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NP Reasoning in the Monotone μ-Calculus [article]

Daniel Hausmann, Lutz Schröder
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.  ... 
arXiv:2002.05075v2 fatcat:zsrpwu225fdwjkmukubw3pmtva

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

Daniel Hausmann, Lutz Schröder
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.  ... 
doi:10.48550/arxiv.2002.05075 fatcat:wwahyiwzcbbqbefs6naqbhrifu

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

Daniel Hausmann, Lutz Schröder
2020 Lecture Notes in Computer Science  
We recall basics on fixpoints and games 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.  ... 
doi:10.1007/978-3-030-51074-9_28 fatcat:sb3lc777s5ba5onnylztv4esne

A 'Natural Logic' inference system using the Lambek calculus

Anna Zamansky, Nissim Francez, Yoad Winter
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  ... 
doi:10.1007/s10849-006-9018-x fatcat:nbz6nujskrexjbyt365maom3qu

Continuation semantics for the Lambek–Grishin calculus

Raffaella Bernardi, Michael Moortgat
2010 Information and Computation  
We present a Curry-Howard interpretation for LG(/, \, , ) derivations, based on Curien and Herbelin's λµ 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.  ... 
doi:10.1016/j.ic.2009.11.005 fatcat:msdvi2jid5f5pkrg7uzvblsnti

Symmetric categorial grammar: residuation and Galois connections [article]

Michael Moortgat
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.  ... 
arXiv:1008.0170v1 fatcat:7ce74ftpbfauvedtp3coubx2ve

A Frobenius Algebraic Analysis for Parasitic Gaps [article]

Michael Moortgat, Mehrnoosh Sadrzadeh, Gijs Wijnholds
2020 arXiv   pre-print
Our analysis is formulated 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  ... 
arXiv:2005.05639v2 fatcat:mvc26s5z7rhrnbnwiwjdq5aanq


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  ... 
doi:10.1017/s1755020316000137 fatcat:wulqblhirnacxpgi3iw4dfizde

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

Arno Bastenhof
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 µcalculus.  ... 
arXiv:1101.5757v1 fatcat:ofrm6l2mbra5xget5vktnzsh34

Tractable Probabilistic mu-Calculus That Expresses Probabilistic Temporal Logics

Pablo Castro, Cecilia Kilmurray, Nir Piterman, Marc Herbstritt
2015 Symposium on Theoretical Aspects of Computer Science  
Its game semantics is very similar to 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.  ... 
doi:10.4230/lipics.stacs.2015.211 dblp:conf/stacs/CastroKP15 fatcat:whgrktaxbrf3lir3tqrunw7ur4

Three notes on the complexity of model checking fixpoint logic with chop

Martin Lange
2007 RAIRO - Theoretical Informatics and Applications  
This paper analyses 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.  ... 
doi:10.1051/ita:2007011 fatcat:qj7kuj4r6rcslbp6yoewgk5huq

Classical Copying versus Quantum Entanglement in Natural Language: The Case of VP-ellipsis

Gijs Wijnholds, Mehrnoosh Sadrzadeh
2018 Electronic Proceedings in Theoretical Computer Science  
Following 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.  ... 
doi:10.4204/eptcs.283.8 fatcat:h7b5buuhpjfedazeqnm35arsce

A Semantic Theory for Value-Passing Processes Based on the Late Approach

Anna Ingólfsdóttir
2003 BRICS Report Series  
We also show that 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.  ... 
doi:10.7146/brics.v10i15.21785 fatcat:a2v5okrs7fbxji4n7bot3l4prm

Generalising Automaticity to Modal Properties of Finite Structures [chapter]

A. Dawar, S. Kreutzer
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.  ... 
doi:10.1007/3-540-36206-1_11 fatcat:tgw2ps565rditmiphtx7uxomta

Fixpoint Logics on Hierarchical Structures [chapter]

Stefan Göller, Markus Lohrey
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 µcalculus over transition systems that are represented by SLPs is PSPACE-complete  ... 
doi:10.1007/11590156_39 fatcat:kjcmhlv3evbrpjej7yp4ndin7q
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