76 research outputs found
Global Numerical Constraints on Trees
We introduce a logical foundation to reason on tree structures with
constraints on the number of node occurrences. Related formalisms are limited
to express occurrence constraints on particular tree regions, as for instance
the children of a given node. By contrast, the logic introduced in the present
work can concisely express numerical bounds on any region, descendants or
ancestors for instance. We prove that the logic is decidable in single
exponential time even if the numerical constraints are in binary form. We also
illustrate the usage of the logic in the description of numerical constraints
on multi-directional path queries on XML documents. Furthermore, numerical
restrictions on regular languages (XML schemas) can also be concisely described
by the logic. This implies a characterization of decidable counting extensions
of XPath queries and XML schemas. Moreover, as the logic is closed under
negation, it can thus be used as an optimal reasoning framework for testing
emptiness, containment and equivalence
Partial Derivative Automaton for Regular Expressions with Shuffle
We generalize the partial derivative automaton to regular expressions with
shuffle and study its size in the worst and in the average case. The number of
states of the partial derivative automata is in the worst case at most 2^m,
where m is the number of letters in the expression, while asymptotically and on
average it is no more than (4/3)^m
A Tree Logic with Graded Paths and Nominals
Regular tree grammars and regular path expressions constitute core constructs
widely used in programming languages and type systems. Nevertheless, there has
been little research so far on reasoning frameworks for path expressions where
node cardinality constraints occur along a path in a tree. We present a logic
capable of expressing deep counting along paths which may include arbitrary
recursive forward and backward navigation. The counting extensions can be seen
as a generalization of graded modalities that count immediate successor nodes.
While the combination of graded modalities, nominals, and inverse modalities
yields undecidable logics over graphs, we show that these features can be
combined in a tree logic decidable in exponential time
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
LNCS
We provide a procedure for detecting the sub-segments of an incrementally observed Boolean signal ω that match a given temporal pattern ϕ. As a pattern specification language, we use timed regular expressions, a formalism well-suited for expressing properties of concurrent asynchronous behaviors embedded in metric time. We construct a timed automaton accepting the timed language denoted by ϕ and modify it slightly for the purpose of matching. We then apply zone-based reachability computation to this automaton while it reads ω, and retrieve all the matching segments from the results. Since the procedure is automaton based, it can be applied to patterns specified by other formalisms such as timed temporal logics reducible to timed automata or directly encoded as timed automata. The procedure has been implemented and its performance on synthetic examples is demonstrated
Deciding Definability by Deterministic Regular Expressions
International audienceWe investigate the complexity of deciding whether a given regular language can be defined with a deterministic regular expression. Our main technical result shows that the problem is Pspace-complete if the input language is represented as a regular expression or nondeterministic finite automaton. The problem becomes Expspace-complete if the language is represented as a regular expression with counters
Succinctness and Formula Size Games
Tämä väitöskirja tutkii erilaisten logiikoiden tiiviyttä kaavan pituuspelien avulla. Logiikan tiiviys viittaa ominaisuuksien ilmaisemiseen tarvittavien kaavojen kokoon. Kaavan pituuspelit ovat hyväksi todettu menetelmä tiiviystulosten todistamiseen. Väitöskirjan kontribuutio on kaksiosainen. Ensinnäkin väitöskirjassa määritellään kaavan pituuspeli useille logiikoille ja tarjotaan näin uusia menetelmiä tulevaan tutkimukseen. Toiseksi näitä pelejä ja muita menetelmiä käytetään tiiviystulosten todistamiseen tutkituille logiikoille.
Tarkemmin sanottuna väitöskirjassa määritellään uudet parametrisoidut kaavan pituuspelit perusmodaalilogiikalle, modaaliselle μ-kalkyylille, tiimilauselogiikalle ja yleistetyille säännöllisille lausekkeille. Yleistettyjen säännöllisten lausekkeiden pelistä esitellään myös variantit, jotka vastaavat säännöllisiä lausekkeita ja uusia “RE over star-free” -lausekkeita, joissa tähtiä ei esiinny komplementtien sisällä.
Pelejä käytetään useiden tiiviystulosten todistamiseen. Predikaattilogiikan näytetään olevan epäelementaarisesti tiiviimpi kuin perusmodaalilogiikka ja modaalinen μ-kalkyyli. Tiimilauselogiikassa tutkitaan systemaattisesti yleisten riippuvuuksia ilmaisevien atomien määrittelemisen tiiviyttä. Klassinen epäelementaarinen tiiviysero predikaattilogiikan ja säännöllisten lausekkeiden välillä osoitetaan uudelleen yksinkertaisemmalla tavalla ja saadaan tähtien lukumäärälle “RE over star-free” -lausekkeissa hierarkia ilmaisuvoiman suhteen.
Monissa yllämainituista tuloksista hyödynnetään eksplisiittisiä kaavoja peliargumenttien lisäksi. Tällaisia kaavoja ja tyyppien laskemista hyödyntäen saadaan epäelementaarisia ala- ja ylärajoja yksittäisten sanojen määrittelemisen tiiviydelle predikaattilogiikassa ja monadisessa toisen kertaluvun logiikassa.This thesis studies the succinctness of various logics using formula size games. The succinctness of a logic refers to the size of formulas required to express properties. Formula size games are some of the most successful methods of proof for results on succinctness. The contribution of the thesis is twofold. Firstly, we define formula size games for several logics, providing methods for future research. Secondly, we use these games and other methods to prove results on the succinctness of the studied logics.
More precisely, we develop new parameterized formula size games for basic modal logic, modal μ-calculus, propositional team logic and generalized regular expressions. For the generalized regular expression game we introduce variants that correspond to regular expressions and the newly defined RE over star-free expressions, where stars do not occur inside complements.
We use the games to prove a number of succinctness results. We show that first-order logic is non-elementarily more succinct than both basic modal logic and modal μ-calculus. We conduct a systematic study of the succinctness of defining common atoms of dependency in propositional team logic. We reprove a classic non-elementary succinctness gap between first-order logic and regular expressions in a much simpler way and establish a hierarchy of expressive power for the number of stars in RE over star-free expressions.
Many of the above results utilize explicit formulas in addition to game arguments. We use such formulas and a type counting argument to obtain non-elementary lower and upper bounds for the succinctness of defining single words in first-order logic and monadic second-order logic
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