Journal of Automated Reasoning

This article is the seventh of a series of articles discussing various open research problems in automated reasoning. Here we focus on self-analysis and the set of support strategy. The problem proposed for research asks for one to find a means for an automated reasoning program to self-analytically determine which clauses to be in the set of support at various points during the program's attempt to complete a given assignment. For evaluating a proposed solution to this research problem, we include suggestions concerning possible test problems.

There are many papers describing problems solved using the Boyer-Moore theorem prover, as well as papers describing new tools and functionalities added to it. Unfortunately, so far there has been no tutorial paper describing typical interactions that a user has with this system when trying to solve a nontrivial problem, including a discussion of issues that arise in these situations. In this paper we aim to fill this gap by illustrating how we have proved an interesting theorem with the Boyer-Moore theorem prover: a formalization of the assertion that the arithmetic mean of a sequence of natural numbers is greater than or equal to their geometric mean. We hope that this report will be of value not only for (non-expert) users of this system, who can learn some approaches (and tricks) to use when proving theorems with it, but also for implementors of automated deduction systems. Perhaps our main point is that, at least in the case of Nqthm, the user can interact with the system without knowing much about how it works inside. This perspective suggests the development of theorem provers that allow interaction that is user oriented and not system developer oriented.

We propose a general way of combining background reasoners in theory reasoning. Using a restricted version of the Craig interpolation lemma, we show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures. In this case, constraint information is propagated across reasoners eexchanging residues that are, in essence, disjunctions of ground literals over a common signature. As an application of our approach, we describe a multitheory version of the semantic tableau calculus, and we prove it sound and complete.

This article is the twelfth of a series of articles discussing various open research problems in automated reasoning. Here we focus on finding criteria for guaranteeing the absence of a complete set of reductions. We include a suggestion for evaluating a proposed solution to this research problem.

In this paper we prove the decidability of the class of unquantified formulae of set theory involving the operators ϕ, ∪, ∩, \, {·}, pred < and the predicates =, ∈, $$ \subseteq $$ , Finite, where pred <(s) denotes the collection of all sets having rank strictly less than the rank of s. This work generalizes and combines earlier results published in the same series of papers.

For close to a century, despite the efforts of fine minds that include Hilbert and Ackermann, Tarski and Bernays, Łukasiewicz, and Rose and Rosser, various proofs of a number of significant theorems have remained missing – at least not reported in the literature – amply demonstrating the depth of the corresponding problems. The types of such missing proofs are indeed diverse. For one example, a result may be guaranteed provable because of being valid, and yet no proof has been found. For a second example, a theorem may have been proved via metaargument, but the desired axiomatic proof based solely on the use of a given inference rule may have eluded the experts. For a third example, a theorem may have been announced by a master, but no proof was supplied. The finding of missing proofs of the cited types, as well as of other types, is the focus of this article. The means to finding such proofs rests with heavy use of McCune's automated reasoning program OTTER, reliance on a variety of powerful strategies this program offers, and employment of diverse methodologies. Here we present some of our successes and, because it may prove useful for circuit design and program synthesis as well as in the context of mathematics and logic, detail our approach to finding missing proofs. Well-defined and unmet challenges are included.

Waldmeister is a high-performance theorem prover for unit equational first-order logic. In the making of Waldmeister, we have applied an engineering approach, identifying the critical points with respect to efficiency in time and space. Our logical three-level system model consists of the basic operations on the lowest level, where we put great stress on efficient data structures and algorithms. For the middle level, where the inference steps are aggregated into an inference machine, flexible adjustment has proven essential during experimental evaluation. The top level holds control strategy and reduction ordering. Although at this level only standard strategies are employed, really large proof tasks have been managed in reasonable time.

The goal of knowledge compilation is to enable fast queries. Prior approaches had the goal of small (i.e., polynomial in the size of the initial knowledge bases) compiled knowledge bases. Typically, query–response time is linear, so that the efficiency of querying the compiled knowledge base depends on its size. In this paper, a target for knowledge compilation called the ri-trie is introduced; it has the property that even if the knowledge bases are large, they nevertheless admit fast queries. Specifically, a query can be processed in time linear in the size of the query regardless of the size of the compiled knowledge base.