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19
A Tutorial on (Co)Algebras and (Co)Induction
 EATCS Bulletin
, 1997
"... . Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition pr ..."
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Cited by 228 (34 self)
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. Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and nonwellfounded sets) are generally of this kind. In general, dynamical systems with a hidden, blackbox state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...
A characterization of Sturmian words by return words
"... : We present a new characterization of Sturmian words using return words. Considering each occurrence of a word w in a recurrent word, we define the set of return words over w to be the set of all distinct words beginning with an occurrence of w and ending exactly before the next occurrence of w in ..."
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Cited by 19 (6 self)
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: We present a new characterization of Sturmian words using return words. Considering each occurrence of a word w in a recurrent word, we define the set of return words over w to be the set of all distinct words beginning with an occurrence of w and ending exactly before the next occurrence of w in the infinite word. It is shown that an infinite word is a Sturmian word if and only if for each nonempty word w appearing in the infinite word, the cardinality of the set of return words over w is equal to two. 2 1 Introduction Sturmian words are infinite words over a binary alphabet with exactly n+1 factors of length n for each n 0 (see [2, 6, 12]). In fact, the study of the Sturmian words appears in many areas like combinatorics on words ([6]), symbolic dynamics ([3, 1, 7, 21]), theoretical computer science ([5, 17]) and tilings ([8, 14, 20, 22, 24]). The Sturmian words have many equivalent characterizations (see for a complete presentation of Sturmian words [6]) using complexity func...
The Size of Power Automata
 Mathematical Foundations of Computer Science, volume 2136 of SLNCS
, 1994
"... We describe a class of simple transitive semiautomata that exhibit full exponential blowup during deterministic simulation. For arbitrary semiautomata we show that it is PSPACEcomplete to decide whether the size of the accessible part of their power automata exceeds a given bound. 1 Motivation C ..."
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Cited by 13 (6 self)
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We describe a class of simple transitive semiautomata that exhibit full exponential blowup during deterministic simulation. For arbitrary semiautomata we show that it is PSPACEcomplete to decide whether the size of the accessible part of their power automata exceeds a given bound. 1 Motivation Consider the following semiautomaton A = h[n]; ; i where [n] = f1; : : : ; ng, = fa; b; cg and the transition function is given by a a cyclic shift on [n]; b the transposition that interchanges 1 and 2, c sends 1 and 2 to 2, identity elsewhere. It is wellknown that A has a transition semigroup of maximal size n n , see [13]. In other words, every function f : [n] ! [n] is already of the form w for some word w. Note that a ; b can be replaced by any other pair of generators for the symmetric group on n points, and c can be replaced by any function whose range has cardinality n 1. It was shown by Salomaa that, for a threeletter alphabet , those are the only choices that produ...
Enumerative Sequences of Leaves in Rational Trees
, 1997
"... . We prove that any INrational sequence s = (sn)n1 of nonnegative integers satisfying the Kraft strict inequality P n1 snk \Gamman ! 1 is the enumerative sequence of leaves by height of a rational kary tree. Particular cases of this result had been previously proven. We give some partial resu ..."
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Cited by 7 (7 self)
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. We prove that any INrational sequence s = (sn)n1 of nonnegative integers satisfying the Kraft strict inequality P n1 snk \Gamman ! 1 is the enumerative sequence of leaves by height of a rational kary tree. Particular cases of this result had been previously proven. We give some partial results in the equality case. 1 Introduction This paper is a study of problems linked with coding and symbolic dynamics. The results can be considered as an extension of the old results of Huffman, Kraft, McMillan and Shannon on source coding. We actually prove results on rational sequences of integers that can be realized as the enumerative sequence of leaves in a rational tree. Let s be an INrational sequence of nonnegative numbers, that is a sequence s = (s n ) n1 such that s n is the number of paths of length n going from an initial state to a final state in a finite multigraph or a finite automaton. We say that s satisfies the Kraft inequality for a positive integer k if P n1 s n k ...
A note on synchronized automata and Road Coloring Problem
 Developments in Language Theory (5th Int. Conf., Vienna, 2001), Lecture Notes in Computer Science
, 2002
"... Abstract. We consider a problem of labeling a directed multigraph so that it becomes a synchronized finite automaton, as an ultimate goal to solve the famous Road Coloring Conjecture, cf. [1,2]. We introduce a relabeling method which can be used for a large class of automata to improve their “degree ..."
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Cited by 7 (0 self)
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Abstract. We consider a problem of labeling a directed multigraph so that it becomes a synchronized finite automaton, as an ultimate goal to solve the famous Road Coloring Conjecture, cf. [1,2]. We introduce a relabeling method which can be used for a large class of automata to improve their “degree of synchronization”. This allows, for example, to formulate the conjecture in several equivalent ways. 1
PeriodicFiniteType Shift Spaces
 in IEEE Int. Symp. Information Theory
, 2001
"... We introduc e thec lass of periodic finitetype (PFT) shift spac es. They are the subc lass of shift spac es defined by a finite set of periodic ally forbidden words. Examples of PFT shifts arise naturally in thec ontext of distanc eenhanc ingc odes for partialresponsec hannels. We show that thec ..."
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Cited by 7 (2 self)
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We introduc e thec lass of periodic finitetype (PFT) shift spac es. They are the subc lass of shift spac es defined by a finite set of periodic ally forbidden words. Examples of PFT shifts arise naturally in thec ontext of distanc eenhanc ingc odes for partialresponsec hannels. We show that thec lass of PFT shifts represent a proper superset of the finitetype shift spac es and a proper subset of almostfinitetype shift spac8 . We prove several properties of labeled graphs that present PFT shifts. For a given PFT shift spac e, we identify a finite set of forbidden words  referred to as "periodic first o#enders"  that define the shift spac e and that satisfyc8 tain minimality properties. Finally, we present ane#c4k t algorithm for c onstruc ting labeled graphs that present PFT shift spac es. 1 Introducti7 Magnet i# recordi#( systems often make use ofbi#5)) codes that di# allow the appearance of certai# sequences that are problemat i# i# the data recordi#5 or retri## al process. In systems usi#6 parti# lresponse equa li#a t i# n and sequence detecti# n, socalled "di#L6 nceenhanci #( " constrai#tr codes have been proposed toi#p5( se the mi#66 umdi#6# nce at the output of the underlyi#4 i# tersymbol i# terference channel by forbi#)6p4 afini#( set ofbi#L65 patterns, e.g., [1, 2, 3]. The set of allowable code sequences are generated by paths i# a labeled, di#(L75# graph. Such sets of constra i#ra sequences are referred to as # B.E. Moision was with the MathematicA Sc0119k Researc h Center, Luc8 t Tec hnologies, Murray Hill, NJ. He is now with the Communic tions Systems and Researc h Sec18 n, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 911018099 (email: bmoision@shannon.jpl.nasa.gov). P. H. Siegel is with the Center for Magnetic Rec rding Rese...
Enumerative Sequences of Leaves and Nodes in Rational Trees
, 1997
"... We prove that any INrational sequence s = (s n ) n1 of nonnegative integers satisfying the Kraft strict inequality P n1 s n k \Gamman ! 1 is the enumerative sequence of leaves by height of a rational kary tree. We give an efficient algorithm to get a kary rational tree. Particular cases of th ..."
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Cited by 6 (5 self)
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We prove that any INrational sequence s = (s n ) n1 of nonnegative integers satisfying the Kraft strict inequality P n1 s n k \Gamman ! 1 is the enumerative sequence of leaves by height of a rational kary tree. We give an efficient algorithm to get a kary rational tree. Particular cases of this result had been previously proven. We give some partial results in the case of equality. Especially we solve this question when the associated sequence of internal nodes has a primitive linear representation. 1 Introduction This paper is a study of problems linked with coding and symbolic dynamics. The results can be considered as an extension of the old results of Huffman, Kraft, McMillan and Shannon on source coding. We actually prove results on rational sequences of integers that can be realized as the enumerative sequence of leaves or nodes in a rational tree. Let s be an INrational sequence of nonnegative numbers, that is a sequence s = (s n ) n1 such that s n is the number of pat...
automata, a Hybrid System for Computational Automata Theory
 CIAA 2002
, 2002
"... We present a system that performs computations on nite state machines, syntactic semigroups, and onedimensional cellular automata. ..."
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Cited by 6 (3 self)
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We present a system that performs computations on nite state machines, syntactic semigroups, and onedimensional cellular automata.
Minimizing local automata
"... Abstract — We design an algorithm that minimizes irreducible deterministic local automata by a sequence of state mergings. Two states can be merged if they have exactly the same outputs. The running time of the algorithm is O(min(m(n −r +1), m log n)), where m is the number of edges, n the number of ..."
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Cited by 6 (2 self)
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Abstract — We design an algorithm that minimizes irreducible deterministic local automata by a sequence of state mergings. Two states can be merged if they have exactly the same outputs. The running time of the algorithm is O(min(m(n −r +1), m log n)), where m is the number of edges, n the number of states of the automaton, and r the number of states of the minimized automaton. In particular, the algorithm is linear when the automaton is already minimal and contrary to Hopcroft’s minimisation algorithm that has a O(kn log n) running time in this case, where k is the size of the alphabet, and that applies only to complete automata. (Note that kn ≥ m.) While Hopcroft’s algorithm relies on a “negative strategy”, starting from a partition with a single class of all states, and partitioning classes when it is discovered that two states cannot belong to the same class, our algorithm relies on a “positive strategy”, starting from the trivial partition for which each class is a singleton. Two classes are then merged when their leaders have the same outputs. The algorithm applies to irreducible deterministic local automata, where all states are considered both initial and final. These automata, also called covers, recognize symbolic dynamical shifts of finite type. They serve to present a large class of constrained channels, the class of finite memory systems, used for channel coding purposes. The algorithm also applies to irreducible deterministic automata that are leftclosing and have a synchronizing word. These automata present shifts that are called almost of finite type. Almostoffinitetype shifts make a meaningful class of shifts, intermediate between finite type shifts and sofic shifts.
Computational Complementarity and Sofic Shifts
, 1997
"... Finite automata (with outputs but no initial states) have been extensively used as models of computational complementarity, a property which mimics the physical complementarity. All this work was focussed on "frames", i.e., on fixed, static, local descriptions of the system behaviour. In this paper ..."
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Cited by 4 (4 self)
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Finite automata (with outputs but no initial states) have been extensively used as models of computational complementarity, a property which mimics the physical complementarity. All this work was focussed on "frames", i.e., on fixed, static, local descriptions of the system behaviour. In this paper we are mainly interested in the asymptotical description of complementarity.To this aim we will study the asymptotical behaviour of two complementarity principles by associating to every incomplete deterministic automaton (with outputs, but no initial state) certain sofic shifts: automata having the same behaviour correspond to a unique sofic shift. In this way, a class of sofic shifts reflecting complementarity will be introduced and studied. We will prove that there is a strong relation between "local complementarity", as it is perceived at the level of "frames", and "asymptotical complementarity" as it is described by the sofic shift.