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Discovery and Reasoning in Mathematics

Alan Bundy
1985 International Joint Conference on Artificial Intelligence  
We discuss the automation of mathematical reasoning, surveying the abilities displayed by human mathematicians and the computational techniques available for automating these abilities.  ...  Automated theorem proving has been an important subfield of AI throughout its history. The main techniques required to automate theorem proving are deduction which involves search control.  ...  -The IMPRESS (Inferring Meta-knowledge about PRESS) program, [Sterling & Bundy 82], was a theorem proving program for proving properties of logic programs.  ... 
dblp:conf/ijcai/Bundy85 fatcat:nfwo4w6rajb7djr72pwbje4b3i

Automatic Acquisition of Search Control Knowledge from Multiple Proof Attempts

Jörg Denzinger, Stephan Schulz
2000 Information and Computation  
We present two inference control heuristics for equational deduction that are based on the evaluation of previous successful proof attempts in domains of interest.  ...  The first evaluation function works by symbolic retrieval of generalized patterns from a knowledge base, and the second function compiles the knowledge into abstract term evaluation trees.  ...  AddWeight and extensions of it provide a good basis for learning approaches to control the search of a theorem prover.  ... 
doi:10.1006/inco.1999.2857 fatcat:u2iletlydng47ievjcsiiftr6e

Learning domain knowledge to improve theorem proving [chapter]

Jörg Denzinger, Stephan Schulz
1996 Lecture Notes in Computer Science  
We present two learning inference control heuristics for equational deduction.  ...  Based on data about facts that contributed to previous proofs, evaluation functions learn to select equations that are likely to be of use in new situations.  ...  The task of learning heuristics for theorem provers is highly non-trivial. The rst problem is the selection of the choice points to be controlled by a learning heuristic.  ... 
doi:10.1007/3-540-61511-3_69 fatcat:qh5stt4rfzdevepn27rbda5rku

High Performance ATP Systems by Combining Several AI Methods

Jörg Denzinger, Marc Fuchs, Matthias Fuchs
1997 International Joint Conference on Artificial Intelligence  
We present a design for an automated theorem prover that controls its search based on ideas from several areas of artificial intelligence (AI).  ...  We provide case studies from two domains in pure equational theorem proving.  ...  and to use control knowledge.  ... 
dblp:conf/ijcai/DenzingerFF97 fatcat:nkuorq2wb5h6lem4m5zl3trlui

Page 447 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews  
geometry theorem proving.  ...  (English summary) [Learning rules for feedback control laws for state constrained problems] C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 6, 445-450.  ... 

Combining theorem proving and symbolic mathematical computing [chapter]

Karsten Homann, Jacques Calmet
1995 Lecture Notes in Computer Science  
We describe a model for such a knowledge base mainly consisting of type and algorithm schemata, algebraic algorithms and theorems.  ...  A high level of interaction requires a common knowledge representation of the mathematical knowledge of the two systems.  ...  and learning of theorems out of algebraic algorithms, generation of algorithms from theorems, interaction of the learning component and the theorem prover and applications of the environment.  ... 
doi:10.1007/3-540-60156-2_3 fatcat:gptoyfozcnb73kgiy6mwgus46m

Neural dynamic optimization for control systems.II. Theory

Chang-Yun Seong, B. Widrow
2001 IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics)  
The paper presents neural dynamic optimization (NDO) as a method of optimal feedback control for nonlinear multi-input-multi-output (MIMO) systems.  ...  Index Terms-Dynamic programming, information time shift operator, learning operator, neural dynamic optimization, neural networks, nonlinear systems, optimal feedback control.  ...  Learning Operator In this section, we introduce the learning operator, which characterizes the learning process of the neural network in searching for the optimal feedback solution.  ... 
doi:10.1109/3477.938255 pmid:18244816 fatcat:lv5pihup35eulacrqgt3zkmeaa

Proof planning with multiple strategies

Erica Melis, Andreas Meier, Jörg Siekmann
2008 Artificial Intelligence  
Proof planning is a technique for theorem proving which replaces the ultra-efficient but blind search of classical theorem proving systems by an informed knowledge-based planning process that employs mathematical  ...  knowledge at a human-oriented level of abstraction.  ...  We are grateful for this substantial funding that made this work possible.  ... 
doi:10.1016/j.artint.2007.11.004 fatcat:g4keqaoqjjartcaufcspbv3fz4

Computer supported mathematics with Ωmega

Jörg Siekmann, Christoph Benzmüller, Serge Autexier
2006 Journal of Applied Logic  
Classical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces-often in the range of several billion clauses.  ...  The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive  ...  Secondly, there is mathematical knowledge on how to prove a theorem, which is encoded in tactics and methods, in ANTS agents, in control knowledge and in strategies.  ... 
doi:10.1016/j.jal.2005.10.008 fatcat:bfpsfvprmnfnddfvwwx5cqbjiq

A robust line search for learning control

Brian J. Driessen, Nader Sadegh, Kwan S. Kwok
2001 International Journal of Control  
In this paper a new line search for a Newton Rhapson learning control algorithm is presented.  ...  It is shown that the answer is generally no, at least for gradient-based learning control algorithms.  ...  On the Robustness Against Erroneous Constraint Jacobean of a New Line Search for a Newton Rhaphson Learning Control Algorithm This section will prove that the modifiedline-search Newton Rhaphson algorithm  ... 
doi:10.1080/00207170010025285 fatcat:volndvxqmnecllcmhmihxtcvci

Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Bernhard Gramlich
2005 Electronical Notes in Theoretical Computer Science  
Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems.  ...  Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving.  ...  I'm very grateful to Maria Paola Bonacina and Thierry Boy de la Tour, the Program Co-Chairs of STRATEGIES 2004, for this kind invitation and for their helpful comments on a draft version of this paper.  ... 
doi:10.1016/j.entcs.2005.01.006 fatcat:reml2iecwfexhlnmr4d22a7sj4

When to prove theorems by analogy? [chapter]

Erica Melis
1996 Lecture Notes in Computer Science  
In recent years several computational systems and techniques for theorem proving by analogy have been developed.  ...  This paper addresses this question, identi es situations where analogy is useful, and discusses the merits of theorem proving by analogy in these situations.  ...  Thanks to J org Siekmann and Wolf Schaarschmidt for reading drafts of this paper.  ... 
doi:10.1007/3-540-61708-6_66 fatcat:kyh4djijofc3zfvxdbghf5o2qi

Ωmega: Computer Supported Mathematics [chapter]

Jörg Siekmann, Christoph Benzmüller
2004 Lecture Notes in Computer Science  
Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces -often in the range of several billion clauses.  ...  While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn 1 -more  ...  Secondly, there is mathematical knowledge on how to prove a theorem, which is encoded in tactics and methods, in ΩAnts agents, in control knowledge and in strategies.  ... 
doi:10.1007/978-3-540-30221-6_2 fatcat:ez4gahurujebriliwvbo6ywxue

Reductio ad Absurdum: Planning Proofs by Contradiction [chapter]

Erica Melis, Martin Pollet, Jörg Siekmann
2006 Lecture Notes in Computer Science  
Proof planning is a general technique in automated theorem proving that captures and makes explicit proof patterns and mathematical search control.  ...  Sometimes it is pragmatically useful to prove a theorem by contradiction rather than finding a direct proof.  ...  mathematical control knowledge in automated theorem proving based on proof planning.  ... 
doi:10.1007/11829263_3 fatcat:u6wtqxtd2ncnhdsc4rnfselhqm

ΩMEGA: Resource-Adaptive Processes in an Automated Reasoning System [chapter]

Serge Autexier, Christoph Benzmüller, Dominik Dietrich, Jörg Siekmann
2010 Resource-Adaptive Cognitive Processes  
The motivation is to reduce the combinatorial explosion of the search space in classical automated theorem proving by providing means for a more global search control.  ...  or control knowledge to restrict the search.  ... 
doi:10.1007/978-3-540-89408-7_17 dblp:series/cogtech/AutexierBDS11 fatcat:t6vlyxgwcbe45ctq7oy3mlb7sy
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