The problem of automated theorem finding

This article is the twenty-fifth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to identify appropriate properties to permit an automated reasoning program tofind new and interesting theorems, in contrast toproving conjectured theorems. Such programs are now functioning in many domains as valuable reasoning assistants. A sharp increase in their value would occur if they could also be used as colleagues to (so to speak) produce research on their own.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 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 analogy

This article is the twenty-seventh 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 determine whether to attack a given question with reasoning by analogy. The imprecise term reasoning by analogy refers to a type of reasoning in which the type of proof being sought is sharply influenced by the style of proof that was successfully used to prove related theorems.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 selecting an approach based on prior success

This article is the twenty-sixth of a series of articles discussing various open research problems in automated reasoning. The problem posed for research asks one to find criteria for choosing from among previously answered questions a question that dictates the approach to take for attacking the problem of current interest. Especially because the better automated reasoning programs often offer a wide range of choices for representation, inference rule, and strategy, a solution to the proposed problem would materially reduce the difficulty of using these powerful aids for research.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 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 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 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 hyperparamodulation

This article is the thirty-second of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find a variant of the inference rule hyperparamodulation that avoids generating many of the binary paramodulants ordinarily deduced.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 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 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 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.     

The problem of choosing between logic programming and general-purpose automated reasoning

This article is the thirteenth of a series of articles discussing various open research problems in automated reasoning. Here we focus on finding criteria for correctly choosing between using logic programming and a more general automated reasoning approach to attack a given assignment. The problem proposed for research asks one to find criteria that classify problems as solvable with a well-focused algorithm or as requiring a more general search for new information. We include suggestions 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.     

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 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 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 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 guaranteeing the absence of a complete set of reductions

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.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 guaranteeing the existence of a complete set of reductions

This article is the eleventh of a series of articles discussing various open research problems in automated reasoning. Here we focus on finding criteria for guaranteeing the existence of a complete set of reductions. We include a suggestion 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.