Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Symbolic Unfolding of Multi-adjoint Logic Programs es Moreno 1 Jaime Penabad 2 e Antonio Riaza 1 Gin´ Jos´ { Gines.Moreno, Jaime.Penabad, JoseAntonio.Riaza } @uclm.es 1 Dept. of Computing Systems, UCLM, 02071 Albacete (Spain) 2 Dept. of Mathematics, UCLM, 02071 Albacete (Spain) 9th European Symposium on Computational Intelligence and Mathematics October 4th - 7th, 2017, Faro (Portugal) 1/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Index Multi-Adjoint Lattices and Programs 1 Operational Semantics for sMALP 2 Unfolding of sMALP Programs 3 Conclusions and Further Research 4 2/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Index Multi-Adjoint Lattices and Programs 1 Operational Semantics for sMALP 2 Unfolding of sMALP Programs 3 Conclusions and Further Research 4 3/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Fuzzy Logic Programming Fuzzy Logic + Logic Programming ⇓ Fuzzy Logic Programming 4/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Multi-Adjoint Lattices Multi-adjoint Lattice A tuple ( L , ≤ , ← 1 , & 1 , . . . , ← n , & n ) is a multi-adjoint lattice if it verifies the following claims: i) ( L , ≤ ) is a complete lattice , with ⊥ and ⊤ elements. ii) (& i , ← i ) is an adjoint pair in ( L , ≤ ), i.e.: 1) & i is increasing in both arguments. 2) ← i is increasing in the first argument and decreasing in the second one. 3) Adjoint property : x ≤ ( y ← i z ) ⇐ ⇒ ( x & i z ) ≤ y 5/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Examples of Multi-Adjoint Lattices Boolean values { true , false } with the classical adjoint pair. Real numbers in the unit interval [0 , 1] with adjoint pairs from the fuzzy logics of Product , G¨ odel and � Lukasiewicz : � 1 if y ≤ x & prod ( x , y ) � x · y ← prod ( x , y ) � x / y if 0 < x < y � 1 if y ≤ x & godel ( x , y ) � min( x , y ) ← godel ( x , y ) � otherwise x & luka ( x , y ) � max(0 , x + y − 1) ← luka ( x , y ) � min( x − y + 1 , 1) We can define other connectives like: | disjunctions : | luka ( x , y ) � min (1 , x + y ) @ aggregators : @ aver ( x , y ) � ( x + y ) / 2 6/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Multi-Adjoint Logic Programming We work with a Prolog-like first order language, with variables, function and predicate symbols, but more connectives : & 1 , & 2 , . . . , & n ( conjunctions ) | 1 , | 2 , . . . , | n ( disjunctions ) ← 1 , ← 2 , . . . , ← n ( implications ) @ 1 , @ 2 , . . . , @ n ( aggregations ) Instead of naive { true , false } , we use a multi–adjoint lattice to model truth degrees ( L , ≤ , ← 1 , & 1 , . . . , ← n , & n ). ([0 , 1] , ≤ , ← luka , & luka , ← prod , & prod , ← godel , & godel ) 7/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Symbolic Multi-Adjoint Logic Programming MALP programs contain weighted rules H ← i B with v : R 1 : p ( X ) ← prod q ( X ) & godel @ aver ( r ( X ) , s ( X )) with 0 . 9 . R 2 : q ( a ) ← with 0 . 8 . P = R 3 : r ( X ) ← with 0 . 7 . R 4 : s ( X ) ← with 0 . 5 . sMALP programs also allow symbolic constants : q ( X ) #& s2 @ aver ( r ( X ) , s ( X )) R 1 : p ( X ) # ← s1 with 0 . 9 . R 2 : q ( a ) ← with # s3 . P s = R 3 : r ( X ) ← with 0 . 7 . R 4 : s ( X ) ← with 0 . 5 . 8/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Index Multi-Adjoint Lattices and Programs 1 Operational Semantics for sMALP 2 Unfolding of sMALP Programs 3 Conclusions and Further Research 4 9/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Operational Semantics for sMALP A state is a pair �Q ; σ � , where Q is a goal and σ a substitution. Procedural semantics: Operational phase based on admissible steps ( → AS ). 1 Interpretive phase based on interpretive steps ( → IS ). 2 Given a program P , a goal Q and a substitution σ , we define a state transition system whose transition relations are → AS and → IS . → ∗ → ∗ Q symbolic fuzzy AS IS �Q ; id � �Q i ; σ � �Q j ; σ � computed answer 10/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Operational phase (I) Admissible step using facts �Q [ A ]; σ �→ AS �Q [ A / v ] θ ; σθ � if 1 A is the selected atom in Q , 2 θ = mgu ( H , A ) � = fail , 3 H ← with v ≪ P . Example Let p ( a ) ← with 0 . 7 be a rule: � ( p ( b ) & godel p ( X )) & godel q ( X ); id � → AS � ( p ( b ) & godel 0 . 7) & godel q ( a ); { X / a }� 11/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Operational phase (II) Admissible step using non-empty rules �Q [ A ]; σ �→ AS �Q [ A / v & i B ] θ ; σθ � if 1 A is the selected atom in Q , 2 θ = mgu ( H , A ) � = fail , 3 H ← i B with v ≪ P and B is not empty. Example Let p ( a ) ← prod p ( f ( a )) with 0 . 7 be a rule: � ( p ( b ) & godel p ( X )) & godel q ( X ); id � → AS � ( p ( b ) & godel 0 . 7 & prod p ( f ( a ))) & godel q ( a ); { X / a }� 12/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Operational phase (III) Admissible step not using program rules �Q [ A ]; σ �→ AS �Q [ A / ⊥ ]; σ � if 1 A is the selected atom in Q , 2 there is no rule in P whose head unifies with A . Example This case is introduced to cope with (possible) unsuccessful admissible derivations. So, with program p ( a ) ← with 0 . 7 we have: � p ( b ); id � → AS � 0; id � 13/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Interpretive phase Interpretive step �Q [ ζ ( r 1 , . . . , r n )]; σ �→ IS �Q [ ζ ( r 1 , . . . , r n ) / r n +1 ]; σ � where ζ denotes a connective defined on the lattice L associated to P and ζ ( r 1 , . . . , r n ) � r n +1 . Example Since the truth function associated to & prod is the product operator, then: � (0 . 8 & luka ((0 . 7 & prod 0 . 9) & godel 0 . 7)); { X / a }�→ IS � (0 . 8 & luka (0 . 63 & godel 0 . 7)); { X / a }� 14/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
Multi-Adjoint Lattices and Programs Operational Semantics for sMALP Unfolding of sMALP Programs Conclusions and Further Research Example of derivation q ( X ) #& s2 @ aver ( r ( X ) , s ( X )) R 1 : p ( X ) # ← s1 with 0 . 9 . R 2 : q ( a ) ← with # s3 . P = R 3 : r ( X ) ← with 0 . 7 . R 4 : s ( X ) ← with 0 . 5 . → R 1 � p ( X ); id � AS � 0 . 9 #& s1 ( q ( X 1 ) #& s2 @ aver ( r ( X 1 ) , s ( X 1 ))); { X / X 1 }� → R 2 AS � 0 . 9 #& s1 (# s3 #& s2 @ aver ( r ( a ) , s ( a ))); { X / a }� → R 3 AS � 0 . 9 #& s1 (# s3 #& s2 @ aver (0 . 7 , s ( a ))); { X / a }� → R 4 AS � 0 . 9 #& s1 (# s3 #& s2 @ aver (0 . 7 , 0 . 5)); { X / a }� → IS � 0 . 9 #& s1 (# s3 #& s2 0 . 6); { X / a }� 15/27 G. Moreno, J. Penabad, J. A. Riaza - ESCIM’17 Symbolic Unfolding of Multi-adjoint Logic Programs
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