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Constructing finite least Kripke models for positive logic programs in serial regular grammar logics
 Logic Journal of the IGPL
"... A serial contextfree grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form ✷tϕ → ✷s1... ✷skϕ. Such an inclusion axiom corresponds to the grammar rule t → s1... sk. Thus the inclusion axioms of L capture a contextfree grammar G( ..."
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A serial contextfree grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form ✷tϕ → ✷s1... ✷skϕ. Such an inclusion axiom corresponds to the grammar rule t → s1... sk. Thus the inclusion axioms of L capture a contextfree grammar G(L). If for every modal index t, the set of words derivable from t using G(L) is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positive multimodal logic program P and a set of finite automata specifying a serial regular grammar logic L, constructs a finite least Lmodel of P. (A model M is less than or equal to model M ′ if for every positive formula ϕ, if M  = ϕ then M ′  = ϕ.) A least Lmodel M of P has the property that for every positive formula ϕ, P  = ϕ iff M  = ϕ. The algorithm runs in exponential time and returns a model with size 2 O(n3). We give examples of P and L, for both of the case when L is fixed or P is fixed, such that every finite least Lmodel of P must have size 2 Ω(n). We also prove that if G is a contextfree grammar and L is the serial grammar logic corresponding to G then there exists a finite least Lmodel of ✷sp iff the set of words derivable from s using G is a regular language. 1
Weakening Horn knowledge bases in regular description logics to have PTIME data complexity
 Proceedings of ADDCT’2007
, 2007
"... This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple (R,T,A) consisting of an RBox R of assertions about roles, a TBox T of global assumptions about concepts, and an ABox A of ..."
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Cited by 5 (4 self)
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This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple (R,T,A) consisting of an RBox R of assertions about roles, a TBox T of global assumptions about concepts, and an ABox A of facts about individuals (objects) and roles. The instance checking problem in a DL is to check whether a given individual a is an instance of a concept C w.r.t. a knowledge base (R,T,A), written as (R,T,A)  = C(a). This problem in DLs including the basic description logic ALC (with R = /0) is EXPTIMEhard. From the point of view of deductive databases, A is assumed to be much larger than R and T, and it makes sense to consider the data complexity, which is measured when the query consisting of R, T, C, a is fixed while A varies as input data. It is desirable to find and study fragments of DLs with PTIME data complexity. Several authors have recently introduced a number of Horn fragments of DLs with PTIME data complexity [2,1,3]. The most expressive fragment from those is HornSHIQ introduced by Hustadt et al. [3]. It assumes, however, that the constructor ∀R.C does not occur in bodies of program clauses and goals. The data complexity of the “general Horn fragment of ALC ” is coNPhard [6]. So, to obtain PTIME data complexity one has to adopt some restrictions for the “general Horn fragments of DLs”. The goal is to find as less restrictive conditions as possible. A RBox is a finite set of assertions of the form Rs1 ◦... ◦ Rs ⊑ Rt, whereRs1,..., k Rs, Rt are role names. A regular RBox is an RBox whose set of corresponding grammar k rules t → s1...sk forms a grammar such that the set of words derivable from any symbol s using the grammar is a regular language specified by a finite automaton. We assume that the corresponding finite automata specifying R are given when R is considered. By R eg we denote ALC extended with regular RBoxes. We extend the language of ALC and R eg with the concept constructor ∀∃, which creates a concept ∀∃Rt.C from a role name Rt and a concept C.LetSem1(∀∃Rt.C)={∀Rt.C,∃Rt.⊤} and Sem2,R (∀∃Rt.C)= {∀Rt.C} ∪{∀Rs1...∀Rsi−1∃Rsi.⊤Rs1 ◦ ···◦Rs ⊑ Rt is a consequence of R and 1 ≤
Analytic cutfree tableaux for regular modal logics of agent beliefs
 Proceedings of CLIMA VIII, vol. 5056 of LNAI
, 2008
"... Abstract. We present a sound and complete tableau calculus for a class BReg of extended regular modal logics which contains useful epistemic logics for reasoning about agent beliefs. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. Applying sound ..."
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Abstract. We present a sound and complete tableau calculus for a class BReg of extended regular modal logics which contains useful epistemic logics for reasoning about agent beliefs. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. Applying sound global caching to the calculus, we obtain the first optimal (EXPTime) tableau decision procedure for BReg. We demonstrate the usefulness of BReg logics and our tableau calculus using the wise men puzzle and its modified version, which requires axiom (5) for single agents. 1
Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
Hasse diagrams for classes of deterministic bottomup treetotreeseries transformations
 Theoret. Comput. Sci
, 2005
"... The relationship between classes of treetotreeseries and otreetotreeseries transformations, which are computed by restricted deterministic bottomup weighted tree transducers, is investigated. Essentially, these transducers are deterministic bottomup tree series transducers, except that the ..."
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The relationship between classes of treetotreeseries and otreetotreeseries transformations, which are computed by restricted deterministic bottomup weighted tree transducers, is investigated. Essentially, these transducers are deterministic bottomup tree series transducers, except that the former are defined over monoids whereas the latter are defined over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of nondeletion, linearity, totality, and homomorphism can equivalently be defined for deterministic bottomup weighted tree transducers. Using wellknown results of classical tree transducer theory and also new results on deterministic weighted tree transducers, classes of treetotreeseries and otreetotreeseries transformations computed by restricted deterministic bottomup weighted tree transducers are ordered by set inclusion. More precisely, for every commutative monoid and all sensible combinations of the above mentioned restrictions, the inclusion relation of the classes of treetotreeseries and otreetotreeseries transformations is completely conveyed by means of Hasse diagrams.
The Complexity of Regularity in Grammar Logics
 J. of Logic and Computation
"... A modal reduction principle of the form [i1]... [in]p ⇒ [j1]... [jn ′]p can be viewed as a production rule i1 ·... · in → j1 ·... · jn ′ in a formal grammar. We study the extensions of the multimodal logic Km with m independent K modal connectives by finite addition of axiom schemes of the above for ..."
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A modal reduction principle of the form [i1]... [in]p ⇒ [j1]... [jn ′]p can be viewed as a production rule i1 ·... · in → j1 ·... · jn ′ in a formal grammar. We study the extensions of the multimodal logic Km with m independent K modal connectives by finite addition of axiom schemes of the above form such that the associated finite set of production rules forms a regular grammar. We show that given a regular grammar G and a modal formula φ, deciding whether the formula is satisfiable in the extension of Km with axiom schemes from G can be done in deterministic exponentialtime in the size of G and φ, and this problem is complete for this complexity class. Such an extension of Km is called a regular grammar logic. The proof of the exponentialtime upper bound is extended to PDLlike extensions of Km and to global logical consequence and global satisfiability problems. Using an equational characterization of contextfree languages, we show that by replacing the regular grammars by linear ones, the above problem becomes undecidable. The last part of the paper presents nontrivial classes of exponential time complete regular grammar logics.
A Tableau Calculus with AutomatonLabelled Formulae for Regular Grammar Logics
"... Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automatonlabelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cutfree and has the analytic superf ..."
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Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automatonlabelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar logics using our calculus. 1
doi:10.1093/comjnl/bxm049 FixedParameter Algorithms
"... We survey the use of fixedparameter algorithms in the field of phylogenetics, which is the study of evolutionary relationships. The central problem in phylogenetics is the reconstruction of the evolutionary history of biological species, but its methods also apply to linguistics, philology or archi ..."
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We survey the use of fixedparameter algorithms in the field of phylogenetics, which is the study of evolutionary relationships. The central problem in phylogenetics is the reconstruction of the evolutionary history of biological species, but its methods also apply to linguistics, philology or architecture. A basic computational problem is the reconstruction of a likely phylogeny (genealogical tree) for a set of species based on observed differences in the phenotype like color or form of limbs, based on differences in the genotype like mutated nucleotide positions in the DNA sequence, or based on given partial phylogenies. Ideally, one would like to construct socalled perfect phylogenies, which arise from a very simple evolutionary model, but in practice one must often be content with phylogenies whose ‘distance from perfection ’ is as small as possible. The computation of phylogenies has applications in seemingly unrelated areas such as genomic sequencing and finding and understanding genes. The numerous computational problems arising in phylogenetics often are NPcomplete, but for many natural parametrizations they can be solved using fixedparameter algorithms.
Simple Words in Equality Sets
, 1996
"... It is well known that equality sets between two morphisms possess a remarkable generative capacity: an arbitrary recursively enumerable set is obtained from an equality set by certain simple operations. Interconnections between simplicity of computations and structural primitivity of words in equali ..."
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It is well known that equality sets between two morphisms possess a remarkable generative capacity: an arbitrary recursively enumerable set is obtained from an equality set by certain simple operations. Interconnections between simplicity of computations and structural primitivity of words in equality sets have also been observed. The paper discusses recent work in this area, pointing out certain open problems and emphasizing new directions for research. TUCS Research Group