Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term π-Institutions

George Voutsadakis

Bulletin of the Section of Logic (2016)

  • Volume: 45, Issue: 2
  • ISSN: 0138-0680

Abstract

top
Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the π-institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-F´erez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-F´erez pertaining to arbitrary multi-term π-institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools.

How to cite

top

George Voutsadakis. "Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term π-Institutions." Bulletin of the Section of Logic 45.2 (2016): null. <http://eudml.org/doc/295521>.

@article{GeorgeVoutsadakis2016,
abstract = {Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the π-institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-F´erez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-F´erez pertaining to arbitrary multi-term π-institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools.},
author = {George Voutsadakis},
journal = {Bulletin of the Section of Logic},
keywords = {Behavioral Equivalence; Hidden Logic; Multi-Sorted Logic; Multi-term π-Institutions; Interpretability; Deductive Equivalence},
language = {eng},
number = {2},
pages = {null},
title = {Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term π-Institutions},
url = {http://eudml.org/doc/295521},
volume = {45},
year = {2016},
}

TY - JOUR
AU - George Voutsadakis
TI - Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term π-Institutions
JO - Bulletin of the Section of Logic
PY - 2016
VL - 45
IS - 2
SP - null
AB - Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the π-institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-F´erez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-F´erez pertaining to arbitrary multi-term π-institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools.
LA - eng
KW - Behavioral Equivalence; Hidden Logic; Multi-Sorted Logic; Multi-term π-Institutions; Interpretability; Deductive Equivalence
UR - http://eudml.org/doc/295521
ER -

References

top
  1. [1] S. Babenyshev and M. A. Martins, Behavioral Equivalence of Hidden k-Logics: An Abstract Algebraic Approach, Journal of Applied Logic, Vol. 16 (2016), pp. 72–91. 
  2. [2] M. Barr and C. Wells, Category Theory for Computing Science, Third Edition, Les Publications CRM, Montréal, 1999. 
  3. [3] W. J. Blok and B. Jónsson, Equivalence of Consequence Operations, Studia Logica, Vol. 83, No. 1/3 (2006), pp. 91–110. 
  4. [4] W. J. Blok and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society, Vol. 77, No. 396 (1989). 
  5. [5] C. Caleiro, R. Gon calves and M. Martins, Behavioral Algebraization of Logics, Studia Logica, Vol. 91, No. 1 (2009), pp. 63–111. 
  6. [6] J. M. Font and T. Moraschini, M-Sets and the Representation Problem, Studia Logica, Vol. 103, No. 1 (2015), pp. 21–51. 
  7. [7] Galatos, N., and N. Galatos, J. Gil-Férez, Modules over Quantaloids: Applications to the Isomorphism Problem in Algebraic Logic and -institutions, Journal of Pure and Applied Algebra, Vol. 221, No. 1 (2017), pp. 1–24. 
  8. [8] N. Galatos and C. Tsinakis, Equivalence of Closure Operators: an Order-Theoretic and Categorical Perspective, The Journal of Symbolic Logic, Vol. 74, No. 3 (2009), pp. 780–810. 
  9. [9] J. Gil-Férez, Multi-term -Institutions and their Equivalence, Mathematical Logic Quarterly, Vol. 52, No. 5 (2006), pp. 505–526. 
  10. [10] M. A. Martins, Behavioral Reasoning in Generalized Hidden Logics, Ph.D. Thesis, Faculdade de Ciências, University of Lisbon, 2004. 
  11. [11] M. A. Martins and D. Pigozzi, Behavioural Reasoning for Conditional Equations, Mathematical Structures in Computer Science, Vol. 17 (2007), pp. 1075–1113. 
  12. [12] D. Sannella and A. Tarlecki, Foundations of Algebraic Specification and Formal Software Development, EATCS Monographs in Theoretical Computer Science, Springer 2012. 
  13. [13] G. Voutsadakis, Categorical Abstract Algebraic Logic: Equivalent Institutions, Studia Logica, Vol. 74 (2003), pp. 275–311. 
  14. [14] G. Voutsadakis, Categorical Abstract Algebraic Logic: Behavioral -Institutions, Studia Logica, Vol. 102, No. 3 (2014), pp. 617–646. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.