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From: leff@smu.CSNET (Laurence Leff)
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This is our latest list of abstracts of SEKI REPORTS and SEKI WORKING PAPERS
for both 1986 and 1987.

PhD theses can be ordered at a nominal fee of US$ 10, SEKI REPORTS will cost
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SEKI   REPORTS    SR-86-01 bis SR-86-20





SEKI-REPORT SR-86-01

K.-H.Blaesius

Consruction of Equality Graphs

Abstract: The theoretical and practical problems of equality rea-
soning  in  Automated  Deduction  are  notorious. A new method is
presented to cope with the enormous search space that usually ar-
ises  when  equational axioms are present. Starting from an empty
graph a production system constructs graphs which represent solu-
tions  for  simpler problems defined by abstraction. These graphs
contain global information and are plans for guiding  the  search
for  a  proof  of  the original problem, represented in the final
graph. The construction of equality graphs is based on  the  idea
to search for the differences between two terms by seperating to-
plevel symbol and subterms of a functional term.  The  impact  of
the  explicit  representation  of  information  contained  in the
inference system on the control of inferences is  discussed.  Fi-
nally the method is compared to other equality reasoning methods.



SEKI-REPORT-SR-86-02    Ch. Beierle, A. Voss

Specification, horizontal  composition  and  parameterization  of
algebraic implementations

Abstract: Loose specifications of abstract data types (ADTs) have
many  nonisomorphic algebras as models. An implementation between
two loose specifications should therefore consider many  abstrac-
tion  functions  together  with their source and target algebras.
Just like specifications are stepwise refined to  restrict  their
class  of models, implementations should be stepwise refinable to
restrict the class of abstraction  functions.  In  this  scenario
specifications and implementations can be developed interwovenly.

        We suggest to have implementation  specifications  analo-
gously  to  loose ADT specifications: Implementations have signa-
tures, models, axioms and sentences thus constituting an institu-
tion.  Implementation specifications are the theories of this in-
stitution and refinements between  implementation  specifications
are its theory morphisms.

        In this framework, implementations between  parameterized
specifications and horizontal composition of implementations turn
out  to  be  special  cases  of  the  more  powerful  concept  of
parameterized  implementations, which allow to instantiate an im-
plementation by substituting a subimplementation by  another  im-
plementation.



SEKI-REPORT-SR-86-03    M.  Reinfrank, M. Beetz, H.  Freitag,  J.
Klug

Kapri: A rule based non-monotonic inference engine  with  an  in-
tegrated reason maintenance system

Abstract: KAPRI (Kaiserslautern Augmented Production Rule  Inter-
preter)  uses rules of the form if I unless I then I . For such a
rule to be fired, its monotonic if-antecedents  are  required  to
match  the  current  database,  while  none  of its non-monotonic
unless-antecedents does. The current state of a  KAPRI-system  is
represented  by  a  dependency network composed of assertions and
corresponding justifications for believing in them  in  terms  of
belief  respectively  disbelief in other assertions. This network
is  maintained  by   a   reason   maintenance   system,   KL-DNMS
(Kaiserslautern  Dependency  Network Management System), that re-
vises the current belief status with respect to  every  modifica-
tion due to the firing of a rule or to the addition/retraction of
a basic fact.

In the present paper, we describe a very simple first version  of
KAPRI,  and  discuss  some  key issues that arise from augmenting
production rules by an UNLESS-part, and from integrating  a  pro-
duction  rule-based  inference  engine  with a reason-maintenance
system.









SEKI-REPORT-SR-86-04    H. Boley

A Relational/Functional Integration with valued Clauses

Abstract: The RELFUN programming language is introduced as an at-
tempt  to  integrate the capabilities of the relational and func-
tional styles. Definitions of functions and relations are  speci-
fied  uniformly  by  valued  Horn clauses, where rules return the
value of their rightmost premise.  Functional nestings are  flat-
tened  to  relational  conjunctions,  using a purified version of
PROLOG's is-primitive. RELFUN functions may have non-ground argu-
ments, like relations, and/or return non-ground values; their in-
put and output arguments can be inverted like those of relations.
Higher-order   functions   are   definable  as  (function-)valued
clauses, with funarg  problems  being  avoided  by  the  standard
renaming  of  clause variables. RELFUN's operational semantics is
specified as an abstract machine, which  also  yields  its  first
(e-mailable) FRANZ LISP implementation.



SEKI-REPORT-SR-86-05    H.-J. Buerckert

Some Relationships between  Unification,  Restricted  Unification
and Matching

Abstract: We present restricted T-unification that is unification
of  terms  under a given equational theory T with the restriction
that not all variables are allowed to be substituted. Some  rela-
tionships   between   restriced   T-unification,  unrestriced  T-
unification and T-matching (one-sided  T-unification)  are  esta-
blished.  Our  main result is that, in the case of an almost col-
lapse free equational theory the most general restricted unifiers
and for certain termpairs the most general matchers are also most
general unrestricted unifiers; this does not hold for  more  gen-
eral  theories. Almost collapse free theories are theories, where
only terms starting with projection symbols may collapse (i.e. to
be T-equal) to variables.



SEKI-REPORT-SR-86-06    R. Goebel

Completion of Globally Finite Term Rewriting Systems  for  Induc-
tive Proofs

Abstract: The Knuth-Bendix Algorithm (KBA) is not  able  to  com-
plete term rewriting systems with cyclic rules such as the commu-
tativity. This kind of rules cause cycles in a  reduction  chain.
This problem may be solved by an extension of the KBA for global-
ly finite term rewriting systems.  For  a  globally  finite  term
rewriting  system, cycles may occur in a reduction chain, but for
each term there is only a finite set of reductions.  A  confluent
and  globally  finite  term rewriting sytem R provides a decision
procedure for equality induced by R:  Two  terms  are  equal  iff
there is a common term in their reduction sets.

This extension requires new methods for testing the  global  fin-
iteness  and a new confluence test, because local confluence does
not imply global confluence for  globally  finite  relations.  In
this paper we give a theoretical framework for this extension. We
will show how this theory can be applied to term  rewriting  sys-
tems, if we are mainly interested in the initial algebra which is
induced by the set of rules.



SEKI-REPORT-SR-86-07    M. Schmidt-Schauss

Unification Properties of Idempotent Semigroups

Abstract: Unification in free idempotent semigroups is of  unifi-
cation type zero, i.e. there are unifiable terms s,t but there is
no minimal, complete set of unifiers for these two terms.  Unifi-
cation  in  free idempotent semigroups is strongly complete, i.e.
the unification problem  is always solvable with unifier
{x , t}, even if x occurs in t.

We give a generalization of the usual unification  hierarchy  and
demonstrate   that  the  number  of  independent unifiers in A+I-
unifier sets is not bounded.

It is known that there is a conditional, canonical term rewriting
system  for idempotent semigroups.  To strengthen this result, we
show that there can be no unconditional and finite rewriting sys-
tem.



SEKI-REPORT-SR-86-08    M. Schmidt-Schauss

Some Undecidable Classes of Clause Sets

Abstract: The undecidability of finitely generated stable transi-
tive  relations  on free terms is proved in an elementary way and
then used to show the undecidability of the  implication  A^B  of
two  clauses  A and B. The implication is decidable in the case A
has at most two literals that are complementary unifinable  after
renaming.  Application  of the undecidability of transitive rela-
tions yields more classes of clause sets that have an undecidable
satisfiability  problem:  clause  sets  consisting of two clauses
with two literals each (2-clauses) and in addition 2  ground  un-
its, clause sets consisting of a single 3-clausse and arbitrarily
many (non-ground) units and clause sets conssting of  a  4-clause
and  arbitrarily many ground units. Finally we show the undecida-
bility of the so-called D-clause sets,  where  a  D-clause  is  a
Horn-clause of the form

Q1(x1) Y