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Multi-adjoint algebras versus non-commutative residuated structures

M. Eugenia Cornejo, Jesús Medina, Eloisa Ramírez-Poussa
2015 International Journal of Approximate Reasoning  
Furthermore, a comparison between adjoint negations and the negations introduced in extended-order algebras is presented.  ...  Adjoint triples are an interesting generalization of t-norms and their residuated implications, since they increase the flexibility in the framework where they are used.  ...  . ⊓ ⊔ As a consequence, adjoint negations defined from multi-adjoint algebras are more general than the ones given from symmetrical algebras.  ... 
doi:10.1016/j.ijar.2015.08.003 fatcat:q27ko7up75an7g3fjnxn4oogfi

Algebraic Structure of Adjoint Triples Generating a Weak Negation on the Unit Interval [chapter]

M. Eugenia Cornejo, Jesús Medina, Eloísa Ramírez-Poussa
2020 Lecture Notes in Computer Science  
Adjoint negations are a generalization of residuated negations built from the implications of an adjoint triple.  ...  Specifically, this work shows how to build adjoint triples on the unit interval such that their adjoint negations coincide with a given weak negation.  ...  The following result, which was proven in [14] , shows that adjoint negations are more general than weak negations. Theorem 1 [14] .  ... 
doi:10.1007/978-3-030-52705-1_25 fatcat:gzcakcjkpvblbecfasija5sgui

On the measure of incoherent information in extended multi-adjoint logic programs

Nicolas Madrid, Manuel Ojeda-Aciego
2013 2013 IEEE Symposium on Foundations of Computational Intelligence (FOCI)  
In this paper we continue analyzing the introduction of negation into the framework of residuated logic programming [8], [10] ; specifically, we focus on extended programs, that is we consider programs  ...  with strong negation.  ...  However, the pair (p, ∼p) seems to be more incoherent w.r.t. I 2 than w.r.t. I 1 since I 2 makes the pair (p, ∼p) to break the coherence condition to a "bigger degree" than I 1 .  ... 
doi:10.1109/foci.2013.6602452 dblp:conf/foci/MadridO13 fatcat:jaflixrgojd25iftli2byl2vzy

Residuation in non-associative MV-algebras [article]

Ivan Chajda, Helmut Länger
2018 arXiv   pre-print
It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law.  ...  Further, we study a bit more stronger version of an algebra where the binary operation is even monotonous.  ...  Next we introduce a concept which is weaker than that of a residuated poset. Namely, adjointness will be replaced by the conditions occurring in Lemmata 4 and 5.  ... 
arXiv:1810.02405v1 fatcat:rizpjdxt75altmw3tfe4fdm6xq

On an Open Problem of U. Höhle - A Characterization of Conditionally Cancellative T-Subnorms [chapter]

Balasubramaniam Jayaram
2010 Communications in Computer and Information Science  
Further, we give an equivalence condition for a conditionally cancellativite t-subnorm to be a t-norm and hence show that conditionally cancellativite t-subnorms whose natural negations are strong are,  ...  Its natural negation n M is given by n M (x) = sup{t ∈ [0, 1] | M (x, t) = 0}, x ∈ [0, 1] . (4) Note that though n M (0) = 1, it need not be a fuzzy negation, since n M (1) can be greater than 0.  ...  Thus the adjoint pair (I, M ) satisfies (1). Example 1.  ... 
doi:10.1007/978-3-642-14055-6_71 fatcat:yrd6olmuanewpd7outynb3mxpm

Residuated structures derived from commutative idempotent semirings

Ivan Chajda, Helmut L¤nger
2019 Discussiones Mathematicae - General Algebra and Applications  
Then the resulting residuated lattice even satisfies the double negation law.  ...  To a residuated lattice L which does not satisfy the double negation law there can be assigned a so-called augmented semiring. This can be used for reconstruction of the so-called core C(L) of L.  ...  Hence the commutative semiring mentioned in Theorem 2.3 is more general than the MV-semiring from [1] .  ... 
doi:10.7151/dmgaa.1300 fatcat:pt5i4xubovdizkta2ect5cx24u

General coupled semirings of residuated lattices

Ivan Chajda, Helmut Länger
2016 Fuzzy sets and systems (Print)  
We generalize this correspondence to residuated lattices satisfying the double negation law.  ...  In order to prove that L(C) is a residuated lattice we have to check the adjointness property.  ...  We will work with residuated lattices having one more property. Definition 3. Let L = (L, ∨, ∧, ⊗, →, 0, 1) be a residuated lattice.  ... 
doi:10.1016/j.fss.2015.12.009 fatcat:lvoigp55dzdypn5hhm2ezx3nla

Abductive reasoning in normal residuated logic programming via bipolar max-product fuzzy relation equations

David Lobo, Maria Eugenia Cornejo Piñero, Jesús Medina
2019 Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019)  
Fuzzy relation equations have recently been extended to a more general framework in which the unknown variables appear together with their logical negation connectives, giving rise to the called bipolar  ...  This paper shows an abductive procedure for a special kind of normal residuated logic program by means of bipolar maxproduct fuzzy relation equations.  ...  It is convenient to mention that there might be more than one possibility for these inputs.  ... 
doi:10.2991/eusflat-19.2019.81 dblp:conf/eusflat/LoboPM19 fatcat:xmn7czajqfh3rc3ds77ep257ky

Relating Multi-Adjoint Normal Logic Programs to Core Fuzzy Answer Set Programs from a Semantical Approach

M. Eugenia Cornejo, David Lobo, Jesús Medina
2020 Mathematics  
This paper relates two interesting paradigms in fuzzy logic programming from a semantical approach: core fuzzy answer set programming and multi-adjoint normal logic programming.  ...  Specifically, it is shown how core fuzzy answer set programs can be translated into multi-adjoint normal logic programs and vice versa, preserving the semantics of the starting program.  ...  adjoint pairs and a negation operator.  ... 
doi:10.3390/math8060881 fatcat:3wr4vsbbfvew5ibekvglpsvrlm

Towards a Fuzzy Answer Set Semantics for Residuated Logic Programs

Nicolás Madrid, Manuel Ojeda-Aciego
2008 2008 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology  
In this work we introduce the first steps towards the definition of an answer set semantics for residuated logic programs with negation.  ...  Definition 2 A residuated lattice with negation is an algebra (L, * , ←, ∼, ¬), where (L, * , ←) is a residuated lattice, ∼ is a negation operator and ¬ is a default negation operator.  ...  In this paper, we focus on the initial definitions of stable model and answer set in the framework of residuated logic programs [2] , as a initial step towards an answer set semantics for multi-adjoint  ... 
doi:10.1109/wiiat.2008.357 dblp:conf/iat/MadridO08 fatcat:4a4g7hdpuzeerc5l5agj6six54

Duality vs. adjunction for fuzzy mathematical morphology and general form of fuzzy erosions and dilations

Isabelle Bloch
2009 Fuzzy sets and systems (Print)  
A pair of operators (I, C) are said adjoint if: ∀( , , ) ∈ [0, 1] 3 , C( , ) ⇔ I ( , ). (3) The adjoint of a conjunction is a residual implication (there is an equivalence between the residuation principle  ...  The negation writes c( ) = (1 − p ) 1/ p , the t-norm T ( , ) = max(0, ( p + p − 1) 1/ p ), the dual t-cornom S( , ) = min(1, ( p + p ) 1/ p ) and the residual implication I ( , ) = min(1, ((1 − ) p +  ... 
doi:10.1016/j.fss.2009.01.006 fatcat:uuzjbg7apnfo3nxc6t666vktom

Dual multi-adjoint concept lattices

J. Medina, M. Ojeda-Aciego
2013 Information Sciences  
of the multi-adjoint paradigm is applied and no negation on the lattices is needed.  ...  Whereas the usual approach to the latter is via a negation operator, this paper presents a fuzzy generalization of the dual concept lattice, the dual multi-adjoint concept lattice, in which the philosophy  ...  of inaccurate) information, which will give more flexibility than the existing procedures.  ... 
doi:10.1016/j.ins.2012.10.030 fatcat:z72hqglbgjgzlai6tsttakdywm

On Coherence and Consistence in Fuzzy Answer Set Semantics for Residuated Logic Programs [chapter]

Nicolás Madrid, Manuel Ojeda-Aciego
2009 Lecture Notes in Computer Science  
In this work we recall the first steps towards the definition of an answer set semantics for residuated logic programs with negation, and concentrate on the development of relationships between the notions  ...  The previous example presented some negations which grant more or less restrictive consistence values; however, one would like to be able to construct the corresponding negation to a prescribed consistence  ...  Then we initiate the analysis of the relationships between the notions of coherence of an interpretation with the more common notion of consistence. 2 On fuzzy answer set semantics for residuated programs  ... 
doi:10.1007/978-3-642-02282-1_8 fatcat:hgrbkqsep5e4re4irh2op4jbmu

POSITIVE LOGIC WITH ADJOINT MODALITIES: PROOF THEORY, SEMANTICS, AND REASONING ABOUT INFORMATION

MEHRNOOSH SADRZADEH, ROY DYCKHOFF
2010 The Review of Symbolic Logic  
Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.)  ...  For such a logic we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus.  ...  For example, in category Sup of sup-lattices with join-preserving maps, every such map is residuated, i.e. has a right adjoint.  ... 
doi:10.1017/s1755020310000134 fatcat:byi5bppldzeebnbe3i6cro5yzi

Positive Logic with Adjoint Modalities: Proof Theory, Semantics and Reasoning about Information

Mehrnoosh Sadrzadeh, Roy Dyckhoff
2009 Electronical Notes in Theoretical Computer Science  
Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.)  ...  For such a logic we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus.  ...  For example, in category Sup of sup-lattices with join-preserving maps, every such map is residuated, i.e. has a right adjoint.  ... 
doi:10.1016/j.entcs.2009.07.102 fatcat:huz6cihturbubjjfhwpfx3sqsi
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