743 research outputs found
Decidability Results for Saturation-based Model Building
Saturation-based calculi such as superposition can be successfully instantiated to decision procedures for many decidable fragments of first-order logic. In case of termination without generating an empty clause, a saturated clause set implicitly represents a minimal model for all clauses, based on the underlying term ordering of the superposition calculus. In general, it is not decidable whether a ground atom, a clause or even a formula holds in this minimal model of a satisfiable saturated clause set. Based on an extension of our superposition calculus for fixed domains with syntactic disequality constraints in a non-equational setting, we describe models given by ARM (Atomic Representations of term Models) or DIG (Disjunctions of Implicit Generalizations) representations as minimal models of finite saturated clause sets. This allows us to present several new decidability results for validity in such models. These results extend in particular the known decidability results for ARM and DIG representations
Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints
The monadic shallow linear Horn fragment is well-known to be decidable and
has many application, e.g., in security protocol analysis, tree automata, or
abstraction refinement. It was a long standing open problem how to extend the
fragment to the non-Horn case, preserving decidability, that would, e.g.,
enable to express non-determinism in protocols. We prove decidability of the
non-Horn monadic shallow linear fragment via ordered resolution further
extended with dismatching constraints and discuss some applications of the new
decidable fragment.Comment: 29 pages, long version of CADE-26 pape
Worst-case Optimal Query Answering for Greedy Sets of Existential Rules and Their Subclasses
The need for an ontological layer on top of data, associated with advanced
reasoning mechanisms able to exploit the semantics encoded in ontologies, has
been acknowledged both in the database and knowledge representation
communities. We focus in this paper on the ontological query answering problem,
which consists of querying data while taking ontological knowledge into
account. More specifically, we establish complexities of the conjunctive query
entailment problem for classes of existential rules (also called
tuple-generating dependencies, Datalog+/- rules, or forall-exists-rules. Our
contribution is twofold. First, we introduce the class of greedy
bounded-treewidth sets (gbts) of rules, which covers guarded rules, and their
most well-known generalizations. We provide a generic algorithm for query
entailment under gbts, which is worst-case optimal for combined complexity with
or without bounded predicate arity, as well as for data complexity and query
complexity. Secondly, we classify several gbts classes, whose complexity was
unknown, with respect to combined complexity (with both unbounded and bounded
predicate arity) and data complexity to obtain a comprehensive picture of the
complexity of existential rule fragments that are based on diverse guardedness
notions. Upper bounds are provided by showing that the proposed algorithm is
optimal for all of them
Combining decision procedures for the reals
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. Let Tadd[Q] be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant 1, and function symbols for multiplication by
rational constants. Let Tmult[Q] be the analogous theory for the
multiplicative structure, and let T[Q] be the union of the two. We
show that although T[Q] is undecidable, the universal fragment of
T[Q] is decidable. We also show that terms of T[Q]can
fruitfully be put in a normal form. We prove analogous results for theories in
which Q is replaced, more generally, by suitable subfields F
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.Comment: Will appear in Logical Methods in Computer Scienc
Reachability analysis of first-order definable pushdown systems
We study pushdown systems where control states, stack alphabet, and
transition relation, instead of being finite, are first-order definable in a
fixed countably-infinite structure. We show that the reachability analysis can
be addressed with the well-known saturation technique for the wide class of
oligomorphic structures. Moreover, for the more restrictive homogeneous
structures, we are able to give concrete complexity upper bounds. We show ample
applicability of our technique by presenting several concrete examples of
homogeneous structures, subsuming, with optimal complexity, known results from
the literature. We show that infinitely many such examples of homogeneous
structures can be obtained with the classical wreath product construction.Comment: to appear in CSL'1
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