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NP Reasoning in the Monotone μ-Calculus
[article]

2020
*
arXiv
*
pre-print

*monotone*

*μ*-

*calculus*with

*the*universal modality – contains both concurrent propositional dynamic logic (CPDL) and

*the*alternation-free fragment of game logic as fragments. ... Satisfiability checking for

*monotone*modal logic is known to be (only)

*NP*-complete. ...

*The*

*Monotone*

*µ*-

*Calculus*We proceed to recall

*the*syntax and semantics of

*the*

*monotone*

*µ*-

*calculus*. ...

##
###
NP Reasoning in the Monotone $μ$-Calculus
[article]

2020

*monotone*$

*μ*$-

*calculus*with

*the*universal modality -- contains both concurrent propositional dynamic logic (CPDL) and

*the*alternation-free fragment of game logic as fragments. ... Satisfiability checking for

*monotone*modal logic is known to be (only)

*NP*-complete. ...

*The*satisfiability checking problem for

*the*alternation-free

*monotone*

*µ*-

*calculus*

*the*universal modality is

*in*

*NP*(hence

*NP*-complete). Proof. ...

##
###
NP Reasoning in the Monotone $$\mu $$-Calculus
[chapter]

2020
*
Lecture Notes in Computer Science
*

We recall basics on fixpoints and games

doi:10.1007/978-3-030-51074-9_28
fatcat:sb3lc777s5ba5onnylztv4esne
*in*Sect. 2, and*the*syntax and semantics of*the**monotone**μ*-*calculus**in*Sect. 3. We discuss a key technical tool, formula tracking,*in*Sect. 4. ... We adapt*the*standard tableaux system to*the**monotone**μ*-*calculus**in*Sect. 5.*In*Sect. 6, we establish our main results using a game characterization of satisfiability. ...*μ*-*calculus*with*the*universal modality [8] is only*NP*-complete. ...##
###
A 'Natural Logic' inference system using the Lambek calculus

2006
*
Journal of Logic, Language and Information
*

*The*Lambek-based system we propose extends

*the*system by Fyodorov et al. [3] , which is based on

*the*Ajdukiewicz/Bar-Hillel (AB)

*calculus*([6]). ... Basing

*the*system on

*the*Lambek

*calculus*leads to problems with non-normalized proof terms, which are treated by using normalization axioms. ... If ψ = λx.

*µ*then φ is positive (negative)

*in*ψ if φ is positive (negative)

*in*

*µ*.Fact 7.2 [11] If x is positive (negative)

*in*φ then λx.φ denotes an upward (downward)

*monotone*function.Dynamic marking is ...

##
###
Continuation semantics for the Lambek–Grishin calculus

2010
*
Information and Computation
*

We present a Curry-Howard interpretation for LG(/, \, , ) derivations, based on Curien and Herbelin's λµ

doi:10.1016/j.ic.2009.11.005
fatcat:msdvi2jid5f5pkrg7uzvblsnti
*µ**calculus*[10]. ...*The*absence of structural context*in**the*succedent makes*the*analysis of a number of phenomena*in*natural language semantics problematic. ... To adapt*the*λµ*µ**calculus*to λLG, we have to restrict and refine it. ...##
###
Symmetric categorial grammar: residuation and Galois connections
[article]

2010
*
arXiv
*
pre-print

*The*Lambek-Grishin

*calculus*is a symmetric extension of

*the*Lambek

*calculus*:

*in*addition to

*the*residuated family of product, left and right division operations of Lambek's original

*calculus*, one also ... Whereas

*the*(dual) residuated operations are

*monotone*,

*the*Galois connected operations (and their duals) are antitone. ... Acknowledgements My interest

*in*

*the*four unary negation operations was raised by a comment Jim Lambek made on an earlier presentation of

*the*symmetric

*calculus*. ...

##
###
A Frobenius Algebraic Analysis for Parasitic Gaps
[article]

2020
*
arXiv
*
pre-print

Our analysis is formulated

arXiv:2005.05639v2
fatcat:mvc26s5z7rhrnbnwiwjdq5aanq
*in*terms of Lambek*calculus*extended with structural control modalities. ... Existing categorial analyses, both*in**the*typelogical and*in**the*combinatory traditions, rely on explicit forms of syntactic copying. ...*The*∆ and*µ*morphisms satisfy*the*Frobenius condition given below (*µ*⊗ 1 X ) • (1 X ⊗ ∆) = ∆ •*µ*= (1 X ⊗*µ*) • (∆ ⊗ 1 X ) Informally,*the*comultiplication ∆ decomposes*the*information contained*in*one ...##
###
THE LOGIC OF LEIBNIZ'S GENERALES INQUISITIONES DE ANALYSI NOTIONUM ET VERITATUM

2016
*
The Review of Symbolic Logic
*

*The*present paper provides a systematic reconstruction of

*the*

*calculus*developed by Leibniz

*in*theGenerales Inquisitiones. ... Leibniz's central aim

*in*this treatise is to develop a symbolic

*calculus*of terms that is capable of underwriting all valid modes of syllogistic and propositional

*reasoning*. ...

*The*year 1686 marked an exceptionally fruitful period

*in*

*the*intellectual life of Gottfried ...

##
###
Polarized Montagovian Semantics for the Lambek-Grishin calculus
[article]

2011
*
arXiv
*
pre-print

*In*this paper, we adapt Girard's polarity-sensitive double negation embedding for classical logic to extract a compositional Montagovian semantics from a display

*calculus*for focused proof search

*in*LG ... Grishin proposed enriching

*the*Lambek

*calculus*with multiplicative disjunction (par) and coresiduals. ... ) ⌈·⌉ (CBV) p ¬p p A/B, B\A ¬⌊B⌋ ⌊A⌋ ¬(⌈B⌉ ¬⌈A⌉) B A, A B ¬(⌊B⌋ ¬⌊A⌋) ¬⌈B⌉ ⌈A⌉ B&M code their derivations inside a variation of Curien and Herbelin'sλµ

*µ*

*calculus*. ...

##
###
Tractable Probabilistic mu-Calculus That Expresses Probabilistic Temporal Logics

2015
*
Symposium on Theoretical Aspects of Computer Science
*

Its game semantics is very similar to

doi:10.4230/lipics.stacs.2015.211
dblp:conf/stacs/CastroKP15
fatcat:whgrktaxbrf3lir3tqrunw7ur4
*the*game semantics of*the*classical*µ*-*calculus*(using parity obligation games instead of parity games). ... An important feature of this extended version of pctl is that its model checking is only exponential w.r.t.*the*alternation depth of fixed points, one of*the*main characteristics of Kozen's*µ*-*calculus*. ...*In*Section 5 we show that a well-known "hard" problem*in**NP*∩co-*NP*can be reduced to model checking formulas of*µ*p -*calculus*with only one fixpoint operator. ...##
###
Three notes on the complexity of model checking fixpoint logic with chop

2007
*
RAIRO - Theoretical Informatics and Applications
*

This paper analyses

doi:10.1051/ita:2007011
fatcat:qj7kuj4r6rcslbp6yoewgk5huq
*the*complexity of model checking Fixpoint Logic with Chop -an extension of*the*modal*µ*-*calculus*with a sequential composition operator. ... This is already*the*case for its alternation-free fragment.*The*expression complexity of FLC is trivially P-hard and limited from above by*the*complexity of solving a parity game, i.e.*in*UP∩co-UP. ... Finally, we would also like to thank an anonymous referee for very useful comments -*in*particular*the*observation that led to Corollary 5.2*in*its simplified present form. ...##
###
Classical Copying versus Quantum Entanglement in Natural Language: The Case of VP-ellipsis

2018
*
Electronic Proceedings in Theoretical Computer Science
*

Following

doi:10.4204/eptcs.283.8
fatcat:h7b5buuhpjfedazeqnm35arsce
*the*line of research*in*compositional distributional semantics set out by (Coecke et al., 2010) we develop an extension of*the*Lambek*calculus*which admits a controlled form of contraction to ...*In**the*first model, we follow*the*categorical approach of (Coecke et al., 2013)*in*which a functorial passage sends*the*proofs of*the*grammar to linear maps on vector spaces and we use Frobenius algebras ... Acknowledgements*The*authors are thankful for*the*anonymous reviewers' comments.*The*first author gratefully acknowledges support from a Queen Mary Principal Studentship. ...##
###
A Semantic Theory for Value-Passing Processes Based on the Late Approach

2003
*
BRICS Report Series
*

We also show that

doi:10.7146/brics.v10i15.21785
fatcat:a2v5okrs7fbxji4n7bot3l4prm
*in*general*the*omega-bisimulation preorder is strictly included*in**the*model induced preorder. ...*The*first one, which is value-finitary, i. e. only*reasons*about a finite number of values at each time, is shown to be sound and complete with respect to this model. ... Acknowledgment*The*author would like to thank*the*anonymous referees for many useful comments that led to a thorough revision of*the*paper. ...##
###
Generalising Automaticity to Modal Properties of Finite Structures
[chapter]

2002
*
Lecture Notes in Computer Science
*

*The*function that takes Ò to

*the*size of

*the*smallest labelling system doing this is called

*the*labelling index of

*the*property. ... We define a measure of

*the*size of a structure that we call rank, and show that any modal property of structures can be approximated up to any fixed rank Ò by a labelling system. ... We now consider two fixed-point extensions of modal logic:

*the*modal -

*calculus*and

*the*modal iteration

*calculus*´ÅÁ

*µ*. ...

##
###
Fixpoint Logics on Hierarchical Structures
[chapter]

2005
*
Lecture Notes in Computer Science
*

*In*order to analyze

*the*modal

*µ*-

*calculus*, parity games on hierarchically defined graphs are studied. ...

*In*this paper,

*the*modelchecking problem for

*the*modal

*µ*-

*calculus*and (monadic) least fixpoint logic on hierarchically defined graphs is investigated. ...

*The*modal

*µ*-

*calculus*over SLP-defined graphs

*In*this section, we show that both

*the*data and combined complexity of

*the*modal

*µ*

*calculus*over transition systems that are represented by SLPs is PSPACE-complete ...

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