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Circular (Yet Sound) Proofs [chapter]

Albert Atserias, Massimo Lauria
2019 Lecture Notes in Computer Science  
We call such proofs circular. We show that, for all sets of standard inference rules, circular proofs are sound.  ...  Contrary to the case of circular resolution, for Frege we show that circular proofs can be converted into tree-like ones with at most polynomial overhead.  ...  We acknowledge the work of Jordi Coll who conducted experimental results for finding and visualizing actual circular resolution proofs of small instances of the sparse pigeonhole principle.  ... 
doi:10.1007/978-3-030-24258-9_1 fatcat:qzclgx5oyjgt7bviyim3koz4vu

Circular (Yet Sound) Proofs [article]

Albert Atserias, Massimo Lauria
2018 arXiv   pre-print
We call such proofs circular. We show that, for all sets of standard inference rules, circular proofs are sound.  ...  For Frege we show that circular proofs can be converted into tree-like ones with at most polynomial overhead.  ...  We acknowledge the work of Jordi Coll who conducted experimental results for finding and visualizing actual circular resolution proofs of small instances of the sparse pigeonhole principle.  ... 
arXiv:1802.05266v1 fatcat:h3wvw7tm3raslh3cgnpmdmdxay

Circular Proofs as Session-Typed Processes: A Local Validity Condition [article]

Farzaneh Derakhshan, Frank Pfenning
2022 arXiv   pre-print
corresponding to valid circular proofs as introduced by Fortier and Santocanale.  ...  Proof theory provides a foundation for studying and reasoning about programming languages, most directly based on the well-known Curry-Howard isomorphism between intuitionistic logic and the typed lambda-calculus  ...  are prepared to accept circular proofs, and (c) impose conditions under which recursively defined processes correspond to valid circular proofs.  ... 
arXiv:1908.01909v5 fatcat:uoj3dcjwcjfxfgb2eglbnk5qny

Local Validity for Circular Proofs in Linear Logic with Fixed Points

Rémi Nollet, Alexis Saurin, Christine Tasson, Michael Wagner
2018 Annual Conference for Computer Science Logic  
Among all representations of valid circular proofs, a new fragment is described, based on a stronger validity criterion.  ...  Circular (ie. non-wellfounded but regular) proofs have received increasing interest in recent years with the simultaneous development of their applications and meta-theory: infinitary proof theory is now  ...  Proposition 30 (Soundness of labelling). If π is an extended labelled circular representation then π is a circular representation of a valid µMALL ω proof. Proof. See proof in appendix C, p. 21.  ... 
doi:10.4230/lipics.csl.2018.35 dblp:conf/csl/NolletST18 fatcat:rni5ihtukngejmxd2f3co5opve

Extracting Proofs from Tabled Proof Search [chapter]

Dale Miller, Alwen Tiu
2013 Lecture Notes in Computer Science  
We design a proof system with tables and show that by managing tabled entries carefully, one would still be able to obtain a sound proof system.  ...  In general, tabling involves storing previously proved subgoals and reusing (instead of reproving) them in proof search.  ...  If this were admitted as a proof, then one can prove ⊥. Indeed, this kind of loop is forbidden in sound circular proof systems [4, 14] and is an example of a non-progressing loop.  ... 
doi:10.1007/978-3-319-03545-1_13 fatcat:2py5ovqc3zb2tksrojeodg77t4

Optimal Proximity Proofs [chapter]

Ioana Boureanu, Serge Vaudenay
2015 Lecture Notes in Computer Science  
Provably secure distance-bounding is a rising subject, yet an unsettled one; indeed, very few distance-bounding protocols, with formal security proofs, have been proposed.  ...  Matters like security, alongside with soundness, or added tolerance to noise do not always coexist in the (new) distance-bounding designs.  ...  Some open challenges remain: 1. identify an optimal and sound protocol for num c = num r = 2; 2. study the optimality of soundness; 3. implement these protocols.  ... 
doi:10.1007/978-3-319-16745-9_10 fatcat:vjhim54hgbakhagkvk35lt2ar4

Constructive truth and circularity [article]

Nik Weaver
2010 arXiv   pre-print
to be sound.  ...  But explicitly asserting A ↔ T (A) is not acceptable because it implies a global assurance that our reasoning is sound which we are not able to supply and which is in fact viciously circular.  ... 
arXiv:0905.1681v3 fatcat:fv5ilxtgt5fizoljom3qxnuo6e

Investigating Students' Proof Reasoning: Analyzing Students' Oral Proof Explanations and their Written Proofs in High School Geometry

Michael L. Winer, Michael T. Battista
2022 IEJME-Mathematics Education  
We found that the majority of the students utilized sound reasoning in their oral explanations, but struggled to capture that reasoning in their written two-column proofs.  ...  Constructing formal geometry proofs in is an important topic in the mathematics curriculum. But students' difficulties with proof are well documented.  ...  their proof of that theorem (direct circular), or they justified a deduction using a theorem which is true, but had not yet been proven in the axiomatic system (indirect circular).  ... 
doi:10.29333/iejme/11713 fatcat:33nahvorqzbozbsdoo4364pjvi

Proofs, pictures, and Euclid

John Mumma
2009 Synthese  
However, if part of the proof is itself geometric, then it looks like there is some kind of circularity.  ...  Although these abstract diagrams may allow for shorter proofs than Hilbert's formal system, and may also be sound and complete, it's not totally clear in what sense they explicate Euclid's proofs if they're  ... 
doi:10.1007/s11229-009-9509-9 fatcat:xc6t6bsv35c2heglfeo5fki2ly

Circular Saws

1852 Scientific American  
For proof of this your" Nameless" Having stated so much over my name , I Scientific American , in relation to Circular friend is referred to the Report of a Selecb now take leave of ,; Nameless , " intending  ...  Copeland , Jame� Bogardus , and and sensitive, that she suffered by the sound file in the same way as described above. I Mr. James Stone (who was Gurney's machi-and noise in common conversation!  ... 
doi:10.1038/scientificamerican01241852-146b fatcat:4bstlgh3jna65pe3uygm2pyk5q

Substructural Proofs as Automata [chapter]

Henry DeYoung, Frank Pfenning
2016 Lecture Notes in Computer Science  
We present subsingleton logic as a very small fragment of linear logic containing only , 1, least fixed points and allowing circular proofs.  ...  We show that cut-free proofs in this logic are in a Curry-Howard correspondence with subsequential finite state transducers.  ...  Already, the proof term P sounds vaguely like an SFT. The logic's inference rules now become typing rules for proof terms.  ... 
doi:10.1007/978-3-319-47958-3_1 fatcat:qf3q4jms4zaalaezcsinkkw6xq

Frege's Proof of Referentiality

�ystein Linnebo
2004 Notre Dame Journal of Formal Logic  
I argue that Frege's proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle.  ...  Although Frege's proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which it may successfully be applied  ...  What's good is that the proof fails only at a relatively late stage and that Frege's contextual account of reference and his procedure for extending the sphere of denoting names are logically sound and  ... 
doi:10.1305/ndjfl/1095386645 fatcat:6s3gsrxc5rbu5owywijvi54kv4

Circular Saws

1856 Scientific American  
Lucie's Sound, on account of the frost.  ...  It is therefore advisable to connect the teeth by a circular line whose radius should be about one-fourth or one-third the length of the tooth.  ... 
doi:10.1038/scientificamerican03151856-211a fatcat:6ai2sid4vbcjvdjexgytdkqs3u

Non-Classical Circular Definitions

Shawn Standefer
2017 The Australasian Journal of Logic  
I present systems of circular denitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them.  ...  Circular denitions have primarily been studied in revision theory in the classical scheme.  ...  Proof. The proof is by induction on the construction of the tableau. The classical rules are immediate from the standard soundness proofs.  ... 
doi:10.26686/ajl.v14i1.4030 fatcat:nltcjz57jfacfklwnclvjrxpii

Towards a Better Understanding of (Partial Weighted) MaxSAT Proof Systems [article]

Javier Larrosa, Emma Rollon
2021 arXiv   pre-print
We show that each addition makes the resulting proof system stronger, with the virtual rule capturing the recently proposed concept of circular proof.  ...  In this paper we want to address MaxSAT from the more formal point of view of Proof Complexity. With that aim we start providing basic definitions and proving some basic results.  ...  Relation between ResSV and circular proofs Now we show that the MaxSAT ResSV proof system is an extension of circular proofs from SAT to MaxSAT.  ... 
arXiv:2003.02718v3 fatcat:ovm3s3sui5fddiqpzfcmm4jrt4
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