A view of automated proof checking and proving
Christian Lengauer · Annals of Operations Research · 1988
Cited by 1
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About 333 papers for “Automated proof checking”
Automated proof checking in introductory discrete mathematics classes
Andrew J. Haven · DSpace@MIT (Massachusetts Institute of Technology) · 2013
Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.
Cited by 1Open access
Sociomathematical Norms and Automated Proof Checking in Mathematical Education: Reflections and Experiences
Merlin Carl · Journal of Humanistic Mathematics · 2024
According to a widely held view, mathematical proofs are essentially (indications of) formal derivations, and thus in principle mechanically checkable (this view is defended, for example, by Azzouni [3]). This should in particular hold for the kind of simple proof exercises typically given to students of mathematics learning to write proofs. If that is so, then automated proof checking should be an attractive option for math education at the undergraduate level. An opposing view would be that mathematical proofs are social objects and that what constitutes a mathematical proof can thus not be separated from the social context in which it arises. In particular, such “sociomathematical norms” play a role in teaching situations, see, for example, Stephan [37]. There thus seems to be a tension between the inherent social nature of “natural” mathematical proofs and attempts at their automated verification. In this paper, we explore this tension both theoretically and on the basis of our experiences with developing the Diproche system, a program for the automated verification of natural language proofs in college-level introduction to proof classes (modelled after the example of the Naproche system by Cramer, Koepke et al. [14]) and its use at the Europa-Universität Flensburg during the winter term 2020/2021.
Cited by 0Open access
The Proof Checkers Pacheck and Pastèque for the Practical Algebraic Calculus
Kaufmann, Daniela, Fleury, Mathias, Biere, Armin · reposiTUm (TU Wien) · 2020
Generating and checking proof certificates is important to increase the trust in automated reasoning tools.In recent years formal verification using computer algebra became more important and is heavily used in automated circuit verification.An existing proof format which covers algebraic reasoning and allows efficient proof checking is the practical algebraic calculus.In this paper we present two independent proof checkers PACHECK and PAST QUE.The checker PACHECK checks algebraic proofs more efficiently than PAST QUE, but the latter is formally verified using the proof assistant Isabelle/HOL.Furthermore, we introduce extension rules to simulate essential rewriting techniques required in practice.For efficiency we also make use of indices for existing polynomials and include deletion rules too.
Cited by 0Open access
An Approach to Link HOL and MDG for Hardware Verification
V. K. Pisini, S. Tahar, O. Ait-Mohamed et al. · 2007
In order to overcome the limitations of automated tools and the cumbersome proof process of interactive theorem proving, we adopt an hybrid approach for formal hardware verification by linking HOL and MDG. This approach uses the strengths of theorem proving (HOL) with its powerful mathematical tools such as induction and abstraction, and the advantages of automated tools (MDG) which support equivalence checking and model checking.
Cited by 0
Craig Interpolation, Proof-Theoretically via Nested Sequents
Alessio Guglielmi, Richard Mckinley, Willem Heijltjes et al. · 2013
Formal proofs are becoming increasingly important in a number of domains in computer science and mathematics. The topic of the colloquium is structural proof theory, broadly construed. Some examples of relevant topics: Structure Sequential and parallel structure in proofs; sharing and duplication of proofs; permu-tation of proof steps; canonical forms; focusing; polarities; graphical proof syntax; proof complexity; cut-eliminiation strategies; CERes Proof Checking Generating, transmitting, translating, and checking proof objects; universal proof lan-guages; proof certificates; proof compression; cut-introduction; certification of high-performance systems (SMT, resolution, etc.) Proof Search Automated and interactive proof search in constrained logics (linear, temporal, bunched, probabilistic, etc.); mixing deduction and computation; induction and co-induction; cyclic proofs; computational interpretations List of talks (in presentation order)
Cited by 0
Four Decades of Mizar
Adam Grabowski, Artur Korniłowicz, Adam Naumowicz · Journal of Automated Reasoning · 2015
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.
Cited by 131Open access
Data gathered in study: Moving Automated Compliance Checking to the Operational Phase of the Building Life-cycle: Analysis and Feasibility Study in the UK
Beach, Thomas, Yeung, Jonathan, Ghoroghi, Ali · Figshare · 2024
Construction regulations ensure a base level of safety of built assets. Formal checking of built assets against regulations happens at defined points in their life-cycle, and often there is limited automation involved. The movement towards digitising and automation elements of physical site inspections paves the way towards a whole life-cycle approach for compliance checking. This paper to which this research data refers explores whole life-cycle compliance checking. It does this by extending current checking processes which are common in the design and construction phases into the operational phase of the building life-cycle, proposing that the best way to achieve this is through automated/semi-automated data capture and analysis. To test the feasibility of this concept, this paper conducted proof of concept testing. One of these proof of concept demonstrators was a LIDAR scanner that could generate floor plan data and test it against regulations. This data-set contains the data collected in the testing of the demonstrator.
Cited by 0Open access
Abstract Testing, Proof and Automation. An Integrated Approach
Simon Burton, John Clark, John Mcdermid et al. · 2008
This paper presents a discussion on the complementary roles of testing and proof within automated software verification and validation processes. We demonstrate how a combination of the two approaches can lead to greater levels of automation and integrity. In particular we discuss the use of automated counter-example generation to support proof, and automated proof as a means of automating and checking test case generation. The high levels of automation are made possible by identifying repeating structures in the proofs, restricting the specification to a subset of an otherwise expressive formal notation and exploiting a general–purpose theorem proving tool with built-in constraint solvers. 1
Cited by 0
, O. Ait-Mohamed
V. K. Pisini, S. Tahar, P. Curzon et al. · 2015
In order to overcome the limitations of automated tools and the cumbersome proof process of interactive theorem proving, we adopt a hybrid approach for formal hardware verication which uses the strengths of the-orem proving (HOL) with powerful mathematical tools such as induction and abstraction, and the advantages of automated tools (MDG) which support equivalence checking and model checking. The MDG system is a decision diagram based verication tool, primarily de-signed for hardware verication. HOL is a theorem prover built on higher-order logic. 1
Cited by 0
Counting Constraints in Flat Array Fragments
Ghilardi, S., Pagani, E. · Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano) · 2016
We identify a fragment of Presburger arithmetic enriched with free function symbols and cardinality constraints for interpreted sets, which is amenable to automated analysis. We establish decidability and complexity results for such a fragment and we implement our algorithms. The experiments run in discharging proof obligations coming from invariant checking and bounded model-checking benchmarks show the practical feasibility of our decision procedure.
Cited by 0Open access
Another look at automated theorem-proving
Neal Koblitz · Journal of Mathematical Cryptology · 2007
I examine the use of automated theorem-proving for reductionist security arguments in cryptography and discuss three papers that purport to show the potential of computer-assisted proof-writing and proof-checking. I look at the proofs that the authors give to illustrate the “game-hopping” technique — for Full-Domain Hash signatures, ElGamal encryption, and Cramer-Shoup encryption — and ask whether there is evidence that automated theorem-proving can contribute anything of value to the security analysis of cryptographic protocols.
Cited by 10Open access
Terms for Efficient Proof Checking and Parsing
Michael Färber · 2023
Proofs automatically generated by interactive or automated theorem provers are often several orders of magnitude larger than proofs written by hand. This implies significant challenges for processing such proofs efficiently. It turns out that the data structures used to encode terms have a high impact on performance. This article proposes several term data structures; in particular, heterogeneous terms for proof checking that distinguish long- and short-lived terms, and abstract terms for proof parsing. Both term data structures are implemented in the proof checker Kontroli, enabling it to parse and check proofs both sequentially and concurrently without overhead. The evaluation on three datasets exported from interactive theorem provers shows that the new term data structures significantly improve proof checking performance.
Cited by 0Open access
Applications of Foundational Proof Certificates in theorem proving
Roberto Blanco Martínez · 2017
Formal trust in an abstract property, be it a mathematical result or a quality of the behavior of a computer program or a piece of hardware, is founded on the existence of a proof of its correctness. Many different kinds of proofs are written by mathematicians or generated by theorem provers, with the common problem of ascertaining whether those claimed proofs are themselves correct. The recently proposed Foundational Proof Certificate (FPC) framework harnesses advances in proof theory to define the semantics of proof formats, which can be verified by an independent and trusted proof checking kernel written in a logic programming language. This thesis extends initial results in certification of first-order proofs in several directions. It covers various essential logical axes grouped in meaningful combinations as they occur in practice: first,classical logic without fixed points and proofs generated by automated theorem provers; later, intuitionistic logic with fixed points and equality as logical connectives and proofs generated by proof assistants. The role of proof certificates is no longer limited to representing complete proofs to enable independent checking, but is extended to model proof transformations where details can be added to or subtracted from a certificate. These transformations yield operationally simpler certificates, around which increasingly trustworthy and performant proof checkers are constructed. Another new role of proof certificates is writing high-level proof outlines, which can be used to represent standard proof patterns as written by mathematicians, as well as automated techniques like property-based testing. We apply these developments to fully certify results produced by two families of standard automated theorem provers: resolution- and satisfiability-based. Another application is the design of programmable proof description languages for a proof assistant.
Cited by 1
Applications of Foundational Proof Certificates in theorem proving
Roberto Blanco Martínez · theses.fr (ABES) · 2017
Formal trust in an abstract property, be it a mathematical result or a quality of the behavior of a computer program or a piece of hardware, is founded on the existence of a proof of its correctness. Many different kinds of proofs are written by mathematicians or generated by theorem provers, with the common problem of ascertaining whether those claimed proofs are themselves correct. The recently proposed Foundational Proof Certificate (FPC) framework harnesses advances in proof theory to define the semantics of proof formats, which can be verified by an independent and trusted proof checking kernel written in a logic programming language. This thesis extends initial results in certification of first-order proofs in several directions. It covers various essential logical axes grouped in meaningful combinations as they occur in practice: first,classical logic without fixed points and proofs generated by automated theorem provers; later, intuitionistic logic with fixed points and equality as logical connectives and proofs generated by proof assistants. The role of proof certificates is no longer limited to representing complete proofs to enable independent checking, but is extended to model proof transformations where details can be added to or subtracted from a certificate. These transformations yield operationally simpler certificates, around which increasingly trustworthy and performant proof checkers are constructed. Another new role of proof certificates is writing high-level proof outlines, which can be used to represent standard proof patterns as written by mathematicians, as well as automated techniques like property-based testing. We apply these developments to fully certify results produced by two families of standard automated theorem provers: resolution- and satisfiability-based. Another application is the design of programmable proof description languages for a proof assistant.
Cited by 0Open access
Verification Column
Christel Baier · ACM SIGLOG News · 2023
Automated theorem proving can be seen as a logic-based approach for generating mathematical proofs mechanically. Important applications are hardware and software verification, general-purpose proof assistants and proof checking. Most state-of-the-art theorem provers rely on proof calculi for higher-order logic that incorporate superposition techniques or satisfiability modulo theories. The article by Bentkamp, Blanchette, Nummelin, Tourret and Waldmann provides an overview of the recently developed λ-superposition approach. It takes inspirations of classical superposition for first-order logic and extends it for higher-order constructs. This new approach paves the way for a new generation of automated theorem provers for powerful logics, still being impressively efficient.
Cited by 0
Certified interpolant generation for EUF
Andrew Reynolds, Cesare Tinelli, Liana Hadarean · 2011
Logical interpolants have found a wide array of applications in automated verification, including symbolic model checking and predicate abstraction. It is often critical to these applications that reported interpolants exhibit desired properties, correctness being first and foremost. In this paper, we introduce a method in which interpolants are computed by type inference within the trusted core of a proof checker. Interpolants produced this way from a proof of the joint unsatisfiability of two formulas are certified as correct by construction. We focus our attention to the quantifier-free theory of equality and uninterpreted functions (EUF) and present an interpolant generating proof calculus that can be encoded in the LFSC proof checking framework with limited reliance upon computational side conditions. Our experimental results show that our method generates certified interpolants with small overhead with respect to solving. 1
Cited by 0
Inspection of concurrent systems: combining tables, theorem proving and model checking
Vera Pantelic · University of Limerick Institutional Repository (University of Limerick) · 2006
A process for rigorous inspection of concurrent systems using tabular specification was developed and applied to the classic Readers/Writers concurrent program by Jin in [15]. The process involved rewriting the program into a table and then performing a manual "column-by-column" inspection for safety and clean completion properties. The key element in the process is obtaining an invariant strong enough to prove the properties of interest. This thesis presents partial automation of the proposed approach by combining theorem proving and model checking. Model checking is first used to validate a formal model of the system with a small, xed number of concurrent process instances. The verification of the system for an arbitrary number of processes is then performed using theorem proving together with model checking on the earlier model to quickly validate potential invariants before they are used in the formal proof. This method was used to check the manual proof of the Readers/Writers problem given in [15], discovering several random and one systematic mistake of the proof. Then, a new, significantly automated proof was performed.
Cited by 8Open access
A Sceptic's Approach to Combining HOL and Maple
J. Harrison, L. Théry, Inria Sophia Antipolis · 1997
. We contrast theorem provers and computer algebra systems, pointing out the advantages and disadvantages of each, and suggest a simple way to achieve a synthesis of some of the best features of both. Our method is based on the systematic separation of search for a solution and checking the solution, using a physical connection between systems. We describe the separation of proof search and checking in some detail, relating it to proof planning and to the complexity class NP, and discuss different ways of exploiting a physical link between systems. Finally, the method is illustrated by some concrete examples of computer algebra results proved formally in the HOL theorem prover with the aid of Maple: the evaluation of trigonometric integrals. Key words: Proof checking, automated theorem proving, computer algebra, complexity theory JEL codes: ? 1. Theorem provers vs. computer algebra systems Computer algebra systems (CASs) seem superficially similar to computer theorem provers: both a...
Cited by 0
A Shallow Embedding of Resolution and Superposition Proofs into the ??-Calculus Modulo
Guillaume Burel · HAL (Le Centre pour la Communication Scientifique Directe) · 2013
The ??-calculus modulo is a proof language that has been proposed as a proof standard for (re-)checking and interoperability. Resolution and superposition are proof-search methods that are used in state-of-the-art first-order automated theorem provers. We provide a shallow embedding of resolution and superposition proofs in the ??-calculus modulo, thus offering a way to check these proofs in a trusted setting, and to combine them with other proofs. We implement this embedding as a backend of the prover iProver Modulo.
Cited by 0