Unification modulo Lists with Reverse, Relation with Certain Word Equations

International audienceDecision procedures for various list theories have been investigated in the literature with applications to automated verification. Here we show that the unifiability problem for some list theories with a \emph{reverse} operator is NP-complete. We also give a unifiability algorithm for the case where the theories are extended with a \emph{length} operator on lists

Unification modulo Lists with Reverse as Solving Simple Sets of Word Equations

Decision procedures for various list theories have been investigated in the literature with applications to automated verification. Here we show that the unifiability problem for some list theories with a reverse operator is NP-complete. We also give a unifiability algorithm for the case where the theories are extended with a length operator on lists

Unification modulo Lists with Reverse as Solving Simple Sets of Word Equations

Decision procedures for various list theories have been investigated in the literature with applications to automated verification. Here we show that the unifiability problem for some list theories with a reverse operator is NP-complete. We also give a unifiability algorithm for the case where the theories are extended with a length operator on lists

E-Generalization Using Grammars

We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.Comment: 49 pages, 16 figures, author address given in header is meanwhile outdated, full version of an article in the "Artificial Intelligence Journal", appeared as technical report in 2003. An open-source C implementation and some examples are found at the Ancillary file

A Unification Algorithm For The First Order Theory of Quandles

The long-range goal of this project is to develop an algorithm to decide whether two terms are unifiable over the theory of quandles. First, it is shown that the general E-unification reduces to the E-matching problem due to the right-cancellation axioms of quandles. The E-matching process takes the general narrowing approach to equational matching. However, a naive application of narrowing is, at best, recursively enumerable and hence will not terminate given terms that do not match. This modification of narrowing places a hard limit on the use of the delta rules of the term rewriting system for quandles to ensure termination. It is implemented in the SWI-Prolog logic programming language. The question remains open as to whether the imposed limits still allow the program to find a unifier for all matching pairs

An axiomatic approach for solving geometric problems symbolically

technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design

Four Lessons in Versatility or How Query Languages Adapt to the Web

Exposing not only human-centered information, but machine-processable data on the Web is one of the commonalities of recent Web trends. It has enabled a new kind of applications and businesses where the data is used in ways not foreseen by the data providers. Yet this exposition has fractured the Web into islands of data, each in different Web formats: Some providers choose XML, others RDF, again others JSON or OWL, for their data, even in similar domains. This fracturing stifles innovation as application builders have to cope not only with one Web stack (e.g., XML technology) but with several ones, each of considerable complexity. With Xcerpt we have developed a rule- and pattern based query language that aims to give shield application builders from much of this complexity: In a single query language XML and RDF data can be accessed, processed, combined, and re-published. Though the need for combined access to XML and RDF data has been recognized in previous work (including the W3C’s GRDDL), our approach differs in four main aspects: (1) We provide a single language (rather than two separate or embedded languages), thus minimizing the conceptual overhead of dealing with disparate data formats. (2) Both the declarative (logic-based) and the operational semantics are unified in that they apply for querying XML and RDF in the same way. (3) We show that the resulting query language can be implemented reusing traditional database technology, if desirable. Nevertheless, we also give a unified evaluation approach based on interval labelings of graphs that is at least as fast as existing approaches for tree-shaped XML data, yet provides linear time and space querying also for many RDF graphs. We believe that Web query languages are the right tool for declarative data access in Web applications and that Xcerpt is a significant step towards a more convenient, yet highly efficient data access in a “Web of Data”

On the number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function $f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.Comment: 17 pp, 4 figs, accepted for Physical Review