The problem of hyperparamodulation and nuclei

This article is the thirty-third of a series of articles discussing various open research problems in automated reasoning. The problem for research asks one to establish criteria for allowing certain — but not all — new clauses to become nuclei when using the inference rule hyperparamodulation or hyperresolution.This work was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of demodulation during inference rule application

This article is the twenty-second of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find criteria for deciding when to permit and when to avoid demodulation during the application of inference rules, focusing mainly on hyperresolution, UR-resolution, and hyperparamodulation. Since these three inference rules admit natural points at which one or more demodulators (rewrite rules) could be applied-for example, after the removal of a literal or the replacement of a term-and since the dominant practice is to demodulate only after an inference rule has been completely applied, the proposed research focuses on an intriguing alternative. For evaluating a proposed solution to this research problem, we suggest problems from mathematics, logic, circuit design, program verification, and the world of puzzles.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Eneregy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of finding an inference rule for set theory

This article is the ninth of a series of articles discussing various open research problems in automated reasoning. Here we focus on finding an inference rule for set theory. The problem proposed for research asks one to find a means for building in set-theoretic reasoning in a manner similar to the way paramodulation builds in equality-oriented reasoning. For evaluating a proposed solution to this research problem, we include suggestions concerning possible test problems.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

The problem of choosing the representation,inference rule,and strategy

This article is the nineteenth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find guidelines for simultaneously choosing the most effective representation, inference rule, and strategy. Since the three areas are tightly coupled, formulation of such guidelines requires understanding the precise nature of this coupling. For evaluating a proposed solution to this research problem, we suggest possible test problems from group theory, Gödel's finite axiomatization of set theory, and program verification.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of induction

This article is the thirtieth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks for criteria for accurately determining when an induction argument is the appropriate form of argument for an automated reasoning program to employ. This research problem also asks for criteria for choosing well the property on which to conduct the induction argument.This work was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of demodulator adjunction

This article is the twenty-third of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find criteria for deciding which e1uality unit clauses to adjoin with regard to CPU time and may affect refutation completeness. For evaluating a proposed solution to this research problem, we suggest problems from mathematics, logic, circuit design, program verification, and the world of puzzles.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W     

The problem of reasoning by case analysis

This article is the twenty-ninth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find criteria that an automated reasoning program can apply to decide when to conduct a case analysis argument and to decide which cases are appropriate. When the choice of employing a case analysis argument is wise and the cases are well chosen, the likelihood that the reasoning program — or, for that matter, a person — will find an answer to the given question is sharply increased.This work was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of reasoning from inequalities

This article is the twenty-first of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find an inference rule that performs as paramodulation does, but with the focus on inequalities rather than on equalities. Since, too often, inequalities that are present in the input play a passive role during a reasoning program's attempt to complete an assignment, such an inference rule would markedly add to program effectiveness by giving inequalities the potential of playing a key role. For evaluating a proposed solution to this research problem, we suggest as possible test problems theorems from group theory and theorems from ring theory. This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-13-109-Eng-38.     

The problem of finding a semantic strategy for focusing inference rules

This article is the fifteenth of a series of articles discussing various open research problems in automated reasoning. Here we focus on finding a strategy — other than a simple syntactic one — that can be used for focusing the attention of the inference rule(s) in use. The problem proposed for research asks one to find semantic criteria for selecting the clauses needed to complete the application of an inference rule. Suggestions for test problems are included. We note that Yuan Yu has solved two of these problems using a rather different approach with the Boyer-Moore theorem prover; his work is reported elsewhere in this issue of the Journal of Automated Reasoning.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of choosing between using and avoiding equality predicates

This article is the twentieth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find criteria for effectively choosing between using and avoiding equality predicates. Since there exist inference rules (such as paramodulation) that enable an automated reasoning program to treat equality as understood, the discovery of such criteria would mark an important advance for the field. For evaluating a proposed solution to this research problem, we suggest possible test problems from group theory, ring theory, and Boolean algebra.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of strategy and hyperresolution

This article is the thirty-first of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find a strategy that can be coupled with the inference rule hyperresolution to control the behavior of an automated reasoning program as effectively as does paramodulation.This work was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of choosing the type of subsumption to use

This article is the eighteenth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find criteria for choosing the type of subsumption (if any) to use for completing a specific assignment. Since subsumption can be expensive (in terms of computer time), the application of such criteria to select the most effective variant of subsumption for attacking a specific problem might materially affect the liklihood of success. As possible test problems for evaluating a proposed solution to this research problem, we suggest SAM's lemma, theorems from Tarskian geometry, and theorems from equivalential calculus.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of naming and function replacement

This article is the twenty-eighth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find criteria that an automated reasoning program can profitably use to remove functions present in the representation and replace them with appropriate predicates or constants thatname the entities that werenamed by the functions. The notation used to present a problem to a reasoning program can have a profound effect on the likelihood of the program's success.This work was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

The problem of explaining the disparate performance of hyperresolution and paramodulation

This article is the sixth of a series of articles discussing various open research problems in automated reasoning. Here we focus on the effectiveness of hyperresolution versus that of paramodulation. The problem proposed for research asks one to find the properties that explain why paramodulation is so much more effective than hyperresolution is for solving various problems from algebra. Fore evaluating a proposed solution to this research problem, we include suggestions concerning possible test problems.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

The problem of choosing the inference rule to employ

This article is the second of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research focuses on finding criteria or metarules for choosing the inference rule(s) to be employed by the automated reasoning program in use when instructing the program to complete a specific assignment. We include actual experiments that illustrate the difficulty of making an appropriate choice.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

The problem of self-analytically choosing the weights

This article is the eighth of a series of articles discussing various open research problems in automated reasoning. Here we focus on self-analysis and the weighting strategy. The problem proposed for research asks one to find a means for an automated reasoning program to self-analytically determine which weights or priorities to give to clauses 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.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

The problem of extending the set of support strategy

This article is the third of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research focuses on redefining or extending the definition of the set of support strategy so that the resulting strategy is as effective at all clause levels as it is at level 1. We include specific test problems for evaluating a proposed solution to this research problem.     

The problem of definition expansion and contraction

This article is the fourth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research focuses on finding criteria that would enable an automated reasoning program to expand or contract definitions wisely. For evaluating a proposed solution to this research problem, we include suggestions concerning possible test problems.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

The problem of self-analytically choosing the set of support

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.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

The problem of choosing between predicate and function notation for problem representation

This article is the sixteenth of a series of articles discussing various open research problems in automated reasoning. Here we focus on choosing between the use of predicates and the use of functions for representing information to an automated reasoning program. The problem proposed for research asks one to find criteria that dictate the most effective choice of notation-functions or predicates-for presenting the question to be studied. The criteria must reflect the close coupling between inference rule, strategy, and representation. We suggest possible test problems for evaluating a proposed solution to this research problem.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.