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Examining Fragments of the Quantified Propositional Calculus

2008
*
Journal of Symbolic Logic (JSL)
*

When restricted to proving formulas,

doi:10.2178/jsl/1230396765
fatcat:file2pnqujgvliksajephdbn34
*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. ...##
###
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]

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*. ...

##
###
On the universal and existential fragments of the μ-calculus

2006
*
Theoretical Computer Science
*

We therefore study

doi:10.1016/j.tcs.2005.11.015
fatcat:zulrihp2krb6bgjxfworh4upwm
*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. ...##
###
Sequent Calculi for the classical fragment of Bochvar and Halldén's Nonsense Logics

2013
*
Electronic Proceedings in Theoretical Computer Science
*

These calculi are obtained by restricting in an appropriate way

doi:10.4204/eptcs.113.12
fatcat:pwvpvsv7frbgbahxqnkazkib2q
*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. ...##
###
A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond

2019
*
Electronic Proceedings in Theoretical Computer Science
*

We consider a ZX-

doi:10.4204/eptcs.287.18
fatcat:nj5va3jysfcvnfuvicusni2qrq
*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. ...##
###
Polynomial Approximations for Model Checking
[chapter]

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. ...

##
###
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]

2017
*
arXiv
*
pre-print

axiom that cannot be derived from

arXiv:1702.01945v2
fatcat:yj7ep6hjpvb2bb2g6ix3iz3eny
*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. ...##
###
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
[article]

2018
*
arXiv
*
pre-print

We prove

arXiv:1705.11151v2
fatcat:qj3fahvg5ncwrndq4awlohw6ie
*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. ...##
###
Specialization of antecedent negation loop-rule for a fragment of propositional intuitionistic logic sequent calculus

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. ...

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

2016
*
The Review of Symbolic Logic
*

Moreover, we show that Leibniz's

doi:10.1017/s1755020316000137
fatcat:wulqblhirnacxpgi3iw4dfizde
*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 ...##
###
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

2018
*
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '18
*

All diagrams were written with

doi:10.1145/3209108.3209131
dblp:conf/lics/JeandelPV18
fatcat:eoixhygq7nfsraj25qctolozeu
*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. ...##
###
An integration of model checking with automated proof checking
[chapter]

1995
*
Lecture Notes in Computer Science
*

We describe an approach for such an integration where a BDD-based model checker for

doi:10.1007/3-540-60045-0_42
fatcat:inwlv6nsyzdgzjbzrk3qhrrjri
*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*. ...##
###
ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T Quantum Mechanics

2017
*
International Symposium on Mathematical Foundations of Computer Science
*

We exhibit another new simple axiom that cannot be derived from

doi:10.4230/lipics.mfcs.2017.11
dblp:conf/mfcs/JeandelPVW17
fatcat:cx3hip6azvcvxe4xsh2nrtd7oi
*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. ...
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