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Jochen Burghardt

Publications and source records attributed to Jochen Burghardt.

14 recordsLinked to original sources
arXiv2021

An Algebra of Properties of Binary Relations

We consider all 16 unary operations that, given a homogeneous binary relation R, define a new one by a boolean combination of xRy and yRx. Operations can be composed, and connected by pointwise-defined logical junctors. We consider the usual properties of relations, and allow them to be lifted by prepending an operation. We investigate extensional equality between lifted properties (e.g. a relation is connex iff its complement is asymmetric), and give a table to decide this equality. Supported by a counter-example generator and a resolution theorem prover, we investigate all 3-atom implications between lifted properties, and give a sound and complete axiom set for them (containing e.g. "if R's complement is left Euclidean and R is right serial, then R's symmetric kernel is left serial").

math.LO↗
arXiv2019

Regular Substitution Sets: A Means of Controlling E-Unification

A method for selecting solution constructors in narrowing is presented. The method is based on a sort discipline that describes regular sets of ground constructor terms as sorts. It is extended to cope with regular sets of ground substitutions, thus allowing different sorts to be computed for terms with different variable bindings. An algorithm for computing signatures of equationally defined functions is given that allows potentially infinite overloading. Applications to formal program development are sketched.

cs.FL↗
arXiv2018

Simple Laws about Nonprominent Properties of Binary Relations

We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like "not irreflexive, but asymmetric". We considered the non-trivial laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties.

math.LO↗
arXiv2018

A Scheme-Driven Approach to Learning Programs from Input/Output Equations

We describe an approach to learn, in a term-rewriting setting, function definitions from input/output equations. By confining ourselves to structurally recursive definitions we obtain a fairly fast learning algorithm that often yields definitions close to intuitive expectations. We provide a Prolog prototype implementation of our approach, and indicate open issues of further investigation.

cs.LO↗
arXiv2017

An Improved Algorithm for E-Generalization

E-generalization computes common generalizations of given ground terms w.r.t. a given equational background theory E. In 2005 [arXiv:1403.8118], we had presented a computation approach based on standard regular tree grammar algorithms, and a Prolog prototype implementation. In this report, we present algorithmic improvements, prove them correct and complete, and give some details of an efficiency-oriented implementation in C that allows us to handle problems larger by several orders of magnitude.

cs.LO↗
arXiv2017

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.

cs.LO↗
arXiv2016

Weight Computation of Regular Tree Languages

We present a general framework to define an application-dependent weight measure on terms that subsumes e.g. total simplification orderings, and an O(n log n) algorithm for the simultaneous computation of the minimal weight of a term in the language of each nonterminal of a regular tree grammar, based on Barzdins' liquid-flow technique.

cs.FL↗
arXiv2014

Eine entscheidbare Klasse n-stelliger Horn-Prädikate

Similar to a tree grammar, a Horn theory can be used to describe an infinite set of terms. In this paper, we present a class of Horn theories such that the set of definable predicates is closed wrt. conjunction and such that the satisfiability of a predicate is decidable. This extends previous results on Horn clauses with unary predicates.

cs.LO↗
arXiv2014

Implementing Anti-Unification Modulo Equational Theory

We present an implementation of E-anti-unification as defined in Heinz (1995), where tree-grammar descriptions of equivalence classes of terms are used to compute generalizations modulo equational theories. We discuss several improvements, including an efficient implementation of variable-restricted E-anti-unification from Heinz (1995), and give some runtime figures about them. We present applications in various areas, including lemma generation in equational inductive proofs, intelligence tests, diverging Knuth-Bendix completion, strengthening of induction hypotheses, and theory formation about finite algebras.

cs.LO↗
arXiv2014

Experiences in Developing Time-Critical Systems - The Case Study "Production Cell"

Starting from an informal requirements description of a toy production cell used in an intra-project competition in 1994, we give a formal specification that is as close as possible to requirements. We use the deductive program synthesis approach by Manna and Waldinger (1980) to obtain a verified TTL-like circuitery to control the cell. The formal specification also covers mechanical aspects and thus allows to reason not only about software issues but also about issues of mechanical engineering. Besides an approach confined to first order predicate logic with explicit, continuous time, an attempt is presented to employ application specific user-defined logical operators to get a more concise specification as well as proof.

cs.SE↗
arXiv2014

Formale Entwicklung einer Steuerung für eine Fertigungszelle mit SYSYFOS

Using the synthesis approach of Manna and Waldinger, a formally specified and verified control circuitery for a production cell was developped. Building an appropriate formal language level, we could achieve a requirements specification to the informal description. We demonstrated that the paradigm of deductive synthesis can be applied to the development of complete verified systems, including hardware and mechanics. We defined two domain-specific logical operators that schematise frequent patterns in specification and proof and hence allow a more concise and expressive presentation. In Burghardt (1995), an english short version of this paper, without appendices, can be found.

cs.LO↗
arXiv2014

Axiomatization of Finite Algebras

We show that the set of all formulas in n variables valid in a finite class A of finite algebras is always a regular tree language, and compute a finite axiom set for A. We give a rational reconstruction of Barzdins' liquid flow algorithm (Barzdin+Barzdin, 1991). We show a sufficient condition for the existence of a class A of prototype algebras for a given theory T. Such a set allows us to prove T |= p simply by testing whether p holds in A.

cs.LO↗