Struth, G., \Canonical Transformations in Algebra, Universal Algebra and Logic," Ph.D. thesis, Institut Fur Informatik, University of Saarlandes (1998). 21  Home/Search Document Not in Database Summary Related Articles Check |
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Modal Kleene Algebra and Applications - Survey Jules Desharnais Self-citation (Struth) (Correct)
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
Modal Kleene Algebra And Applications - A Survey - Desharnais, Möller, Struth Self-citation (Struth) (Correct)
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
Termination of Ground Non-Symmetric Knuth-Bendix Completion - Struth (2002) Self-citation (Struth) (Correct)
....Keywords: Knuth Bendix completion, transitive relations, quasiorderings, reachability, termination, homeomorphic embedding. 1 Introduction Non symmetric Knuth Bendix completion is a method for reasoning about reachability in transitive relations, orderings, graphs and constraint systems [13,16], in particular, when these systems have in nitely many states. Although this procedure generalizes concepts and techniques of equational completion, there are signi cant di erences [16] There are, for instance, no term normal forms and decision procedures are not don t care non deterministic, ....
....We also consider syntactic orderings on (ground) terms. We assume that is total and noetherian and that it contains the proper subterm relation. 3 Non Symmetric Rewriting and Completion We presuppose the basic concepts and notation of equational and non symmetric rewriting and completion [5,9,13,16]. A combination algorithm of non symmetric and equational completion with applications to graph algorithms has been presented in [12] We therefore brie y recall only the most important notions. For the sake of simplicity, we restrict our attention to quasiorderings. The adaption to transitive ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
Declarative Graph Algorithms via Knuth-Bendix Completion - Struth (2002) Self-citation (Struth) (Correct)
....classes analogously to the de nition of on . v] w] i v w for some v 2 [v] and w 2 [w] 3 Non Symmetric Rewriting We presuppose the basic concepts and notation of equational rewriting (c.f. 13] Introductions to non symmetric rewriting can be found in [17,20]. Non symmetric rewriting addresses reachability of binary relations. Let I be a binary relation and some noetherian syntactic ordering on a set A. Let I be partitioned into three sets R = I , S = I and (non orientable) R and S steps in paths can be separated, if the ....
....rules decreasing from left to right and from right to left respectively with respect to . Let I , R and S be the associated rewrite relations. Then, semicommutation depends of the relative positions in a term where consecutive rewrite rules apply. This is analyzed in critical pair lemmata [20]. Intuitively, a critical pair is an element of S R that prevents semicommutation. As in equational rewriting, critical pairs arise when one rewrite step takes place at a pre x position of the other and the lower one at a skeleton position. Applicability of the second step then depends on the ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
Knuth-Bendix Completion for Non-Symmetric Transitive Relations - Struth (2001) Self-citation (Struth) (Correct)
....S R R S i (R [ S) R S . ii) SR R S implies (R [ S) R S , if (R [ S 1 ) is well founded. Lemma 1 (i) expresses a generalized Church Rosser property, lemma 1 (ii) a generalized Newman s lemma that is appropriate for quasiorderings. See [11,15,16] for proofs. Similar properties hold for transitive relations. Lemma 2. Let R and S be binary relations on some set A. i) SR R S [ S i (R [ S) R S [ S . ii) SR R S [S implies (R[S) R S [S , if (R[S 1 ) is well founded. 3 ....
.... , respectively, rewrite proofs. Those of the rst shape are appropriate for quasiorderings, those of the second shape for transitive relations. However, there is no concept of term normal form for nonsymmetric rewriting. Lemma 1 and lemma 2 extend to non symmetric rewriting modulo a relation E [11,15,16]. Here we only consider the two relations theory. The abstract results in lemma 1 and lemma 2 can be re ned to critical pair lemmata at the term level. They introduce drastic di erences to the equational case, as has rst been noted in [11] We assume two rewrite relations R and S induced by the ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
Calculating Church-Rosser Proofs in Kleene Algebra - Struth (2002) (1 citation) Self-citation (Struth) (Correct)
....with the Kleene star. Like in the proof of proposition 2, proposition 3 yields base cases for inductive proofs of Church Rosser statements with more than two generators. Theorem 4 and theorem 5 are the basis for non symmetric rewriting with quasiorderings and non symmetric transitive relations [23]. The inequalities b a a b are examples of semi commutation properties (as opposed to commutation properties like ab = ba) As we will see in the next section, their interpretations in the relational model generalize the diamond property. Non symmetric Church Rosser theorems ....
....that do not involve well foundedness. 9 Discussion The Church Rosser theorems in section 5 are interesting for the foundations of rewriting. Theorem 4 is the Church Rosser theorem for rewriting with quasiorderings [16] proposition 3 that for rewriting with non symmetric transitive relations [23]. The commutation and semicommutation relations in the hypotheses are not only relevant to Church Rosser theorems. They can be used to express independence or precedence in execution sequences or Mazurkiewicz traces in imperative and concurrent programs [14,4,7] Kleene algebra is related to ....
G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
Deriving Focused Calculi For Transitive Relations - Struth (2001) Self-citation (Struth) (Correct)
....closed, the derived chaining rules are more restrictive and transparent than previous ones and also the completeness proof is conceptually clear and simple. Chaining rules and their ordering constraints can be naturally motivated by Knuth Bendix procedures for non symmetric transitive relations [16]. In particular, a restricted variant of the calculus is such a Knuth Bendix procedure. Brie y, the main contributions of this text are the following: We propose a general method for deriving focused calculi based on ordered resolution. We use our method for novel syntactic completeness ....
.... So far, we have derived the rules, but where do the ordering constraints come from We now show that the chaining rules and their ordering constraints nd a natural explanation in terms of concepts of rewriting and the Knuth Bendix procedure for non symmetric transitive relations, as speci ed in [16] 3 . In a real synthesis situation, the semantic information is needed a priori to encode the desired properties of the inference rules via the ordering. Here, a rst step in the development process would be to model the Positive Chaining rule as an extension of the critical pair computation ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998.
How to Synthesize a Tableau - Struth Self-citation (Struth) (Correct)
....The systematic development of programs from specifications is an important area of research. Analogously, it seems interesting to investigate methods for synthesizing procedurally optimized and domain specific reasoning facilities from a given knowledge basis. We have presented such a method in [15] in the context of superposition calculi [9, 1] for algebraic theories. It uses ordered resolution as a procedural framework at the meta level and transfers the rewriting approach to solving uniform word problems by construction of a canonical term rewrite system and symmetrization [11] to ....
....a solution to the word problem of the associated free theory. For simplicity and lack of space we restrict to the basic features of the procedure. The relation to a variant of rewriting and Knuth Bendix completion for non symmetric transitive relations and symmetrization is formally presented in [15]. To derive a tableau, integration of theory specific knowledge essentially means encoding the subformula property in the syntactic ordering. The resolution basis now is a set of Horn clauses and the query of the word problem a negative atom, which helps to encode not only disjunctive, but also ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998. 10
Ordered Chaining Calculi For Finite Distributive Lattices - Struth Self-citation (Struth) (Correct)
....and suprema of pairs of elements. Their word problems are closely related to logical proof and decision procedures: an algebraic variant of ordered resolution, for instance, solves the uniform word problem for distributive lattices by KnuthBendix completion for non symmetric binary relations [10]. Analogously, the Buchberger algorithm solves the word problem for finitely presented polynomial rings by equational completion [3] In the present text, the correspondence between resolution and the Buchberger algorithm motivates a novel synthesis method for deriving superposition calculi [6, ....
....the object level of lattices. Resolution, for instance, is synthesized as a solution to the uniform word problem for distributive lattices at the object level within resolution at the metalevel. Another ordering yields a tableau calculus solving the word problem for the free distributive lattice [10]. Besides the above theoretical contributions, our lattice calculi are a basis for reasoning about set theories, modal logics as boolean algebras with operations, relational algebras and lattice ordered groups, rings or fields. Our method should also apply to further lattices and equational ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Insitut fur Informatik, Universitat des Saarlandes, 1998.
A New Approach to Ordered Chaining - Struth Self-citation (Struth) (Correct)
....relation . Restricting variable chainings is simple only for ordered chaining calculi [2, 3] where inferences are constrained by a syntactic ordering on terms, atoms and clauses. Procedurally, this further prunes the search space. Conceptually, chaining can be connected with rewriting techniques [11, 15] and the standard approach to equational superposition calculi [9, 1] Inference rules are postulated and a posteriori justified in semantic completeness proofs by a model construction, that is strongly intertwined with the deductive process and the syntactic ordering. This may still lead to ....
....Resolution, which we will discuss below. We present a novel synthesis method for deriving an ordered chaining calculus from the axiom based approach. In this text, we can only sketch the basics of the procedure. Its motivation in terms of standard algebraic concepts and techniques is discussed in [15]. Here are the basic ideas. As already mentioned, chaining rules simulate two resolution inferences of two non theory clauses on the transitivity axiom and replace the latter. Therefore, by completeness of the chaining calculi, every proof to the empty clause in the chaining calculus can be ....
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G. Struth. Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut fur Informatik, Universitat des Saarlandes, 1998. 15
Unknown - An Way (Correct)
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Struth, G., \Canonical Transformations in Algebra, Universal Algebra and Logic," Ph.D. thesis, Institut Fur Informatik, University of Saarlandes (1998). 21
On a Generalised Logicality Theorem - Marc Aiguier Diane (Correct)
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G.Struth, Canonical Transformations in Algebra, Universal Algebra and Logic. PhD thesis, Institut Fur Informatik, University of Saarlandes, 1998.
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