Comparing Approaches to the Exploration of the Domain of Residue Classes

Meier, Andreas; Pollet, Martin; Sorge, Volker

doi:10.1006/jsco.2002.0550

Cited by 14 publications

(8 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A means to reduce the branching factor is the extensive use of mathematical knowledge to guide proof planning. This guidance has, for instance, been realized by rippling heuristics for difference reduction (Bundy et al, 1993) and by declarative control rules for many other types of proofs (Melis, 1998a;Meier et al, 2002).…”

Section: Guided Search In Proof Planningmentioning

confidence: 99%

Constraint Solving for Proof Planning

Zimmer

Melis

2004

J Autom Reasoning

View full text Add to dashboard Cite

Proof planning is an application of AI planning to theorem proving that employs plan operators that encapsulate mathematical proof techniques. Many proofs require the instantiation of variables; that is, mathematical objects with certain properties have to be constructed. This is particularly difficult for automated theorem provers if the instantiations have to satisfy requirements specific for a mathematical theory, for example, for finite sets or for real numbers, because in this case unification is insufficient for finding a proper instantiation. Often, constraint solving can be employed for this task. We describe a framework for the integration of constraint solving into proof planning that combines proof planners and stand-alone constraint solvers. Proof planning has some peculiar requirements that are not met by any off-the-shelf constraint-solving system. Therefore, we extended an existing propagation-based constraint solver in a generic way. This approach generalizes previous work on tackling the problem. It provides a more principled way and employs existing AI technology.

show abstract

Section: Guided Search In Proof Planningmentioning

confidence: 99%

Constraint Solving for Proof Planning

Zimmer

Melis

2004

J Autom Reasoning

View full text Add to dashboard Cite

show abstract

“…This is not uncommon for automated problem solvers when some analysis has to be carried out separately for a set of values, and it ultimately turns out that the inference lines differ among each other only in terms of references to these values. We encountered such a situation for categorizing residue classes (Meier et al, 2002) in terms of their algebraic structures. In the course of these proofs, case analyses have been carried out, distinguishing according to all residue classes modulo a given integer n (0 .…”

Section: Application Conditions Of Operations and Their Effectsmentioning

confidence: 99%

Handling Dependencies in Reorganizing Content Specifications – A Case Study of Case Analysis

Horacek

2006

Res Lang Comput

View full text Add to dashboard Cite

Handling the dependencies among alternatives in composing expressions in an efficient and qualitatively accurate manner is a fundamental problem of NLG. To pursue this goal effectively, simplifications are put forward in practical approaches, but also ambitious control regimes are tried out occasionally. However, neither of these is able to operate adequately on larger and involved structures. Approaching this issue in a methodological way, we present a case study from the area of mathematical proofs that illustrates the rhetorically motivated reorganization of machine-generated case analyses. Ingredients of this investigation are the design of optimization operations, orderings on groups of operations that take their dependencies into account, and tentative applications of local operations to test the effects of crucial dependencies. Our approach conceives NLG as a standard pipe-line architecture putting emphasis on orderings, with local revisions as a minor extension. This is particularly effective when text planning is organized as an optimization rather than as a construction process, such as for the presentation of mathematical proofs.

show abstract

“…In one of these strategies, the TryAndError strategy (see [12]), the Analyze MV-Dependencies pattern is crucial since Multi has to deal with nested existential quantifiers, which result in 'nested' meta-variables shared by several goals. Hence, dependencies among the meta-variables and the goals have to be analyzed.…”

Section: Residue Class Problemsmentioning

confidence: 99%

“…The suitable backtracking is guided by prefer-constraints-deletion, which overwrites the default backtracking in Multi in this case. A domain, where prefer-constraints-deletion is used for such a trial and error of metavariable instantiations are residue class problems (see [12,9]). …”

Section: Analyzing Meta-variable Dependenciesmentioning

confidence: 99%

Failure Reasoning in Multiple-Strategy Proof Planning

Meier

Melis

2005

Electronic Notes in Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

Monitoring a solution process and applying the right action at the right moment are at the heart of intelligent problem solving by humans. This includes the analysis of failure events and the development of "recommendations" to overcome typical failures. In this article, we present how meta-reasoning on failures is used in multiple-strategy proof planning with the Multi system. Multi allows for a flexible traversal of the search space and a flexible construction of the proof plan guided by mathematically motivated heuristics. Because of the flexible control in Multi failures can be exploited to guide subsequent proof plan manipulations and refinements. The failure reasoning cannot only ease the derivation of a solution proof plan but is required for some problems to find a solution at all.

show abstract

Comparing Approaches to the Exploration of the Domain of Residue Classes

Cited by 14 publications

References 19 publications

Constraint Solving for Proof Planning

Constraint Solving for Proof Planning

Handling Dependencies in Reorganizing Content Specifications – A Case Study of Case Analysis

Failure Reasoning in Multiple-Strategy Proof Planning

Contact Info

Product

Resources

About