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A formal framework for specifying sequent calculus proof systems
, 2012
"... Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequ ..."
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Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequences of the specified sequent calculus proof systems. In particular, derivability of an inference rule from a set of inference rules can be decided by bounded (linear) logic programming search on the specified rules. We also present two simple and decidable conditions that guarantee that the cut rule and non-atomic initial rules can be eliminated.
Using LJF as a Framework for Proof Systems
, 2009
"... In this work we show how to use the focused intuitionistic logic system LJF as a framework for encoding several different intuitionistic and classical proof systems. The proof systems are encoded in a strong level of adequacy, namely the level of (open) derivations. Furthermore we show how to prove ..."
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In this work we show how to use the focused intuitionistic logic system LJF as a framework for encoding several different intuitionistic and classical proof systems. The proof systems are encoded in a strong level of adequacy, namely the level of (open) derivations. Furthermore we show how to prove relative completeness between the different systems. By relative completeness we mean that the systems prove the same formulas. The proofs of relative completeness exploit the encodings to give, in most cases, fairly simple proofs. This work is heavily based on the recent work by Nigam and Miller, which uses the focused linear logic system LLF to encode the same proof systems as we do. Our work shows that the features of linear logic are not needed for the full adequacy result, and furthermore we show that even though encoding in LLF is more generic and streamlined, the encoding in LJF sometimes gives simpler, more natural encodings and easier proofs.
Communicating and trusting proofs: The case for broad spectrum proof certificates. Available from author’s website
, 2011
"... Abstract. Proofs, both formal and informal, are documents that are intended to circulate within societies of humans and machines distributed across time and space in order to provide trust. Such trust might lead one mathematician to accept a certain statement as true or it might help convince a cons ..."
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Abstract. Proofs, both formal and informal, are documents that are intended to circulate within societies of humans and machines distributed across time and space in order to provide trust. Such trust might lead one mathematician to accept a certain statement as true or it might help convince a consumer that a certain software system is secure. Using this general characterization of proofs, we examine a range of perspectives about proofs and their roles within mathematics and computer science that often appear contradictory. We then consider the possibility of defining a broad spectrum proof certificate format that is intended as a universal language for communicating formal proofs among computational logic systems. We identify four desiderata for such proof certificates: they must be (i) checkable by simple proof checkers, (ii) flexible enough that existing provers can conveniently produce such certificates from their internal evidence of proof, (iii) directly related to proof formalisms used within the structural proof theory literature, and (iv) permit certificates to elide some proof information with the expectation that a proof checker can reconstruct the missing information using bounded and structured proof search. We consider various consequences of these desiderata, including how they can mix computation and deduction and what they mean for the establishment of marketplaces and libraries of proofs. In a companion paper we proposal a specific framework for achieving all four of these desiderata. 1
3.5. Deep Inference and Categorical Axiomatizations 4 3.6. Proof Nets and Combinatorial Characterization of Proofs 4 3.7. A Systematic Approach to Cut Elimination 5
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Programs, Verification and Proofs
"... Proof search and reasoning with logic specifications IN COLLABORATION WITH: Laboratoire d’informatique de l’école polytechnique (LIX) ..."
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Proof search and reasoning with logic specifications IN COLLABORATION WITH: Laboratoire d’informatique de l’école polytechnique (LIX)
Communicating and trusting proofs: The case for foundational proof certificates
"... It is well recognized that proofs serve two different goals. On one hand, they can serve the didactic purpose of explaining why a theorem holds: that is, a proof has a message that is meant to describe the “why ” behind a theorem. On the other hand, proofs can serve as certificates of validity. In t ..."
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It is well recognized that proofs serve two different goals. On one hand, they can serve the didactic purpose of explaining why a theorem holds: that is, a proof has a message that is meant to describe the “why ” behind a theorem. On the other hand, proofs can serve as certificates of validity. In this case, once a certificate
Mathematical practice, crowdsourcing, and social machines
"... Abstract. The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowd-sourc ..."
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Abstract. The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowd-sourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. The Study of Mathematical Practice is an emerging interdisciplinary field which draws on philoso-phy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice- for example the community question-answering system mathoverflow contains around 40,000 mathe-matical conversations, and polymath collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of “soft ” aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a “social machine”, a new paradigm, identi-fied by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathe-matics, and to transform the reach, pace, and impact of mathematics research. 1
