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Understanding LEOII’s proofs
 International Workshop on the Implementation of Logics (IWIL2012
, 2012
"... The Leo and LeoII provers have pioneered the integration of higherorder and firstorder automated theoremproving. To date, the LeoII system is, to our knowledge, the only automated higherorder theoremprover which is capable of generating joint higherorder–firstorder proof objects in TPTP for ..."
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Cited by 5 (4 self)
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The Leo and LeoII provers have pioneered the integration of higherorder and firstorder automated theoremproving. To date, the LeoII system is, to our knowledge, the only automated higherorder theoremprover which is capable of generating joint higherorder–firstorder proof objects in TPTP format. This paper discusses LeoII’s proof objects. The target audience are practitioners with an interest in using LeoII proofs within other systems. 1
A Lost Proof
 IN TPHOLS: WORK IN PROGRESS PAPERS
, 2001
"... We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which t ..."
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Cited by 3 (2 self)
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We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which the proof is carried through to the order of the logic in which the problem is formulated in the first place can result in unmanageable long proofs, although there are short proofs in a logic of higher order. Our motivation in this paper is of practical nature and its aim is to sketch the implications of this example to current technology in automated theorem proving, to point to related questions about the foundational character of type theory (without explicit comprehension axioms) for mathematics, and to work out some challenging aspects with regard to the automation of this proof – which, as we belief, nicely illustrates the discrepancy between the creativity and intuition required in mathematics and the limitations of state of the art theorem provers.
Automation of HigherOrder Logic
 THE HANDBOOK OF THE HISTORY OF LOGIC, EDS. D. GABBAY & J. WOODS; VOLUME 9: LOGIC AND COMPUTATION, EDITOR JÖRG SIEKMANN
, 2014
"... ..."
How to Use a FirstOrder Model Generator for Adjusting Problem Formulations of HigherOrder Logic
, 1994
"... Introduction Model generators play an increasing role in automated theorem proving. The reasons range from the recognition of illformulated problems or the suggestion of lemmas to semantically driven strategies. The arguments in favor of a model generator hold for a firstorder logic as well as fo ..."
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Introduction Model generators play an increasing role in automated theorem proving. The reasons range from the recognition of illformulated problems or the suggestion of lemmas to semantically driven strategies. The arguments in favor of a model generator hold for a firstorder logic as well as for a higherorder logic. The main problem of efficient model finding can be seen in the complexity of the search space. This problem will drastically increase if we try to transfer the standard methods to higherorder logic. If you have n elements in a firstorder universe D ' of individuals you will have n n functions in D ('!') , the universe for the unary functions, which have to be searched heuristically. For the theorem proving in higherorder logic there are three main approaches: first build higherorder theorem provers, second use an axiomatic set theory and
Theoretical Foundations for Practical ‘Totally Functional Programming’
, 2007
"... Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt ..."
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Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt to redress the problem of interpretation. It incorporates an awareness of the undesirability
of interpretation with observations that definitions and a certain style of programming appear to
offer alternatives to it. Application of TFP is expected to lead to a number of significant outcomes,
theoretical as well as practical. Primary among these are novel programming languages to lessen or
eliminate the use of interpretation in programming, leading to betterquality software. However,
TFP contains a number of lacunae in its current formulation, which hinder development of these
outcomes. Among others, formal semantics and typesystems for TFP languages are yet to be
discovered, the means to reduce interpretation in programs is to be determined, and a detailed
explication is needed of interpretation, definition, and the differences between the two. Most
important of all however is the need to develop a complete understanding of the nature of
interpretation. In this work, suitable typesystems for TFP languages are identified, and guidance
given regarding the construction of appropriate formal semantics. Techniques, based around the
‘fold’ operator, are identified and developed for modifying programs so as to reduce the amount of
interpretation they contain. Interpretation as a means of languageextension is also investigated.
v
Finally, the nature of interpretation is considered. Numerous hypotheses relating to it considered in
detail. Combining the results of those analyses with discoveries from elsewhere in this work leads
to the proposal that interpretation is not, in fact, symbolbased computation, but is in fact something
more fundamental: computation that varies with input. We discuss in detail various implications of
this characterisation, including its practical application. An often moreuseful property, ‘inherent
interpretiveness’, is also motivated and discussed in depth. Overall, our inquiries act to give
conceptual and theoretical foundations for practical TFP.
Generalisation in Mathematical Induction from a SecondOrder Point of View
, 1997
"... Introduction Two of the most essential problems in inductive theorem proving are finding an induction scheme and if necessary generalising the induction hypothesis. In this contribution certain general assumptions of this problem will be clarified from the point of view of secondorder logic. This ..."
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Introduction Two of the most essential problems in inductive theorem proving are finding an induction scheme and if necessary generalising the induction hypothesis. In this contribution certain general assumptions of this problem will be clarified from the point of view of secondorder logic. This adds to the knowledge about the search problem in induction theorem proving. The main contributions are that we firstly show how induction schemes and generalisations can systematically be obtained, namely by applying socalled comprehension axioms, and secondly that the search space for the general problem of mathematical induction can be ordered in a way that it is finitely branching. This corresponds to the fact that the a priori infinitely many comprehension axioms can be replaced by a finite subset. 2 Generalisation and Comprehension A standard example for the necessity of generalisation is given by the formula 8x IN P<F48.0
Sound And Complete Translations From Sorted HigherOrder Logic Into Sorted FirstOrder Logic
 Proceedings of PRICAI94, Third Pacific Rim International Conference on Artificial Intelligence
, 1994
"... Extending existing calculi by sorts is a strong means for improving the deductive power of firstorder theorem provers. Since many mathematical facts can be more easily expressed in higherorder logic  aside the greater power of higherorder logic in principle , it is desirable to transfer the a ..."
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Extending existing calculi by sorts is a strong means for improving the deductive power of firstorder theorem provers. Since many mathematical facts can be more easily expressed in higherorder logic  aside the greater power of higherorder logic in principle , it is desirable to transfer the advantages of sorts in the firstorder case to the higherorder case. One possible method for automating higherorder logic is the translation of problem formulations into firstorder logic and the usage of firstorder theorem provers. For a certain class of problems this method can compete with proving theorems directly in higherorder logic as for instance with the TPS theorem prover of Peter Andrews or with the Nuprl proof development environment of Robert Constable. There are translations from unsorted higherorder logic based on Church's simple theory of types into manysorted firstorder logic, which are sound and complete with respect to a Henkinstyle general models semantics. In thi...