set of the Fibonacci word is the set of Fibonacci numbers. A characterization of the Fibonacci word in terms of its abelian period set is obtained as a corollary.

    更新日期:2020-01-04
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