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Multi-focused Proofs with Different Polarity Assignments

Elaine Pimentel, Vivek Nigam, João Neto
2016 Electronical Notes in Theoretical Computer Science  
In this work, we will reason on how a given focused proof, where atoms are assigned with some polarity, can be transformed into another focused proof, where the polarity assignment to atoms is changed.  ...  polarity into another focused proof where the same atom is assigned negative polarity and viceversa.  ...  atom is assigned with positive polarity into another focused proof where the same atom is assigned negative polarity and vice-versa.  ... 
doi:10.1016/j.entcs.2016.06.011 fatcat:hlxstnqpdrecdlahq6rzyhwnm4

Imogen: Focusing the Polarized Inverse Method for Intuitionistic Propositional Logic [chapter]

Sean McLaughlin, Frank Pfenning
2008 Lecture Notes in Computer Science  
In this paper we describe Imogen, a theorem prover for intuitionistic propositional logic using the focused inverse method.  ...  We represent fine-grained control of the search behavior by polarizing the input formula.  ...  Different assignments of polarities can yield dramatically different performance.  ... 
doi:10.1007/978-3-540-89439-1_12 fatcat:qddjqbvoh5czjolu5lzqwaahse

A Framework for Proof Systems

Vivek Nigam, Dale Miller
2010 Journal of automated reasoning  
In particular, we show that by providing different polarizations within a focused proof system for linear logic, one can account for natural deduction (normal and non-normal), sequent proofs (with and  ...  Armed with just a few, simple variations to the linear logic encodings, more proof systems can be accommodated, including proof system using generalized elimination and generalized introduction rules.  ...  This polarity assignment may differ between different proof system encodings.  ... 
doi:10.1007/s10817-010-9182-1 fatcat:6e2wtfstabglncb5zgxqe6ccvy

Focusing Strategies in the Sequent Calculus of Synthetic Connectives [chapter]

Kaustuv Chaudhuri
2008 Lecture Notes in Computer Science  
maximally multi-focused proofs.  ...  Different focusing strategies arise from different polarisations and arrangements of synthetic inference rules, which are shown to be complete by synthetic rule permutations.  ...  cut-elimination with a priority assignment preserves maximality.  ... 
doi:10.1007/978-3-540-89439-1_33 fatcat:l2mch27bffflfizdp36axdcubq

Dual-context Modal Logic as Left Adjoint of Fitch-style Modal Logic

Yoshihiko Kakutani, Yuito Murase, Yuichi Nishiwaki
2019 Journal of Information Processing  
Moreover, the adjunction is refined with polarity and the adjoint calculus is extended to polarized logic.  ...  In order to show the statement, we embed both the two systems, which are refined with levels, into the adjoint calculus that equips an adjunction a priori.  ...  Proof. For ( 1 ), similarly to Levy's proof, perform induction primarily on M ⇓ V and secondarily on M → n M . The main difference with the original proof is the case of ,P.  ... 
doi:10.2197/ipsjjip.27.77 fatcat:k5zts37c35deffthvdtvrs3jhe

A general proof certification framework for modal logic

2019 Mathematical Structures in Computer Science  
By relying on that idea and by using a classical focused sequent calculus as a kernel, we propose here a general framework for checking modal proofs.  ...  One of the main issues in proof certification is that different theorem provers, even when designed for the same logic, tend to use different proof formalisms and produce outputs in different formats.  ...  Formulas in LKF can have either positive or negative polarity and are constructed from atomic formulas, whose polarity has to be assigned, and from logical connectives whose polarity is pre-assigned.  ... 
doi:10.1017/s0960129518000440 fatcat:e4kaw5r4hvel7mnsfdspppzxgy

Multi-Focusing on Extensional Rewriting with Sums

Gabriel Scherer, Marc Herbstritt
2015 International Conference on Typed Lambda Calculus and Applications  
It relies on maximally multi-focused proofs, a notion of canonical derivations introduced for linear logic.  ...  We propose a logical justification for the rewriting-based equivalence procedure for simply-typed lambda-terms with sums of Lindley [Lin07].  ...  Acknowledgements The author thanks his advisor, Didier Rémy, for his directions and advice, and for the freedom of making an overly long detour through the proof search literature; Alexis Saurin for his  ... 
doi:10.4230/lipics.tlca.2015.317 dblp:conf/tlca/Scherer15 fatcat:epwdawnvgjhxpojczyrb56cqq4

Lambek-Grishin Calculus: Focusing, Display and Full Polarization [article]

Giuseppe Greco, Valentin D. Richard, Michael Moortgat, Apostolos Tzimoulis
2020 arXiv   pre-print
We introduce a novel focused display calculus fD.LG and a fully polarized algebraic semantics FP.LG for Lambek-Grishin logic by generalizing the theory of multi-type calculi and their algebraic semantics  ...  Focused cut-free proofs exhibit a special normal form that is used for defining identity of sequent calculi proofs.  ...  polarity (cfr. derivation (35) in [27] where the polarity assignment is called 'bias').  ... 
arXiv:2011.02895v1 fatcat:vw7hdkt73rc2bj5aahqnwc4ir4

P-band Polarimetric Ice Sounder: Concept and First Results

Jorgen Dall, Carlos Cilla Hernandez, Steen Savstrup Kristensen, Viktor Krozer, Anders Kusk, Jens Vidkjoer, Jan Balling, Niels Skou, Sten Schmidl Sobjerg, Erik Lintz Christensen
2008 IGARSS 2008 - 2008 IEEE International Geoscience and Remote Sensing Symposium  
In May 2008, a proof-of-concept campaign was organized in Greenland, where data were acquired over the ice sheet.  ...  ESA has assigned the Technical University of Denmark to develop an airborne P-band ice sounding radar demonstrator. 1 The intention is to obtain a better understanding of the electromagnetic properties  ...  In the along track direction, data have been focused with the PDP algorithm [11] . CONCLUSIONS DTU has built and tested an airborne P-band ice sounder for ESA.  ... 
doi:10.1109/igarss.2008.4779766 dblp:conf/igarss/DallHKKKVBSSC08 fatcat:3lo5jvsdvjeefkj4hbuuf4ddl4

Algorithmic specifications in linear logic with subexponentials

Vivek Nigam, Dale Miller
2009 Proceedings of the 11th ACM SIGPLAN conference on Principles and practice of declarative programming - PPDP '09  
We also give some general conditions for when a focused proof step can be executed in constant time.  ...  To illustrate this new expressiveness, we show that focused proof search can be precisely linked to a simple algorithmic specification language that contains while-loops, conditionals, and insertion into  ...  Remember that atoms, such as l(m), are assigned with negative polarity and hence, l(m) ⊥ is assigned with positive polarity.  ... 
doi:10.1145/1599410.1599427 dblp:conf/ppdp/NigamM09 fatcat:2xzv7s5wlzh4polnubupd6divq

Focusing Over Optical Fiber Using Time Reversal

Molly Piels, Edson Porto Da Silva, Jose Estaran, Robert Borkowski, Idelfonso Tafur Monroy, Darko Zibar
2015 IEEE Photonics Technology Letters  
Focusing using a two-transmitter array and 44 km of singlemode fiber is demonstrated experimentally for 3 GBd QPSK signals with a bit error rate below the forward error correction limit.  ...  Simulations were verified with proof-of-principle experiments over standard single mode fiber, and effective focusing was demonstrated.  ...  Time-reversal-based focusing can also be used to send different data to multiple users.  ... 
doi:10.1109/lpt.2014.2387092 fatcat:mbivf4h3qjfbrllbc32v5f2s4e

Focused and Synthetic Nested Sequents [chapter]

Kaustuv Chaudhuri, Sonia Marin, Lutz Straßburger
2016 Lecture Notes in Computer Science  
Focusing is a general technique for transforming a sequent proof system into one with a syntactic separation of non-deterministic choices without sacrificing completeness.  ...  This not only improves proof search, but also has the representational benefit of distilling sequent proofs into synthetic normal forms.  ...  A polarized nested sequent is the same as in the non-focused setting, with the difference that all formulas are polarized.  ... 
doi:10.1007/978-3-662-49630-5_23 fatcat:rsrxkdvlqrbe5nvqgnjnhu6q3u

Polarised Intermediate Representation of Lambda Calculus with Sums

Guillaume Munch-Maccagnoni, Gabriel Scherer
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
The theory of the -calculus with extensional sums is more complex than with only pairs and functions.  ...  A structure of alternating phases naturally emerges through the study of normal forms, offering a reconstruction of focusing. Considering further purity assumption, we obtain maximal multifocusing.  ...  One way to do this is to introduce an explicit multi-focusing syntax that performs a series of independent plays simultaneously on different variables and optionally on ⋆; this corresponds to a multilet  ... 
doi:10.1109/lics.2015.22 dblp:conf/lics/Munch-Maccagnoni15 fatcat:rsuvx3pj35ec3of7xax5nzuv4e

Proof Certificates for Equality Reasoning

Zakaria Chihani, Dale Miller
2016 Electronical Notes in Theoretical Computer Science  
Our proof checker's design is based on the implementation of focused proof search and on making calls to (user-supplied) clerks and experts predicates that are tied to the two phases found in focused proofs  ...  We shall illustrate here how it is possible to check various equality-based proof systems with a programmable proof checker (the kernel checker) for first-order logic.  ...  If a formula is non-atomic, the assignment of polarity is negative if the right introduction rule for its top-level logical connective is invertible, otherwise that formula is positive (the polarity of  ... 
doi:10.1016/j.entcs.2016.06.007 fatcat:kl3wk3vzoffsllajv2b72ub5bq

On the unity of duality

Noam Zeilberger
2008 Annals of Pure and Applied Logic  
In this paper we develop a proof-theoretic framework for analyzing the interaction of types with evaluation order, based on the notion of polarity.  ...  We revisit Dummett's analysis using the tools of Martin-Löf's judgmental method, and then show how to extend it to a unified polarized logic, with Girard's "shift" connectives acting as intermediaries.  ...  I have also greatly benefited from discussions with Steve Awodey, Peter Hancock, Bob Harper, Neel Krishnaswami, Peter Lee, William Lovas, Paul-André Melliès, Jason Reed, Rob Simmons, and Kevin Watkins.  ... 
doi:10.1016/j.apal.2008.01.001 fatcat:6eq3ecmas5etdl3mmzrfgtqdt4
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