This paper deals with decision problems related to the star problem in trace monoids, which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by G. Richomme from 1994 [32, 33], we know that the star problem is decidable in trace monoids which do not contain a submonoid of the form {a,c}* x {b,d}*. Here, we consider a more general problem: Is it decidable whether for some recognizable trace language and some recognizable or finite trace language P the intersection R ∩ P* is recognizable? If P is recognizable, then we show that this problem is decidale iff the underlying trace monoid does not contain a submonoid of the form {a,c}* x b*. In the case of finite languages P, we show several decidability and undecidability results.

Automated proof system for logic of correlated knowledge is presented in the dissertation. The system consists of the sequent calculus GS-LCK and the proof search procedure GS-LCK-PROC. Sequent calculus is sound, complete and satisfy the properties of invertibility of rules, admissibility of weakening, contraction and cut. The procedure GS-LCK-PROC is terminating and allows to check if the sequent is provable. Also decidability of logic of correlated knowledge has been proved. Using the terminating procedure GS-LCK-PROC the validity of all formulas of logic of correlated knowledge can be checked. / Automatinė įrodymų sistema koreliatyvių žinių logikai yra pristatoma disertacijoje. Sistemą sudaro sekvencinis skaičiavimas GS-LCK ir įrodymo paieškos procedūra GS-LCK-PROC. Sekvencinis skaičiavimas yra pagrįstas, pilnas ir tenkina taisyklių apverčiamumo, silpninimo, prastinimo ir pjūvio leistinumo savybes. Procedūra GS-LCK-PROC yra baigtinė ir leidžia patikrinti, ar sekvencija yra išvedama. Taip pat buvo įrodytas koreliatyvių žinių logikos išsprendžiamumas. Naudojant baigtinę procedūra GS-LCK-PROC, visų koreliatyvių žinių logikos formulių tapatus teisingumas gali būti patikrintas.

Apresentaremos uma classe de energias potenciais $\\Pi \\in C^{\\infty}(\\Omega,R)$ que são s-decidíveis e que admitem funções auxiliares de Cetaev da forma $\\langle abla j^s\\Pi(q),q angle$, $q\\in \\Omega \\subset R^n$ que são s-resistentes. / We will present a class of potential energies $\\Pi \\in C^{\\infty}(\\Omega,R)$ that are s-decidable and that admit auxiliary functions of Cetaev of the form $\\langle abla j^s\\Pi(q),q angle$, $q \\in \\Omega \\subset R^n$ which are s-resistant.

Previous results for combining decision procedures for the word problem in the non-disjoint case do not apply to equational theories induced by modal logics - which are not disjoint for sharing the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other types of equational theories. In this paper, we present a new approach for combining decision procedures for the word problem in the non-disjoint case that applies to equational theories induced by modal logics, but is not restricted to them. The known fusion decidability results for modal logics are instances of our approach. However, even for equational theories induced by modal logics our results are more general since they are not restricted to so-called normal modal logics. / This report has also appeared as Report No. 03-03, Department of Computer Science, The University of Iowa.

The word problem for a finite set of ground identities is known to be decidable in polynomial time using congruence closure, and this is also the case if some of the function symbols are assumed to be commutative. We show that decidability in P is preserved if we add the assumption that certain function symbols f are extensional in the sense that f(s1,…,sn) ≈ f(t1,…,tn) implies s1 ≈ t1,…,sn ≈ tn. In addition, we investigate a variant of extensionality that is more appropriate for commutative function symbols, but which raises the complexity of the word problem to coNP.

In a recent research paper, we have proposed an extension of the lightweight Description Logic (DL) EL in which concepts can be defined in an approximate way. To this purpose, the notion of a graded membership function m, which instead of a Boolean membership value 0 or 1 yields a membership degree from the interval [0; 1], was introduced. Threshold concepts can then, for example, require that an individual belongs to a concept C with degree at least 0:8. Reasoning in the threshold DL T EL(m) obtained this way of course depends on the employed graded membership function m. The paper defines a specific such function, called deg, and determines the exact complexity of reasoning in T EL(deg). In addition, it shows how concept similarity measures (CSMs) ~ satisfying certain properties can be used to define graded membership functions m~, but it does not investigate the complexity of reasoning in the induced threshold DLs T EL(m~). In the present paper, we start filling this gap. In particular, we show that computability of ~ implies decidability of T EL(m~), and we introduce a class of CSMs for which reasoning in the induced threshold DLs has the same complexity as in T EL(deg).

Description Logics form a family of powerful ontology languages widely used by academics and industry experts to capture and intelligently manage knowledge about the world. A key advantage of Description Logics is their amenability to automated reasoning that enables the deduction of knowledge that has not been explicitly stated. However, in order to ensure decidability of automated reasoning algorithms, suitable restrictions are usually enforced on the shape of structures that are expressible using Description Logics. As a consequence, Description Logics fall short of expressive power when it comes to representing cyclic structures, which abound in life sciences and other disciplines. The objective of this thesis is to explore ontology languages that are better suited for the representation of structured objects. It is suggested that an alternative approach which relies on nonmonotonic existential rules can provide a promising candidate for modelling such domains. To this end, we have built a comprehensive theoretical and practical framework for the representation of structured entities along with a surface syntax designed to allow the creation of ontological descriptions in an intuitive way. Our formalism is based on nonmonotonic existential rules and exhibits a favourable balance between expressive power and computational as well as empirical tractability. In order to ensure decidability of reasoning, we introduce a number of acyclicity criteria that strictly generalise many of the existing ones. We also present a novel stratification condition that properly extends `classical' stratification and allows for capturing both definitional and conditional aspects of complex structures. The applicability of our formalism is supported by a prototypical implementation, which is based on an off-the-shelf answer set solver and is tested over a realistic knowledge base. Our experimental results demonstrate improvement of up to three orders of magnitude in comparison with previous evaluation efforts and also expose numerous modelling errors of a manually curated biochemical knowledge base. Overall, we believe that our work lays the practical and theoretical foundations of an ontology language that is well-suited for the representation of structured objects. From a modelling point of view, our approach could stimulate the adoption of a different and expressive reasoning paradigm for which robustly engineered mature reasoners are available; it could thus pave the way for the representation of a broader spectrum of knowledge. At the same time, our theoretical contributions reveal useful insights into logic-based knowledge representation and reasoning. Therefore, our results should be of value to ontology engineers and knowledge representation researchers alike.

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g. we show the decidability of these problems for recognizable languages which contain at most one non-connected trace.

This thesis explores the significance of Godel's Theorem for an understanding of law as rules, and of legal adjudication as rule-following. It argues that Godel's Theorem, read through Wittgenstein's understanding of rules and language as a contextual activity, and through Derrida's account of 'undecidability,' offers an alternative account of the relationship of judging to justice. Instead of providing support for the 'indeterminacy' claim, Godel's Theorem illuminates the predicament of undecidability that structures any interpretation and every legal decision, and which constitutes the opening to justice. The first argument in this thesis examines Godel's proof, Wittgenstein's views on rules, and Derrida's undecidability, as manifestations of a common concern with the limits of what can be formalized. The meta-argument examines their misinterpretation and misappropriation within legal theory as a case study of just what they mean about meaning, context, and justice as necessarily co-implicated.

This thesis is concerned with aspects of the model theory of infinite soluble groups. The results proved lie on the border between group theory and model theory: the questions asked are of a model-theoretic nature but the techniques used are mainly group-theoretic in character. We present a characterization of those groups contained in the universal closure of a restricted wreath product U wr G, where U is an abelian group of zero or finite square-free exponent and G is a torsion-free soluble group with a bound on the class of its nilpotent subgroups. For certain choices of G we are able to use this characterization to prove further results about these groups; in particular, results related to the decidability of their universal theories. The latter part of this work consists of a number of independent but related topics. We show that if G is a finitely generated abelian-by-metanilpotent group and H is elementarily equivalent to G then the subgroups gamma_n(G) and gamma_n(H) are elementarily equivalent, as are the quotient groups G/gamma_n(G) and G/gamma_n(H). We go on to consider those groups universally equivalent to F_2(VN_c), where the free groups of the variety V are residually finite p-groups for infinitely many primes p, distinguishing between the cases when c = 1 and when c > 2. Finally, we address some important questions concerning the theories of free groups in product varieties V_k · · ·V_1, where V_i is a nilpotent variety whose free groups are torsion-free; in particular we address questions about the decidability of the elementary and universal theories of such groups. Results mentioned in both of the previous two paragraphs have applications here.