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Topic: Theorem proving

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	Theorem Proving
		The theorem proving tools consist of a powerful collection of inference steps that can be used to reduce a proof goal to simpler sub-goals that can be discharged automatically by the primitive proof steps of the prover.
		Theorem proving is generally harder requiring considerable technical expertise and understanding of the specification.
		One of the major disadvantages of using theorem provers is that if you fail to complete the proof of a property, the tool will not tell you whether the property is indeed unprovable or whether the user is not providing it with enough information to complete the proof.
	crisys.cs.umn.edu /theorem-proving.html (488 words)

	Automated theorem proving - Wikipedia, the free encyclopedia
		A good example of this was the machine-aided proof of the four color theorem, which was very controversial as the first claimed mathematical proof which was essentially impossible to verify by humans due to the enormous size of the program's calculation (such proofs are called non-surveyable proofs).
		First-order theorem proving is one of the most mature subfields of automated theorem proving.
		J Strother Moore, co-author of the Boyer-Moore theorem prover, co-recipient of the Herbrand Award 1999.
	en.wikipedia.org /wiki/Automated_theorem_proving (1258 words)

	Automated Theorem Proving (Site not responding. Last check: 2007-09-10)
		He proves that if a sentence is refutable in T, then it is refutable in R, by using Smullyan's Unifying Principle applied to the class of sets of wff's which are not refutable in R (after variables are replaced by new constants).
		Furthermore, he proves that a set of clauses S has no model iff there is a set of occurrences T of clauses selected from S and an acceptable mating of the literal occurrences in T. Thus, an exhaustive search for such a mating is a complete refutation procedure.
		When trying to prove a theorem (they concentrate on equality of functional programs) by mathematical induction, it is often necessary (in order for the induction step to work) to prove a more general result from which the specific theorem follows.
	www.andrew.cmu.edu /user/cebrown/notes/ATP.html (7701 words)

	Theorem Proving with the Real Numbers (Site not responding. Last check: 2007-09-10)
		We discuss the desirability of combining the rigour of theorem provers with the power and convenience of computer algebra systems, and explain a method we have used in practice to achieve this.
		We aim to show that a theory of real numbers is useful in practice and interesting in theory, and that the `LCF style' of theorem proving is well suited to the kind of work we describe.
		We hope also to convince the reader that the kind of mathematics needed for applications is well within the abilities of current theorem proving technology.
	www.cl.cam.ac.uk /users/jrh/papers/thesis.html (356 words)

	Automated Theorem Proving
		However, the collection of "Theorem Proving Haiku" at http://www.cl.cam.ac.uk/Research/HVG/haiku.html suggests that the use of proof assistants can be a somewhat frustrating occupation.
		PrologLanguage is based on automated theorem proving and can be used to create deductive theorem provers ("out of the box" it can't do so; you have to create or download programs written in it.
		Automated theorem proving triggered the invention of the TuringMachine and Church's LambdaCalculus.
	c2.com /cgi/wiki?AutomatedTheoremProving (1687 words)

	Dr. Dobb's \| Theorem Proving and Database Querying \| July 22, 2001
		Automated theorem proving is one such technique that has been heavily used in research labs.
		To prove a theorem in propositional logic, you start with axioms you assume are true, and use a set of inference rules to derive the result you want.
		A common approach to theorem proving is to rewrite all statements as groups of clauses combined with and.
	www.ddj.com /184410644?pgno=1 (3494 words)

	HOL theorem prover - Wikipedia, the free encyclopedia
		This language was originally developed along with LCF to serve the purpose of a meta-language for theorem proving systems; in fact, the name stands for "Meta-Language".
		HOL is a successor of the LCF theorem prover.
		Among the successors of HOL is the Isabelle theorem prover.
	en.wikipedia.org /wiki/HOL_theorem_prover (417 words)

	Theorem Proving by Analogy - A Compelling Example - Melis (ResearchIndex)
		This paper shows how a new approach to theorem proving by analogy is applicable to real maths problems.
		The Heine-Borel theorem is a widely known result in mathematics.
		Theorem proving by analogy -- a compelling example.
	citeseer.ist.psu.edu /melis95theorem.html (651 words)

	Resolution Theorem Proving
		Automated theorem proving is the application of rules of deduction to the problem of proving that a theorem statement follows from a set of axioms which are taken to be true.
		HR conjectured and Otter proved many interesting conjectures, that we probably would never have stumbled across, including: that there must be two different elements on the diagonal of the multiplication table, that anti-associative algebras cannot be quasigroups and that they cannot have an identity element.
		Proof planning has been particularly useful for proving theorems inductively, i.e., using the the principal of mathematical induction (not to be confused with the type of inductive reasoning we will be covering in the lectures on machine learning).
	www.doc.ic.ac.uk /~sgc/teaching/v231/lecture9.html (4853 words)

	Amazon.ca: Symbolic Logic and Mechanical Theorem Proving: Books: Chin-Liang Chang,Richard Char-Tung Lee (Site not responding. Last check: 2007-09-10)
		Chapters 4-9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
		Mechanical theorem proving is an important subject in artificial intelligence.
		Appropriate for both undergraduates and graduate courses in theorem proving, this text is also suitable as a supplementary textbook in artificial intelligence to provide the student with a background in mechanical theorem proving and application areas.
	www.amazon.ca /Symbolic-Logic-Mechanical-Theorem-Proving/dp/0121703509 (339 words)

	EPGY Theorem Proving Environment User Guide
		The EPGY Theorem Proving Environment is a computer program that enables students to write ordinary mathematical proofs.
		Below is a brief description of the various components of the theorem proving environment.
		To prove an existentially quantified statement, you will often need to supply a particular term to substitute for the quantified variable (there are other situations where the existential generalization rule will be useful, but this is the most common).
	epgy.stanford.edu /tpe/Guide/index.html (6996 words)

	Theorem Proving System (Site not responding. Last check: 2007-09-10)
		Disregarding the wit and wisdom of one of the 20th century's greatest mathematicians, a theorem proving system is a computer program (usually) designed to "prove theorems" from a set of facts and axioms (see AutomatedTheoremProving).
		It should be noted that "theorem proving" does not (in general) refer to anything as grandiose as proving FermatsLastTheorem--instead it refers to being able to enter (usually simple) queries against a database of facts and axioms and being given an answer, derivable from the facts and axioms (as well as the basic laws of logic).
		Theorem proving is, in the general case, an undecidable problem (see GoedelsIncompletenessTheorem); though many specific problems are decidable.
	c2.com /cgi/wiki?TheoremProvingSystem (252 words)

	leanCoP: Lean Connection-Based Theorem Proving
		leanCoP is a compact Prolog theorem prover for classical first-order logic which is based on the connection calculus.
		It is a theorem prover for intuitionistic first-order logic.
		ModLeanTAP - a compact Prolog theorem prover for modal logics based on analytic tableaux by Bernhard Beckert and Rajeev Goré.
	www.leancop.de (356 words)

	Automated theorem proving (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		Category:Proofs Automated theorem proving (currently the most important subfield of automated reasoning) is the proving of mathematical theorems by a computer program.
		A good example of this was the machine-aided proof of the four color theorem, which was very controversial as the first claimed mathematical proof which was essentially impossible to verify by humans due to the enourmous size of the program's calculation (such proofs are called non-surveyable proofs).
		In the latest processors from AMD, Intel, and others, automated theorem proving has been used to verify that the divide and other operations are correct.
	automated-theorem-proving.kiwiki.homeip.net.cob-web.org:8888 (660 words)

	Inference-security analysis using resolution theorem-proving
		Our approach involves analyzing facts and inference rules assumed to be known to a compromiser, deriving all their possible consequences using resolution theorem-proving, a technique which we argue is far more appropriate to this problem than rule-based expert systems or information flow analysis.
		These are serious disadvantages when one wants to prove the security of something, when valid conclusions unreachable by the control algorithm compromise security.
		Another redundancy is when the same clause is proved on two overlapping or adjoining intervals, in which case the two occurrences can be merged to span the union of the time intervals.
	www.cs.nps.navy.mil /people/faculty/rowe/security.html (5590 words)

	automatic theorem proving \| Lambda the Ultimate
		Finally, a minor point: Coq is not an automated theorem prover, but rather a proof assistant: it supports interactive, rather than automated, theorem proving.
		The theorems are as expressive as the entirety of constructive mathematics, and if you want, you can even add the axioms necessary to support classical logic, at the expense of extractability.
		Proving a theorem with Coq is kind of like writing code in an IDE with good undo support, interactive evaluation, and so on.
	lambda-the-ultimate.org /node/1661 (2513 words)

	Theorem Prover Notes
		DTP (Don's Theorem Prover) is an inference engine written by Don Geddis for first-order predicate calculus, and it specializes in domain-independent control of reasoning.
		It is not intended for use in the construction of high-performance theorem provers, but rather to provide a modular and customizable system useful for rapid prototyping and experimentation in teaching and research.
		The Otter theorem prover developed at Argonne National Laboratory has been quite successful at proving difficult theorems that are intractable for PTTP.
	www-ksl.stanford.edu /people/neller/theorem-provers.html (644 words)

	Bibliography of Instance-Based Theorem Proving
		The idea of instance-based theorem proving is to show that a set S of first-order clauses is unsatisfiable by constructing a set T of ground instances of clauses in S and testing T for unsatisfiability using an efficient propositional calculus decision procedure.
		Though instance-based theorem proving may not be as well known as other approaches, it is noteworthy that instance-based theorem proving methods often far outperform other strategies, even when implemented in Prolog.
		A search strategy for use with instance based theorem proving that controls the consumption of storage: Controlling the consumption of storage with sliding priority search in a hyper-linking based theorem prover, with Shie-Jue Lee, Computers and Artificial Intelligence 14:6 (1995) 563-578.
	www.cs.unc.edu /Research/mi/ftp/instance.html (1313 words)

	Computer Algebra, Theorem Proving, and Types (Site not responding. Last check: 2007-09-10)
		Many computations a mathematician performs can be described in "algebraic" terms, that is, as dealing with various symbolic entities that are combined in restricted ways and are subject to laws (e.g., equations) specifying which combinations are equivalent.
		A discussion of the roles automatic theorem proving might have in these fields.
		Concluding remarks on the relationship between types and theorem proving and a prospectus for what can (and should) be done in this area.
	www.cs.cornell.edu /NuPrl/PRLSeminar/PRLSeminar94_95/Wilson/Oct4.html (200 words)

	Effective Theorem Proving for Hardware Verification - Cyrluk, Rajan, Shankar, Srivas (ResearchIndex)
		The attractiveness of using theorem provers for system design verification lies in their generality.
		The major practical challenge confronting theorem proving technology is in combining this generality with an acceptable degree of automation.
		We describe an approach for enhancing the effectiveness of theorem provers for hardware verification through the use of efficient automatic procedures for rewriting, arithmetic and equality reasoning, and an off-the-shelf BDD-based propositional...
	citeseer.ist.psu.edu /cyrluk94effective.html (733 words)

	Scottish Theorem Proving (Site not responding. Last check: 2007-09-10)
		Theorem proving research is notably strong in Scottish universities, with active groups and researchers in at least six departments.
		The Scottish Theorem Proving Seminar Series provides a common venue for communication and sharing of ideas by all these researchers.
		At least once a term, one of the departments hosts an informal seminar for the whole Scottish theorem proving community.
	www.macs.hw.ac.uk /stp (239 words)

	Resolution Theorem Proving
		Resolution theorem proving works by negating the assertion that is to be proved, and trying to prove a contradiction...or the empty set.
		The numbers refer to the item in the database which is "resolved" with the expression.
		The contradiction can be eliminated by asserting the negation of what we began with....which itself was the negation of what we wanted to prove.
	www.rci.rutgers.edu /~cfs/472_html/Logic_KR/resolution.html (666 words)

	Inductive Theorem Proving Contest
		The motivations for running such a competition are to evaluate inductive theorem provers, to identify possible improvements and open research problems, and to construct a problem library for inductive theorem provers.
		Although we are aware of the limitations of this contest we see a clear potential for future competitions which will be improved with respect to the problem library and also with respect to a more granular evaluation of the experimental results.
		Examples are program synthesis (or proving existentially quantified formulas respectively), proving the termination of programs, computing appropriate case analyses and induction schemes, or general issues of first-order theorem proving within inductive proofs.
	www.dfki.de /floc-ws13/contest.html (353 words)

	Resolution Theorem Proving (Site not responding. Last check: 2007-09-10)
		Proof by contradiction: If we negate the formula to be proven and add this to a consistent database of axioms, the database will contain a contradiction if and only if the formula is true.
		To prove a formula P given a set of axioms A
		Negate P and add it to A. Convert all of the formulas (included the negated goal) to clause form.
	www.indiana.edu /~gasser/Q351/theorem_proving.html (200 words)

	Amazon.fr : Automated Theorem Proving: Theory and Practice: Livres en anglais: Monty Newborn,Monroe Newborn (Site not responding. Last check: 2007-09-10)
		As the 21st century begins, the power of our new tool and partner, the computer, is increasing at an astonishing rate.
		Increasingly computers are expected to be more intelligent, to reason, to be able to draw conclusions from facts, or abstractly, to prove theorems, the subject of this book.
		An introduction to the use of computers to prove theorems.
	www.amazon.fr /Automated-Theorem-Proving-Theory-Practice/dp/0387950753 (421 words)

	Example of Resolution Theorem Proving
		(Robinson, 1965) is one proof theoretic method for proving theorems in First Order Logic.
		To the right is a database for use in the example of resolution theorem proving.
		The statements are shown in English (in dark blue), in conjunctive normal form (in green), and in the syntax of First Order Logic (in white).
	www.rci.rutgers.edu /~cfs/305_html/Deduction/ResolutionTP.html (350 words)

	EPGY Theorem Proving Environment
		The Theorem Proving Environment is designed to run within the EPGY Course System.
		In the students' version of the software, exercises are served up with a theorem database and Derivation System rules tailored to the position in the course.
		To view most proofs correctly, will also need the correct theorem database; if a theorem's name does not appear in the open database, it will not appear in a justification line for any proof step, some some inferences may seem quite mysterious!
	epgy.stanford.edu /tpe/index.html?download.html (335 words)

	An Overview of Automated Theorem Proving
		Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement (the conjecture) is a logical consequence of a set of statements (the axioms and hypotheses).
		The interaction may be at a very detailed level, where the user guides the inferences made by the system, or at a much higher level where the user determines intermediate lemmas to be proved on the way to the proof of a conjecture.
		Safelogic is a Swedish company that provides ATP based tools for verification of a system's logical functionality.
	www.cs.miami.edu /~tptp/OverviewOfATP.html (1495 words)

	15-815 Automated Theorem Proving
		It consists of a traditional lecture component and a joint project in which we will construct a theorem prover.
		The basic theorem proving paradigms we plan to cover are tableaux and the inverse method, both of which are applicable to classical and non-classical logics.
		(4/6) Paper on An Intuitionistic Predicate Logic Theorem Prover by Sahlin, Franzén, and Haridi, JLC 2(5):619-656, 1992, and earlier technical report are now available.
	www.cs.cmu.edu /~fp/courses/atp (581 words)

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