GEOM-a prolog geometry theorem prover (original) (raw)
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IEEE Computer Graphics and Applications, 1986
Prolog is a~ef vi its primitives, such~~-The fifth-generation logic programming languagẽ ication, math the requirements of many geometric Prologb2 appears appropriate for research in geomealgoithus.0urrg te lst to yars we aveIm-try and graphics. Examples of its use in architectural plemented programs to solve several problemsin design,' in CAD t and in constructing geometric objects from certain constraints' have been dis-Sysem,conex-wItcalultio, pana grph ravrcussed. ,Prolog has several properties important to geopolygos usIng ultiple recision metric and graphics applications. It is a declarative ratonai~uniere ad crtdrahicma,ovrla. Or~i nstead of an imperative language, which means, at least in theory, that you specify the formula satisfied A earliervesinga ofunction toi averytilcelteof a set, by the solution instead of the procedure to execute *xec4lngaprocdure o lon~s certin~to find the solution. In practice, for nontrivial prottrn xiss, nd sin unfictio toprogat a grams you must often, for efficiency, be somewhat tranitie fncto~Tis rtile escrbestheexpri-imperative. Prolog, like pure Lisp, has no destructive enoejnc#dlngaraigmsfproramninghatsern assignments, except for modifications of the global u~fl, nd frali lits ha weseeas headvntaes database. The unification of logical formulas is a and disadvaiitages of Prolog. primitive operator. The only data structure is the list. Lists can contain free variables that are later instantiated or unified with other free or bound vari-'Sumitro Samaddar is now with Siemens Research and Technology Laboratories. Margaret Nichols is now with the North American Philips An earlier version of this article appeared in Proc. Graphics Interface 86. Lighting Co.
SATCHMO: a theorem prover implemented in Prolog
1988
Satchmo is a theorem prover consisting of just a few short and simple Prolog programs. Prolog may be used for representing problem clauses as well. SATCHMO is based on a model-generation paradigm. It is refutation-complete if used in a level-saturation manner. The paper provides a thorough report on experiences with SATCHMO. A considerable amount of problems could be solved with surprising efficiency.
Automated Generation of Formal and Readable Proofs in Geometry Using Coherent Logic
2010
We present a theorem prover ArgoCLP based on coherent logic that can be used for generating both readable and formal (machine verifiable) proofs in various theories, primarily geometry. We applied the prover to various axiomatic systems and proved dozens of theorems from standard university textbooks on geometry. The generated proofs can be used in different educational purposes and can contribute to the growing body of formalized mathematics. The system can be used, for instance, in showing that modifications of some axioms does not change the power of an axiom system. The system can also be used as an assistant for proving appropriately chosen subgoals of complex conjectures.
Towards Ranking Geometric Automated Theorem Provers
Electronic Proceedings in Theoretical Computer Science
The field of geometric automated theorem provers has a long and rich history, from the early AI approaches of the 1960s, synthetic provers, to today algebraic and synthetic provers. The geometry automated deduction area differs from other areas by the strong connection between the axiomatic theories and its standard models. In many cases the geometric constructions are used to establish the theorems' statements, geometric constructions are, in some provers, used to conduct the proof, used as counterexamples to close some branches of the automatic proof. Synthetic geometry proofs are done using geometric properties, proofs that can have a visual counterpart in the supporting geometric construction. With the growing use of geometry automatic deduction tools as applications in other areas, e.g. in education, the need to evaluate them, using different criteria, is felt. Establishing a ranking among geometric automated theorem provers will be useful for the improvement of the current methods/implementations. Improvements could concern wider scope, better efficiency, proof readability and proof reliability. To achieve the goal of being able to compare geometric automated theorem provers a common test bench is needed: a common language to describe the geometric problems; a comprehensive repository of geometric problems and a set of quality measures.
GraATP: A graph theoretic approach for Automated Theorem Proving in plane geometry
The 8th International Conference on Software, Knowledge, Information Management and Applications (SKIMA 2014), 2014
Automated Theorem Proving (ATP) is an established branch of Artificial Intelligence. The purpose of ATP is to design a system which can automatically figure out an algorithm either to prove or disprove a mathematical claim, on the basis of a set of given premises, using a set of fundamental postulates and following the method of logical inference. In this paper, we propose GraATP, a generalized framework for automated theorem proving in plane geometry. Our proposed method translates the geometric entities into nodes of a graph and the relations between them as edges of that graph. The automated system searches for different ways to reach the conclusion for a claim via graph traversal by which the validity of the geometric theorem is examined.
Design of an intelligent system for the automatic demonstration of geometry theorems
International Conference on Telecommunications, 2010
In this work will be presented the design of an intelligent system destined for development process of demonstrating abilities for geometry theorems. This system will make available to user a proof assistant which will allow interactive vizualization of several demonstrations for the same theorem, demonstrations that have been generated by using three specific methods for automatic demonstration of theorems: area method, full-angle method and inferences accomplishment. For the implementation of the component used to represent knowledge and proof mechanisms will be used Prolog language and for the achievement of geometric construction associated to the theorem will be used Java language.
Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry
Annals of Mathematics and Artificial Intelligence, 2015
The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of nontrivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics-a book on Tarski's geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37% of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42%. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge.
Automated Deduction and Knowledge Management in Geometry
Mathematics in Computer Science, 2020
Scientific research and education at all levels are concerned primarily with the discovery, verification, communication, and application of scientific knowledge. Learning, reusing, inventing, and archiving are the four essential aspects of knowledge accumulation in mankind's civilisation process. In this cycle of knowledge accumulation, which has been supported for thousands of years by written books and other physical means, rigorous reasoning has always played an essential role. Nowadays this process is becoming more and more effective due to the availability of new paradigms based on computer applications. Geometric reasoning with such computer applications is one of the most attractive challenges for future accumulation and dissemination of knowledge.
Integrating Dynamic Geometry Software, Deduction Systems, and Theorem Repositories
2006
The axiomatic presentation of geometry fills the gap between formal logic and our spatial intuition. The study of geometry is, and will always be, very important for a mathematical practitioner. GCLCprover, an automatic theorem prover (ATP) integrated with dynamic geometry software (DGS) gives its user a tool to bridge his/her spatial intuition with formal, Euclidean geometry proofs. GeoThms, a system consisting of the mentioned programs and a database geoDB, provides a framework for exploring geometrical knowledge. A GeoThms user can browse through a list of available geometric problems, their statements, illustrations, and proofs. He/she can also interactively produce new geometrical constructions, theorems, and proofs and add new results to the existing ones. GeoThms framework provides an environment suitable for new ways of studying and teaching geometry at different levels. GeoThms also provides a system for storing mathematical knowledge (in a explicit, declarative form) — not only theorem statements, but also their (automatically generated) proofs and corresponding illustrations.