Results 1  10
of
289
Multilanguage Hierarchical Logics (or: How We Can Do Without Modal Logics)
, 1994
"... MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to ..."
Abstract

Cited by 212 (52 self)
 Add to MetaCart
MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to modal logics. The motivations of our proposal are technical, epistemological and implementational. From a technical point of view, we prove, among other things, that the set of theorems of the most common modal logics can be embedded (under the obvious bijective mapping between a modal and a first order language) into that of the corresponding ML systems. Moreover, we show that ML systems have properties not holding for modal logics and argue that these properties are justified by our intuitions. This claim is motivated by the study of how ML systems can be used in the representation of beliefs (more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. Finally, from an implementation point of view, we argue that ML systems resemble closely the current practice in the computer representation of propositional attitudes and metatheoretic theorem proving.
Productive Use of Failure in Inductive Proof
 Journal of Automated Reasoning
, 1995
"... Proof by mathematical induction gives rise to various kinds of eureka steps, e.g. missing lemmata, generalization, etc. Most inductive theorem provers rely upon user intervention in supplying the required eureka steps. ..."
Abstract

Cited by 116 (26 self)
 Add to MetaCart
(Show Context)
Proof by mathematical induction gives rise to various kinds of eureka steps, e.g. missing lemmata, generalization, etc. Most inductive theorem provers rely upon user intervention in supplying the required eureka steps.
Experiments with Proof Plans for Induction
 Journal of Automated Reasoning
, 1992
"... The technique of proof plans, is explained. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research is described to test this technique in the domain of theorem proving by mathematical induction. Heuristics, adapted from the work of Boye ..."
Abstract

Cited by 105 (37 self)
 Add to MetaCart
The technique of proof plans, is explained. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research is described to test this technique in the domain of theorem proving by mathematical induction. Heuristics, adapted from the work of Boyer and Moore, have been implemented as Prolog programs, called tactics, and used to guide an inductive proof checker, Oyster. These tactics have been partially specified in a metalogic, and the plan formation program, clam, has been used to reason with these specifications and form plans. These plans are then executed by running their associated tactics and, hence, performing an Oyster proof. Results are presented of the use of this technique on a number of standard theorems from the literature. Searching in the planning space is shown to be considerably cheaper than searching directly in Oyster's search space. The success rate on the standard theorems is high. Keywords Theorem prov...
A Science of Reasoning
, 1991
"... This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a highlevel understanding of proofs to complement the lowlevel understanding provided by Logic. It proposes a role for computation in providing this high ..."
Abstract

Cited by 88 (22 self)
 Add to MetaCart
This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a highlevel understanding of proofs to complement the lowlevel understanding provided by Logic. It proposes a role for computation in providing this highlevel understanding, namely by the association of proof plans with proofs. Proof plans are defined and examples are given for two families of proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs The understanding of reasoning has interested researchers since, at least, Aristotle. Logic has been proposed by Aristotle, Boole, Frege and others as a way of formalising arguments and understanding their structure. There have also been psychological studies of how people and animals actually do reason. The work on Logic has been especially influential in the automation of reasoning. For instance, resolution...
Reconstructing Proofs at the Assertion Level
, 1994
"... Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND pr ..."
Abstract

Cited by 70 (9 self)
 Add to MetaCart
(Show Context)
Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level , where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domainspecific inference rules as well. Furthermore, these rules can be represented compactly in a tre structure. Finally, we describe a system called PROVERB , which substantially sh...
ΩMEGA: Towards a Mathematical Assistant
, 1997
"... Ωmega is a mixedinitiative system with the ultimate purpose of supporting theorem proving in mainstream mathematics and mathematics education. The current system consists of a proof planner and an integrated collection of tools for formulating problems, proving subproblems, and proof presentati ..."
Abstract

Cited by 69 (30 self)
 Add to MetaCart
Ωmega is a mixedinitiative system with the ultimate purpose of supporting theorem proving in mainstream mathematics and mathematics education. The current system consists of a proof planner and an integrated collection of tools for formulating problems, proving subproblems, and proof presentation.
The Use of Planning Critics in Mechanizing Inductive Proofs
 International Conference on Logic Programming and Automated Reasoning  LPAR 92, St. Petersburg, Lecture Notes in Artificial Intelligence No. 624
, 1992
"... Proof plans provide a technique for guiding the search for a proof in the context of tactical style reasoning. We propose an extension to this technique in which failure may be exploited in the search for a proof. This extension is based upon the concept of planning critics. In particular we ill ..."
Abstract

Cited by 61 (12 self)
 Add to MetaCart
Proof plans provide a technique for guiding the search for a proof in the context of tactical style reasoning. We propose an extension to this technique in which failure may be exploited in the search for a proof. This extension is based upon the concept of planning critics. In particular we illustrate how proof critics may be used to patch proof plans in the domain of inductive proofs. 1 Introduction Proof plans [Bundy 88] guide the search for a proof in the context of tactical style reasoning [Gordon et al 79]. A proof plan contains a tactic together with a proof rationale. The tactic component specifies the lowlevel structure of a proof in terms of the objectlevel logic inference rules and is used to control the theorem prover. In contrast, the proof rationale, which is expressed in a metalogic, captures the highlevel structure of a proof. Proof plans are constructed from tactic specifications called methods. Using the metalogic, a method expresses the preconditions unde...
Proof Planning with Multiple Strategies
 In Proc. of the First International Conference on Computational Logic
, 2000
"... . Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introd ..."
Abstract

Cited by 60 (37 self)
 Add to MetaCart
(Show Context)
. Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introduce multistrategy proof planning that allows for combining a number of strategies and for switching flexibly between strategies in a proof planning process. Thereby proof planning becomes more robust since it does not necessarily fail if one problem solving mechanism fails. Rather it can reason about preference of strategies and about failures. Moreover, our strategies provide a means for structuring the vast amount of knowledge such that the planner can cope with the otherwise overwhelming knowledge in mathematics. 1 Introduction The choice of an appropriate problem solving strategy is a crucial human skill and is typically guided by some metalevel reasoning. Trained mathematicia...