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Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver

Adam Naumowicz
2015 Journal of automated reasoning  
In this paper we present the results of an experiment with employing an external SAT solver to strengthen the notion of obviousness of the MIZAR proof checker.  ...  The SAT-enhanced MIZAR checker is programmed to automatically spawn a new Logic2CNF process whenever it needs to justify any goal that can be solved by reducing it into a corresponding propositional satisfiability  ...  , even though the SAT-enhanced checker is used.  ... 
doi:10.1007/s10817-015-9332-6 fatcat:oq5uq4kls5ellhi6ryoh6una3q

Four Decades of Mizar

Adam Grabowski, Artur Korniłowicz, Adam Naumowicz
2015 Journal of automated reasoning  
This special issue is dedicated to works related to Mizar, the theorem proving project started by Andrzej Trybulec in the 1970s, and other automated proof checking systems used for formalizing mathematics  ...  Mizar 1 is the name of both a formal language in which the mathematics is written, and of an entire software system that checks the texts for logical correctness and manages the data base of Mizar articles  ...  Adam Naumowicz in "Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver" described the possibilities of the better integration of Mizar with external systems  ... 
doi:10.1007/s10817-015-9345-1 fatcat:wclwjz6eqzbvbis7w5tdolu5n4

Special Issue on User Interfaces in Theorem Proving: Preface

David Aspinall, Christoph Lüth
2007 Journal of automated reasoning  
Experts are held back too: many interface operations which could significantly enhance productivity are not supported, although they are commonplace in other modern applications.  ...  Second, we have the critical question of how proofs are shown to the user: presentations of proof should be readily comprehensible to both author and subsequent readers.  ...  theorem prover to check simple facts, and a powerful interactive prover (viz, Coq) to check the proofs built manually using the graphical presentation.  ... 
doi:10.1007/s10817-007-9075-0 fatcat:ikxjp56qozhmbhqthymb7ntr7a

ATP and Presentation Service for Mizar Formalizations

Josef Urban, Piotr Rudnicki, Geoff Sutcliffe
2012 Journal of automated reasoning  
The service provides ATP assistance to Mizar authors in finding and explaining proofs, and offers generation of Mizar problems as challenges to ATP systems.  ...  We present the main features of MizAR services, followed by an account of initial experiments in finding proofs with the ATP assistance.  ...  The rich Mizar type system becomes explicit when translated to untyped first-order logic. The @proof syntax tells Mizar to skip checking of such proof block.  ... 
doi:10.1007/s10817-012-9269-y fatcat:vzvfnzbx4zhlvjtrzs75ybyssm

ATP and Presentation Service for Mizar Formalizations [article]

Josef Urban, Piotr Rudnicki, Geoff Sutcliffe
2012 arXiv   pre-print
The service provides ATP assistance to Mizar authors in finding and explaining proofs, and offers generation of Mizar problems as challenges to ATP systems.  ...  We present the main features of MizAR services, followed by an account of initial experiments in finding proofs with the ATP assistance.  ...  We consider it a good feature that Mizar does not allow complicated, fragile, and slow proof finding procedures as a part of the core proof checking.  ... 
arXiv:1109.0616v2 fatcat:2qtvzayef5gkdgl6o36nrpao2q

A Survey on Theorem Provers in Formal Methods [article]

M. Saqib Nawaz, Moin Malik, Yi Li, Meng Sun, M. Ikram Ullah Lali
2019 arXiv   pre-print
Technically, mathematical formalisms and automated reasoning based-approaches are employed to perform inferences and to generate proofs in theorem provers.  ...  In such way, BDD, DPLL based SAT solvers can be used to automatically check the formulas.  ...  Marić [191] Decision procedures, proof search, theorem provers, software correctness, interactive theorem provers survey, formal deduction, proof checking, logical frameworks, SAT solvers, SMT solvers  ... 
arXiv:1912.03028v1 fatcat:fabxqg75xjhxlh7u3uyypgyzdy

History of Interactive Theorem Proving [chapter]

John Harrison, Josef Urban, Freek Wiedijk
2014 Handbook of the History of Logic  
Indeed the influential proof-checking system Mizar, described later, maintains to this day a batch-oriented style where proof scripts are checked in their entirety per run.  ...  Unlike FOL proofs, SAT proofs tend to be relatively large, but nevertheless it has turned out to be possible to check them in a fully expansive way in a time comparable to the time used to generate them  ... 
doi:10.1016/b978-0-444-51624-4.50004-6 fatcat:sngnxnl3ivectj5taj3dgsaa7m

Large Formal Wikis: Issues and Solutions [chapter]

Jesse Alama, Kasper Brink, Lionel Mamane, Josef Urban
2011 Lecture Notes in Computer Science  
Yet the legal status of these efforts of the Mizar community has never been clarified.  ...  The Mizar Mathematical Library (MML) is a large corpus of formalised mathematical knowledge. It has been constructed over the course of many years by a large number of authors and maintainers.  ...  " but rather as "proof generators", i.e., that the "real proof" is just the low-level HOL proof object (checked by HOL's LCF-like microkernel), which the user typically never sees.  ... 
doi:10.1007/978-3-642-22673-1_10 fatcat:4mx3a7zco5anvc6x6gencjl7iq

The Space of Mathematical Software Systems – A Survey of Paradigmatic Systems [article]

Katja Bercic, Jacques Carette, William M. Farmer, Michael Kohlhase, Dennis Müller, Florian Rabe, Yasmine Sharoda
2020 arXiv   pre-print
guide to this space of systems, we devise a novel conceptualization of mathematical software that focuses on five aspects: inference covers formal logic and reasoning about mathematical statements via proofs  ...  Then type checking is a form of verification (i.e., checking that a particular proof/term inhabits a particular proposition/type).  ...  This is the basic idea of logic programming, where a formalization consists of a 5 EdNote: FR: cite some SAT and SMT solvers here 6 EdNote: add more citations 7 EdNote: check Cynthia Kop's talk on higher-order  ... 
arXiv:2002.04955v1 fatcat:th2h5qrd4rcazhw2eqbqmy2zmq

Formalization and Implementation of Modern SAT Solvers

Filip Marić
2009 Journal of automated reasoning  
The given pseudo-code served as a basis for our SAT solver argo-sat.  ...  Correctness proofs are formalized within the Isabelle theorem proving system and are available in the extended version of this paper.  ...  Some of them are Isabelle, HOL, Coq, PVS, Mizar, etc. A comparison of these tools can be found in [Wie03] . Program verification.  ... 
doi:10.1007/s10817-009-9127-8 fatcat:v4zdqvha7rdyzbabyzgx2tmrpy

SEMANTIC DERIVATION VERIFICATION: TECHNIQUES AND IMPLEMENTATION

GEOFF SUTCLIFFE
2006 International journal on artificial intelligence tools  
checking".  ...  semantic verification of derivations, and reports on their implementation and testing in the GDV verifier. a That is the reason for using the term "derivation verification" rather than the more common "proof  ...  A basic structural check is, e.g., that the specified parents of each inference step do exist in the derivation. Structural checking is the basis for the high level checker of the Mizar system.  ... 
doi:10.1142/s0218213006003119 fatcat:upiw62aoy5hjhpx7iourxqehhi

Applications of real number theorem proving in PVS

Hanne Gottliebsen, Ruth Hardy, Olga Lightfoot, Ursula Martin
2012 Formal Aspects of Computing  
Among the earliest work in computational logic was de Bruijn's 1955 system AUTOMATH, which checks a hand proof input by the user.  ...  Substantial areas of analysis were developed in the MIZAR project, as far as the Hahn-Banach theorem [NT93], and a textbook on lattices [GHK + 80], as well as Fleuriot's application of non-standard analysis  ...  Fig. 1 .Fig. 2 . 12 Layout of a SATS airport [CGBK04] Paths of two aircraft entering Region R [CGBK04] Fig. 4 . 4 Maple procedure using PVS to enhance the built-in differential equations solver  ... 
doi:10.1007/s00165-012-0232-9 fatcat:cttlm47ehreezkk2ikywgwuixu

Automated deduction for verification

Natarajan Shankar
2009 ACM Computing Surveys  
Many practical proof checking systems also allow new proof rules to be added when they can be explicitly justified in terms of existing ones. Logic can be used in all kinds of interesting ways.  ...  Beyond verification, deduction techniques can also be used to support a variety of applications including planning, program optimization, and program synthesis. proof checking.  ...  Isar [Wenzel 1999 ] is a declarative style of proof presentation and verification for Isabelle inspired by the Mizar proof checking system [Rudnicki 1992 ].  ... 
doi:10.1145/1592434.1592437 fatcat:satgicglyneqvl2nw5pjnr3d3a

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Arve Gengelbach, Johannes Åman Pohjola, Tjark Weber
2021 Electronic Proceedings in Theoretical Computer Science  
[18] ) there is no proof for consistency or conservativity of Mizar.  ...  Åman Pohjola and Gengelbach [17] prove HOL with overloading consistent in a machine-checked proof, by constructing models of theories of definitions through a construction that originates from Kunčar  ... 
doi:10.4204/eptcs.332.1 fatcat:cdcs4lxnova5xnjr3gc2tyejvq

Formalization of Forcing in Isabelle/ZF [article]

Emmanuel Gunther
2020 arXiv   pre-print
Related work There are various formalizations of Zermelo-Fraenkel set theory in proof assistants (v.g. Mizar, Metamath, and recently Lean [10] ) that proceed to different levels of sophistication.  ...  Some of our definitions by recursion (check and forces) do not fit in that scheme.  ... 
arXiv:2001.09715v2 fatcat:2xkycxxn3vewhhwhhsg4qgj2aa
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