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Examining Fragments of the Quantified Propositional Calculus

Steven Perron
2008 Journal of Symbolic Logic (JSL)  
When restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory .  ...  Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas.  ...  In [9] , Krajicek and Pudlak introduced the quantified propositional proof system G and its fragments. These fragments have close connections with bounded arithmetic and computational complexity.  ... 
doi:10.2178/jsl/1230396765 fatcat:file2pnqujgvliksajephdbn34

Page 228 of Computational Linguistics Vol. 25, Issue 2 [page]

1999 Computational Linguistics  
,An)> [An}~,...4er A.2 The Calculus A (a Fragment of N-ILL) (proofs) In this section, we give the proofs for the various propositions presented in the paper.  ...  In Figure 19, we give the one-sided sequent calculus for the multiplicative fragment of noncommu- tative linear logic (N-LL), and in Figure 3 in Section 3, the two-sided sequent calculus for the multiplicative  ... 

First-Order Game Logic and Modal Mu-Calculus [article]

Noah Abou El Wafa, André Platzer
2022 arXiv   pre-print
The differential mu-calculus is complete with respect to first-order fixpoint logic and differential game logic is complete with respect to its ODE-free fragment.  ...  This paper investigates first-order game logic and first-order modal mu-calculus, which extend their propositional modal logic counterparts with first-order modalities of interpreted effects such as variable  ...  structures, -calculus embeds into game logic, which embeds into the two-propositional-variable fragment of the modal -calculus.  ... 
arXiv:2201.10012v2 fatcat:u3en3eie7fewdm66ueavq7b4sy

On the universal and existential fragments of the μ-calculus

Thomas A. Henzinger, Orna Kupferman, Rupak Majumdar
2006 Theoretical Computer Science  
We therefore study the problems of satisfiability, validity, model checking, and implication for the universal and existential fragments of the -calculus, in which only one branching mode is allowed.  ...  We also show the corresponding results for the alternation-free fragment of the -calculus, where no alternations between least and greatest fixed points are allowed.  ...  Acknowledgements We thank the anonymous reviewers for their detailed comments on the organization and on the proofs.  ... 
doi:10.1016/j.tcs.2005.11.015 fatcat:zulrihp2krb6bgjxfworh4upwm

Sequent Calculi for the classical fragment of Bochvar and Halldén's Nonsense Logics

Marcelo E. Coniglio, María I. Corbalán
2013 Electronic Proceedings in Theoretical Computer Science  
These calculi are obtained by restricting in an appropriate way the application of the rules of a sequent calculus for classical propositional logic CPL.  ...  In this paper sequent calculi for the classical fragment (that is, the conjunction-disjunction-implication-negation fragment) of the nonsense logics B3, introduced by Bochvar, and H3, introduced by Halld  ...  Acknowledgements: We would like to thank the anonymous referees for their extremely useful comments on an earlier draft, which have helped to improve the paper.  ... 
doi:10.4204/eptcs.113.12 fatcat:pwvpvsv7frbgbahxqnkazkib2q

A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

Renaud Vilmart
2019 Electronic Proceedings in Theoretical Computer Science  
We consider a ZX-calculus augmented with triangle nodes which is well-suited to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics.  ...  We precisely show the form of the matrices it represents, and we provide an axiomatisation which makes the language complete for the Toffoli-Hadamard quantum mechanics.  ...  All diagrams were written with the help of TikZit. The author would like to thank Emmanuel Jeandel and Simon Perdrix for fruitful discussions and suggestions.  ... 
doi:10.4204/eptcs.287.18 fatcat:nj5va3jysfcvnfuvicusni2qrq

Polynomial Approximations for Model Checking [chapter]

Nikolai V. Shilov, Natalya O. Garanina
2004 Lecture Notes in Computer Science  
The µ-Calculus of D. Kozen (1983) is a very powerful propositional program logic with fixpoints. It is widely used for specification and verification.  ...  A.Emerson, C.Jutla, and P.Sistla studied (1993) the first fragment of the µ-Calculus that permits arbitrary nesting and alternations of fixpoints, and polynomial model checking in finite state systems.  ...  But all proofs and the full-body version of the paper are available upon request.  ... 
doi:10.1007/978-3-540-39866-0_39 fatcat:rgvcbox3mvhzto5djc7pacgkiy

Page 2442 of Mathematical Reviews Vol. , Issue 89E [page]

1989 Mathematical Reviews  
A propositional fragment F is to be understood as a set of formulas containing all propositional variables and closed under substitution of formulas from F.  ...  Summary: “A reversible syntactic and semantic translation between m-valued propositional calculus on a superintuitionistic base and intuitionistic propositional calculus is constructed.  ... 

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics [article]

Emmanuel Jeandel, Renaud Vilmart
2017 arXiv   pre-print
axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics.  ...  We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.  ...  Since the diagrams of equation (E) are in the π 4 -fragment, it implies that the π 4 -fragment of the ZX-calculus is not complete. Remark 1.  ... 
arXiv:1702.01945v2 fatcat:yj7ep6hjpvb2bb2g6ix3iz3eny

A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics [article]

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart
2018 arXiv   pre-print
We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices.  ...  We also prove that the Clifford+T fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.  ...  All diagrams were written with the help of TikZit.  ... 
arXiv:1705.11151v2 fatcat:qj3fahvg5ncwrndq4awlohw6ie

Specialization of antecedent negation loop-rule for a fragment of propositional intuitionistic logic sequent calculus

Romas Alonderis
2009 Lietuvos matematikos rinkinys  
The paper deals with specialization of the antecedent negation loop-rule for the negative implication free fragment of the propositional intuitionistic logic.  ...  Introduction We investigate a fragment of the propositional intuitionistic logic without negative implication.  ...  Calculus LJ * 0 The calculus LJ * 0 is a variant of the multisuccedent intuitionistic propositional Gentzen-like sequent calculus. It is defined as follows: 1. Axioms: , E → E, . 2.  ... 
doi:10.15388/lmr.2009.42 fatcat:wlz4hrpr6zay5gymlikfkxhhpy

THE LOGIC OF LEIBNIZ'S GENERALES INQUISITIONES DE ANALYSI NOTIONUM ET VERITATUM

MARKO MALINK, ANUBAV VASUDEVAN
2016 The Review of Symbolic Logic  
Moreover, we show that Leibniz's calculus can reproduce all the laws of classical propositional logic, thus allowing Leibniz to achieve his goal of reducing propositional reasoning to algebraic reasoning  ...  The present paper provides a systematic reconstruction of the calculus developed by Leibniz in theGenerales Inquisitiones.  ...  The year 1686 marked an exceptionally fruitful period in the intellectual life of Gottfried  ... 
doi:10.1017/s1755020316000137 fatcat:wulqblhirnacxpgi3iw4dfizde

A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart
2018 Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18  
All diagrams were written with the help of TikZit.  ...  Acknowledgements The authors acknowledge support from the projects ANR-17-CE25-0009 SoftQPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex, and STIC-AmSud 16-STIC-05 FoQCoSS.  ...  Completeness of the π 4 -fragment of the ZX-Calculus To finish the proof it remains to compose the two interpretations: Proposition 8.  ... 
doi:10.1145/3209108.3209131 dblp:conf/lics/JeandelPV18 fatcat:eoixhygq7nfsraj25qctolozeu

An integration of model checking with automated proof checking [chapter]

S. Rajan, N. Shankar, M. K. Srivas
1995 Lecture Notes in Computer Science  
We describe an approach for such an integration where a BDD-based model checker for the propositional mu-calculus has been used as a decision procedure within the framework of the PVS proof checker.  ...  If the expressiveness is constrained, there are useful logic fragments for which efficient decision procedures can be found.  ...  is provided from the mu-cMculus fragment of the PVS language to propositional mu-calculus.  ... 
doi:10.1007/3-540-60045-0_42 fatcat:inwlv6nsyzdgzjbzrk3qhrrjri

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

Emmanuel Jeandel, Simon Perdrix, Renaud Vilmart, Quanlong Wang, Marc Herbstritt
2017 International Symposium on Mathematical Foundations of Computer Science  
We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics  ...  We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.  ...  Since the diagrams of equation (E) are in the π 4 -fragment, it implies that the π 4 -fragment of the ZX-calculus is not complete. Remark.  ... 
doi:10.4230/lipics.mfcs.2017.11 dblp:conf/mfcs/JeandelPVW17 fatcat:cx3hip6azvcvxe4xsh2nrtd7oi
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