1,808 research outputs found
Logical Reduction of Metarules
International audienceMany forms of inductive logic programming (ILP) use metarules, second-order Horn clauses, to define the structure of learnable programs and thus the hypothesis space. Deciding which metarules to use for a given learning task is a major open problem and is a trade-off between efficiency and expressivity: the hypothesis space grows given more metarules, so we wish to use fewer metarules, but if we use too few metarules then we lose expressivity. In this paper, we study whether fragments of metarules can be logically reduced to minimal finite subsets. We consider two traditional forms of logical reduction: subsumption and entailment. We also consider a new reduction technique called derivation reduction, which is based on SLD-resolution. We compute reduced sets of metarules for fragments relevant to ILP and theoretically show whether these reduced sets are reductions for more general infinite fragments. We experimentally compare learning with reduced sets of metarules on three domains: Michalski trains, string transformations, and game rules. In general, derivation reduced sets of metarules outperform subsumption and entailment reduced sets, both in terms of predictive accuracies and learning times
A Practical View on Renaming
We revisit variable renaming from a practitioner's point of view, presenting
concepts we found useful in dealing with operational semantics of pure Prolog.
A concept of relaxed core representation is introduced, upon which a concept of
prenaming is built. Prenaming formalizes the intuitive practice of renaming
terms by just considering the necessary bindings, where now some passive
"bindings" x/x may be necessary as well. As an application, a constructive
version of variant lemma for implemented Horn clause logic has been obtained.
There, prenamings made it possible to incrementally handle new (local)
variables.Comment: In Proceedings WLP'15/'16/WFLP'16, arXiv:1701.0014
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Efficient recursion termination for function-free horn logic
We present an efficient scheme to terminate infinite recursion in function-free Horn logic. In [BW84], Brough and Walker show that a preorder linear resolution with a goal termination strategy is incomplete, i.e. it must miss some answers. Their theory is true if left-recursion is allowed. The crucial assumption underlying Brough and Walker's theory is that the order of literals in a clause should not be altered. This assumption, however, is not necessary in programs that do not contain any extra-logical features such as the 'cut' symbol of Prolog. This is because the order of literals does not affect the correctness of such programs, only their efficiency. In this paper, we show that left-recursion can always be eliminated. The idea is to transform loops of the input set into safe loops, that are left-recursion free. Consequently, the goal termination strategy guarantees to always terminate properly with all possible answers; thus, it is complete in the domain of safe loops. We further show that all rules in a safe loop can be transformed into rules that begin with a base literal. This permits the implementation of a simple scheme to carry out the goal termination strategy more efficiently. The basic idea of this scheme is to distribute the history containing all executed goals over assertions, rather than maintaining it as a centralized data structure. This reduces the amount of work performed during execution
First-Order Logic Theorem Proving and Model Building via Approximation and Instantiation
In this paper we consider first-order logic theorem proving and model
building via approximation and instantiation. Given a clause set we propose its
approximation into a simplified clause set where satisfiability is decidable.
The approximation extends the signature and preserves unsatisfiability: if the
simplified clause set is satisfiable in some model, so is the original clause
set in the same model interpreted in the original signature. A refutation
generated by a decision procedure on the simplified clause set can then either
be lifted to a refutation in the original clause set, or it guides a refinement
excluding the previously found unliftable refutation. This way the approach is
refutationally complete. We do not step-wise lift refutations but conflicting
cores, finite unsatisfiable clause sets representing at least one refutation.
The approach is dual to many existing approaches in the literature because our
approximation preserves unsatisfiability
Automated Synthesis of a Finite Complexity Ordering for Saturation
We present in this paper a new procedure to saturate a set of clauses with
respect to a well-founded ordering on ground atoms such that A < B implies
Var(A) {\subseteq} Var(B) for every atoms A and B. This condition is satisfied
by any atom ordering compatible with a lexicographic, recursive, or multiset
path ordering on terms. Our saturation procedure is based on a priori ordered
resolution and its main novelty is the on-the-fly construction of a finite
complexity atom ordering. In contrast with the usual redundancy, we give a new
redundancy notion and we prove that during the saturation a non-redundant
inference by a priori ordered resolution is also an inference by a posteriori
ordered resolution. We also prove that if a set S of clauses is saturated with
respect to an atom ordering as described above then the problem of whether a
clause C is entailed from S is decidable
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