Class Of First-order Formulas Research Articles

Quantifier Alternation in First-Order Formulas with Infinite Spectra

The spectrum of a first-order formula is the set of numbers α such that for a random graph in a binomial model where the edge probability is a power function of the number of graph vertices with exponent −α the truth probability of this formula does not tend to either zero or one. In 1990 J. Spenser proved that there exists a first-order formula with an infinite spectrum. We have proved that the minimum quantifier depth of a first-order formula with an infinite spectrum is either 4 or 5. In the present paper we find a wide class of first-order formulas of depth 4 with finite spectra and also prove that the minimum quantifier alternation number for a first-order formula with an infinite spectrum is 3.

Tree-width for first order formulae

We introduce tree-width for first order formulae \phi, fotw(\phi). We show that computing fotw is fixed-parameter tractable with parameter fotw. Moreover, we show that on classes of formulae of bounded fotw, model checking is fixed parameter tractable, with parameter the length of the formula. This is done by translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable fragment L^k of first order logic. For fixed k, the question whether a given first order formula is equivalent to an L^k formula is undecidable. In contrast, the classes of first order formulae with bounded fotw are fragments of first order logic for which the equivalence is decidable. Our notion of tree-width generalises tree-width of conjunctive queries to arbitrary formulae of first order logic by taking into account the quantifier interaction in a formula. Moreover, it is more powerful than the notion of elimination-width of quantified constraint formulae, defined by Chen and Dalmau (CSL 2005): for quantified constraint formulae, both bounded elimination-width and bounded fotw allow for model checking in polynomial time. We prove that fotw of a quantified constraint formula \phi\ is bounded by the elimination-width of \phi, and we exhibit a class of quantified constraint formulae with bounded fotw, that has unbounded elimination-width. A similar comparison holds for strict tree-width of non-recursive stratified datalog as defined by Flum, Frick, and Grohe (JACM 49, 2002). Finally, we show that fotw has a characterization in terms of a cops and robbers game without monotonicity cost.

Proof-complexity results for nonmonotonic reasoning

It is well-known that almost all nonmonotonic formalisms have a higher worst-case complexity than classical reasoning. In some sense, this observation denies one of the original motivations of nonmonotonic systems, which was the expectation taht nonmonotonic rules should help to speed-up the reasoning process, and not make it more difficult. In this paper, we look at this issue from a proof-theoretical perspective. We consider analytic calculi for certain nonmonotonic logis and analyze to what extent the presence of nonmonotonic rules can simplify the search for proofs. In particular, we show that there are classes of first-order formulae which have only extremely long “classical” proofs, i.e., proofs without applications of nonmonotonic rules, but there are short proofs using nonmonotonic inferences. Hence,despite the increase of complexity in the worst case, there are instances where nonmonotonic reasoning can be much simpler than classical (cut-free) reasoning.

The ground-negative fragment of first-order logic is -complete

AbstractWe prove that for a natural class of first-order formulas the validity problem is -complete.

Logic program synthesis from first-order logic specifications

In this paper, a logic program synthesis method from first-order logic specifications is described. The specifications are describe by Horn clauses extended by universally quantified implicational formulas. These formulas are transformed into definite clause programs by unfold/fold transformation. We show that some classes of first-order formulas can be successfully transformed into definite clauses by unfold/fold transformation.

SKEPTICAL ABDUCTION

Abduction is the process of generating the best explanation as to why a fact is observed given what is already known. A real problem in this area is the selective generation of hypotheses that have some reasonable prospect of being valid. In this paper, we propose the notion of skeptical abduction as a model to face this problem. Intuitively, the hypotheses pointed out by skeptical abduction are all the explanations that are consistent with the given knowledge and that are minimal, i.e. not unnecessarily general. Our contribution is twofold. First, we present a formal characterization of skeptical abduction in a logical framework. On this ground, we address the problem of mechanizing skeptical abduction. A new method to compute minimal and consistent hypotheses in propositional logic is put forward. The extent to which skeptical abduction can be mechanized in first—order logic is also investigated. In particular, two classes of first-order formulas in which skeptical abduction is effective are provided. As an illustration, we finally sketch how our notion of skeptical abduction applies as a theoretical tool to some artificial intelligence problems (e.g. diagnosis, machine learning).

Prefix classes of krom formulae with identity

Two small classes of first order formulae without function symbols but with identity, in prenex conjunctive normal form with all disjunctions binary, are shown to have a recursively unsolvable decision problem, whereas for another such class an algorithm is developed which solves the decision problem of that class. This solves the prefix problem for such classes of formulae except for the Godel-Kalmar-Schutte case.