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Topic: Paradox theorem prover

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	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).
		E is a high-performance prover built on a purely equational calculus, developed primarily in the automated reasoning group of Technical University of Munich.
		J Strother Moore Co-Author of the Boyer-Moore theorem prover, co-recipient of the Herbrand Award 1999.
	en.wikipedia.org /wiki/Theorem_prover (1136 words)

	Automated theorem proving - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-18)
		For first-order logic it is recursively enumerable, i.e., given unbounded resources, any true theorem can eventually be proven, but invalid theorems cannot always be recognized.
		Despite these theoretical limits, practical theorem provers can solve many hard problems in these logics.
		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).
	encyclopedia.worldsearch.com /automated_theorem_proving.htm (710 words)

	The Paradox
		This is the paradox of the case study.
		Any theorem prover has a range of application: from simple theorems that it can prove totally automatically, to hard ones at the limit of its capacity, requiring expert guidance over multiple steps.
		Indeed, although the experimenter will usually not attempt any case studies using rival provers, the champions of many rival provers frequently take up the challenge, subsequently showing that their provers are equally, if not more, capable of completing the case study.
	www-unix.mcs.anl.gov /AAR/issuesept04/node2.html (427 words)

	Metamath Proof Explorer Home Page
		We also use the word "theorem" informally to denote the result of a proof that also allows references to local hypotheses and thus has the form of an inference rule (example: a1i); however, strictly speaking, such a theorem should be called a derived inference or deduction.
		When an axiom or theorem with a distinct variable condition is referenced in a proof, the distinct variable conditions attached the theorem being proved must satisfy those of the referenced axiom or theorem after substitutions are made into the referenced axiom or theorem.
		Theorem schemes can also be proved, but their proofs use metalogical techniques that are not part of the axiom system.
	metamath.planetmirror.com /mpegif/mmset.html (9227 words)

	LavaCUBED \Science\Math\Logic_and_Foundations\Software (Site not responding. Last check: 2007-10-18)
		Isabelle - A generic theorem proving environment developed at Cambridge University (Larry Paulson) and TU Munich (Tobias Nipkow).
		Paradox - A tool that processes first-order logic problems and tries to find finite-domain models for them; written by Koen Claessen and Niklas SÃ¶rensson.
		Theorem prover for first-order clause logic, written in ECRC's Prolog-dialect ECLiPSe.
	www.lavacubed.com /new.cats.php?path=/Science/Math/Logic_and_Foundations/Software (641 words)

	Directory - Science: Math: Logic and Foundations: Software
		Isabelle · cached · A generic theorem proving environment developed at Cambridge University (Larry Paulson) and TU Munich (Tobias Nipkow).
		Paradox · cached · A tool that processes first-order logic problems and tries to find finite-domain models for them; written by Koen Claessen and Niklas Sörensson.
		Proof General · cached · Comprehensive Gnu-Emacs and XEmacs interface for several theorem provers including Coq, Isabelle, Lego, and Phox.
	www.incywincy.com /default?p=27023 (646 words)

	Automated Deduction
		A vague notion of tactic appeared in the early writings of McCarthy on theorem proving and was perhaps nurtured in the Stanford LCF project.
		Citation Theorem proving in LCF would be done by writing programs in a metalanguage (ML) that used special primitives to manipulate the formulas, terms and proofs of the object logic.
		They built a theorem prover that was especially good at automating these inductive proofs.
	www.cs.cornell.edu /Info/Projects/NuPrl/Intro/AutoDeduction/autoded.html (1370 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).
		In the higher order setting NUPRL helped to confirm Higman's lemma and Gerard's paradox, both of which were under active investigation by humans at the time.
		The specialist geometry prover Geometry Expert, of Chou, Gao, and Zhang, has been used to obtain new results in Euclidean geometry.
	www.cs.miami.edu /~tptp/OverviewOfATP.html (1495 words)

	Software Science, Directory (Site not responding. Last check: 2007-10-18)
		Paradox A tool that processes first-order logic problems and tries to find finite-domain models for them; written by Koen Claessen and Niklas S+Â¦rensson.
		Isabelle A generic theorem proving environment developed at Cambridge University (Larry Paulson) and TU Munich (Tobias Nipkow).
		Proof General Comprehensive Gnu-Emacs and XEmacs interface for several theorem provers including Coq, Isabelle, Lego, and Phox.
	www.wacofdn.org /d2RjXzI3MDIz.aspx (650 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The main theorem of the course is Morley's theorem.
		Skolem's paradox about countable models for formalizations of set theory in which one can prove the existence of uncountable sets is resolved with the aid of the important distinction between standard and nonstandard models.
		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.
	www.cs.cmu.edu /afs/cs/project/pal/www/pal-courses-s04.txt (2337 words)

	World Culinary Institute
		A generic theorem proving environment developed at Cambridge University (Larry Paulson) and TU Munich (Tobias Nipkow).
		A linear logic prover that searches a cutfree proof for the given twosided sequent of firstorder linear logic.
		The system is capable of following fullyautomated routines for theorem proving and hypotheses formation, as well as operating interactively when these reoutines fail.
	www.worldculinaryinstitute.com /cgi-bin/odp/index.cgi?base=/Science/Math/Logic_and_Foundations/Software (559 words)

	Untitled Document
		The development of mechanical theorem provers has inspired several applications, some practical, some academic, and most falling under both categories.
		QED (and automated theorem provers, in general, if used appropriately) can quicken this filtration process, accelerating the progress of the mathematical field and freeing mathematicians to consider new problems creatively.
		Though a perfect and beautiful solution to the verification paradox does not exist, pragmatics still call for software verification, whether it is 100% accurate or 99%.
	cse.stanford.edu /classes/sophomore-college/projects-99/automatic-theorem-proving/Applicat.htm (1243 words)

	The Formal Methods Group (Site not responding. Last check: 2007-10-18)
		We are developing a novel FOL theorem prover based on Stålmarck's mehtod of proof for propositional logic.
		Together with Prover Technology and Gunnar Stålmarck, we are working with automatic theorem proving in FOL.
		This work is based on a theorem proving algorithm similar to Stålmarck's algorithm which is used for SAT solving in propositional logic.
	www.cs.chalmers.se /Cs/Research/FormalMethods (825 words)

	Science Math Logic and Foundations Software
		Paradox - A tool that processes firstorder logic problems and tries to find finitedomain models for them; written by Koen Claessen and Niklas SÃ¶rensson.
		Proof General - Comprehensive GnuEmacs and XEmacs interface for several theorem provers including Coq, Isabelle, Lego, and Phox.
		Theorem prover for firstorder clause logic, written in ECRC's Prologdialect ECLiPSe.
	www.apartment1a.com /Resources/Science/Math/Logic_and_Foundations/Software (669 words)

	AARNEWS - September 2004
		Shankar's reward for this case study was that he received a Ph.D. Note that the theorem prover was developed by third parties: Shankar's supervisors Bob Boyer and J Moore.
		It avoids the case study paradox since the effort required from the student is now directly proportional to the value of the analysis that is produced from the case study.
		The simplistic hypothesis that the prover can prove hard theorems--or harder ones than its rivals--leads to a paradox in which the success of the student is inversely proportional to the success of the prover, and where the proof process becomes a relentless and unrewarding slog.
	www-unix.mcs.anl.gov /AAR/issuesept04/issuesept04.html (4467 words)

	Search the Internet - InternetDJ.com (Site not responding. Last check: 2007-10-18)
		Isabelle - - A generic theorem proving environment developed at Cambridge University (Larry Paulson) and TU Munich (Tobias Nipkow).
		Paradox - - A tool that processes first-order logic problems and tries to find finite-domain models for them; written by Koen Claessen and Niklas Sörensson.
		Proof General - - Comprehensive Gnu-Emacs and XEmacs interface for several theorem provers including Coq, Isabelle, Lego, and Phox.
	www.internetdj.com /search/search.php?browse=/Science/Math/Logic_and_Foundations/Software (743 words)

	New Foundations Home Page
		Here find the proof of Cantor's theorem (a theoretically human-readable text document of considerable size) recorded by the prover (this was generated as the result of dialogue between myself and the prover; it is not an automatically generated proof).
		His research is mostly on Jensen's NFU and extensions and on development and application of the Watson theorem prover, which uses a lambda-calculus equivalent to an extension of NFU as its higher order logic.
		Cantor's paradox of the largest cardinal: Cardinal numbers are defined in NF as equivalence classes of sets of the same size.
	math.boisestate.edu /~holmes/holmes/nf.html (2104 words)

	Software (on ScienceFizz.com) (Site not responding. Last check: 2007-10-18)
		A linear logic prover that searches a cut-free proof for the given two-sided sequent of first-order linear logic.
		A generic sequent prover for propositional finitely-valued logics.
		Comprehensive Gnu-Emacs and XEmacs interface for several theorem provers including Coq, Isabelle, Lego, and Phox.
	www.sciencefizz.com /Math/Logic_and_Foundations/Software (612 words)

	TPTP Technical Report
		Theorems about this algebra can be proved from a small number of properties, suggesting the definition of an embedding algebra.
		Due to paradoxes that arise from such a naive definition, mathematicians now regard the notion of a set as an undefined, primitive concept [How72].
		A secondary reason is different formulations of the theorem to be proved, e.g., different clausal forms of a FOF problem.
	www.cs.miami.edu /~tptp/TPTP/Archive/TPTP-v2.7.0/TR-v2.7.0/TPTPTR.shtml (12084 words)

	More Recent Achievements
		A theorem prover is used to glue together appropriate routines from a subroutine library, by matching specifications of each routine with specifications of the current demands.
		Plans for the court case included calling the theorem prover as a witness but by then the parties had settled out of court.
		Both embedded interactive theorem provers, with the student providing the major steps through a proof (usually with hints) and the theorem prover handling the details.
	www.cs.duke.edu /AutoDedFD/report/node7.html (1641 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Athena is an interactive theorem prover for multi-sorted first-order logic with equality.
		To prove an implication, Athena provides syntactic constructs that temporarily add the implication hypothesis to the assumption base during the scope of the proof for the implication conclusion.
		paradox, a tool that processes first-order logic problems and tries to find finite-domain models for them.
	athena-dpl.sourceforge.net (302 words)

	Axiom of Choice (Site not responding. Last check: 2007-10-18)
		Thus, the Banach-Tarski Paradox does not violate the Law of Conservation of Mass; it merely tells us that the notion of "volume" is more complicated than we might have expected.
		One or more of the sets in the decomposition must be Lebesgue unmeasurable; thus a corollary of the Banach-Tarski Theorem is the fact that there exist sets that are not Lebesgue measurable.
		Isabelle is a generic theorem prover which can support a wide variety of logics; it is available for free and will run on most Unix systems.
	www.math.vanderbilt.edu /~schectex/ccc/choice.html (3626 words)

	Software
		3TAP - Many-valued tableau-based theorem prover developed at the University of Karlsruhe.
		Church - Program understands the different types of lambda expressions, can extract lists of variables (both free and bound) and subterms, and can simplify complicated by expression by means of application.
		Theorem Provers - A list of (well-known) theorem provers on logic-server at the university of Muenchen.
	www.supercrawler.com /Science/Math/Logic_and_Foundations/Software (843 words)

	[agi] A difficulty with AI reflectivity (Site not responding. Last check: 2007-10-18)
		Example: For the Gödel Machine's current theorem-prover to accept a proposed new theorem-prover, might require establishing that if the new theorem-prover accepts a proof that a proposed newer theorem-prover is useful, then the newer theorem-prover is useful.
		This is precisely what the sentence asserts, and therefore the sentence is true and Santa Claus does exist.) Löb's Theorem eliminates the self-reference in the Santa Claus Paradox via a diagonalization argument, as Gödel did with the Epimenides Paradox.
		I have not been able to find anything on reflectivity that wraps all the way around, the way humans think (but then humans are naively vulnerable to Epimenides and Santa Claus paradoxes), or the way a reflectively consistent seed AI would have to wrap around.
	www.mail-archive.com /agi@v2.listbox.com/msg02036.html (930 words)

	TLA Papers
		Theorem proving, while not bound by the same computational constraints, may not be feasible for routinely coping with the complex, low-level details of a real multiplier.
		Input to the system consists of the correctness properties, expressed in TLA (the temporal logic of actions), and their proofs, written in a humanly readable, hierarchically structured form.
		Checking is now done by LP (the Larch Prover), using two different translations--one for action reasoning and one for temporal reasoning.
	research.microsoft.com /users/lamport/tla/papers.html (1630 words)

	References
		Graphical theorem proving: An approach to reasoning with the help of diagrams.
		Variable elimination and chaining in a resolution-based prover for inequalties.
		Efficiency and completeness of the set of support strategy in theorem proving.
	www.cs.duke.edu /AutoDedFD/report/node20.html (1449 words)

	Prolog hints and news (Site not responding. Last check: 2007-10-18)
		If you are using a lab account for the course this is already done.
		Otter (which includes the model builder mace) is currently the only theorem prover/model builder available.
		6th Mar 04 The theorem prover bliksem and the model builder paradox are now available on GSLT's server mozart in addition to otter and mace.
	www.ling.gu.se /~cooper/compsem-gslt/prologhints.html (169 words)

	Logic on the Web -- Web architecture
		On the semantic web in general, a party must be able to follow a proof of a theorem but is not expected to generate one.
		The paradox problem lies not simply in being able to express a paradox, but the logic being such that merely considering a paradox leads one to be able to deduce a falsehood.
		These tricks limit the logic so that you can't in fact have the Russel paradox set (of all sets which are not members of themselves) as you exclude such things as not well-formed.
	www.w3.org /DesignIssues/Logic.html (3114 words)

	Randall Holmes Home Page
		This is a preliminary set of notes; it describes the theoretical basis for an actual implementation of sequent calculus under the Watson theorem prover.
		Here find the SML source (version of October 21, 2005) for a sequent prover using NFU as its logic, adapted from the sequent calculus for SF in Marcel's cut-elimination paper.
		Here are my latest thoughts on the interpretation of ZF in double extension set theory in which interesting properties of the ordinals in double extension set theory are uncovered and used to correct a technical problem with Kisielewicz's interpretation of ZF in double extension set theory.
	diamond.boisestate.edu /~holmes (1048 words)

	[No title]
		Paradox - Processes first-order logic problems and tries to find finite-domain models for them.
		Protein - PROver with a Theory Extension Interface - Automated theorem prover for first-order clause logic.
		PVS Specification and Verification System - Specification language integrated with support tools and a theorem prover.
	botw.org /top/Science/Math/Logic_and_Foundations/Software (587 words)

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