A connection between Similarity Logic
Programming and Gödel Modal Logic
Luciano Blandi
Dept. Matematica & Informatica
University of Salerno
84084 Fisciano (Salerno) ITALY
lblandi@unisa.it
Lluis Godo
Institut d’Investigació en intel·ligència Artificial
Consejo Superior de Investigaciones Ccientificas
Campus UAB, 08193 Bellaterra, Spain
godo@iiia.csic.es
Ricardo Oscar Rodriguez
Dpto. de Computación
Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
ricardo@dc.uba.ar
Overview
 Two approaches to Similarity-based
Approximate Reasoning:
I. Extended Logic Programming
II. Extended Modal Logic
 Correspondences between the two
approaches
Similarity relation (*)
Definition
A Similarity on a domain U is a fuzzy subset R: U×U[0,1] of U×U
such that the following properties hold:
i)
R(x, x) = 1 for any x є U
(reflexivity)
ii)
R(x, y) = R(y, x) for any x, y є U
(symmetry)
iii) R(x, z)  R(x, y) * R(y, z) for any x, y, z є U
(transitivity)
where * : [0,1] × [0,1]  [0,1] is a t-norm
We say that R is strict if the following implication is also verified
iv)
R(x, z) = 1  x = z
(*) L. Valverde, “On the structure of F-indistinguishability operators”. Fuzzy Sets Syst 1985;17:313–328.
I. Approximate reasoning by extension of the Logic Programming paradigm
Similarity-based SLD Resolution
● SLD Resolution inferential engine as model of logic
deduction
● Similarity relation embedded in the SLD Resolution model
as approach to the approximate reasoning
(*) M.I. Sessa, “Approximate Reasoning by Similarity-based SLD Resolution.”, Theoretical Computer
Science, 275 (2002) 389-426.
I. Approximate reasoning by extension of the Logic Programming paradigm
SLD Resolution: an example
Goal:
 meeting(‘Data Mining’, date(D, M,Y))
MATCH
Program:
meeting(X, date(D, M,Y))

X = ‘Data Mining’
conference(X, date(D, M, Y)).
conference(Information Retrieval’, date(22, 6, 2000)).
conference('Neural Network', date(17, 9, 2000)).
conference('Artificial Intelligence', date(10, 1, 2002)).
conference('Data Mining', date(12, 5, 2001)).
conference('Data Discovery', date(22, 9, 2001)).
Solutions:
meeting(‘Data Mining’, date(12, 5, 2001))
I. Approximate reasoning by extension of the Logic Programming paradigm
Similarity-based SLD Resolution: an example
Goal:
 conference(‘Data Mining’, date(D, M,Y))
Program:
conference(‘Information Retrieval’, date(22, 6, 2000)).
R(‘Data Mining’,‘Information Retrieval’) = 0.6 > 0
conference('Neural Network', date(17, 9, 2000)).
conference('Artificial Intelligence', date(10, 1, 2002)).
conference('Data Mining', date(12, 5, 2001)).
R(‘Data Mining’,‘Data Mining’) = 1 > 0
conference('Data Discovery', date(22, 9, 2001)).
R(‘Data Mining’, ‘Data Discovery’) = 0.4 > 0
Solutions:
Approximation degree:
conference(‘Data Mining’, date(22, 6, 2000))
 = 0.6
conference(‘Data Mining’, date(12, 5, 2001))
=1
conference(‘Data Mining’, date(22, 9, 2001))
 = 0.4
exact solution
I. Approximate reasoning by extension of the Logic Programming paradigm
Similarity-based SLD Resolution
To determine the computed answer substitutions by means
similarity-based SLD derivation
G0 C1,1,U1 G1  … Cm,m,Um Gm  …
Approximation degree associated to the computed answer
 = min{U1, U2, …, Uk}
I. Approximate reasoning by extension of the Logic Programming paradigm
Weak Unification Algorithm (*)
Given two atoms A = p(s1,... , sn) and B = q(t1,... , tn) of the same arity with
no common variables to be unified, construct the associated set of equation
W = {p = q, s1 = t1, ... , sn = tn}.
If R(p,q) = 0, halts with failure, otherwise, set U = R(p, q) and W = W-{p=q}.
Until the current set of equation W does not change, non deterministically
choose from W an equation of a form below and perform the associated action.
(1) f(s1, ..., sn) = g(t1, ...,tn) where R(f, g) > 0: replace by the equations
s1=t1, ..., sn=tn, and set U = min{U, R(f, g)};
(2) f(s1,... , sn) = g(t1,... , tm) where R(f, g) = 0: halts with failure;
(3) x = x: delete the equation;
(4) t = x where t is not a variable; replace by the equation x = t;
(5) x = t where x  t and x has another occurrence in the set of equations: if
x appears in t then halt with failure, otherwise perform the substitution
{x/t} in every other equations.
R.K. Apt, “Logic Programming”, in: J. van Leeuwen (Ed.), Haridbook of Theoretical Computer Science,
vol. B, (Elsevier, Amsterdam, 1990) 492-574.
(*) M.I. Sessa, “Approximate Reasoning by Similarity-based SLD Resolution.”, Theoretical Computer
Science, 275 (2002) 389-426.
I. Approximate reasoning by extension of the Logic Programming paradigm
An extended Prolog system (*)
meeting(‘Data Mining’, date(D, M,Y))
meeting(X, date(D, M,Y)) :conference(X, date(D, M, Y)),
check(D,M,Y,Agenda).
…
conference(‘Information Retrieval', date(12, 5, 01)).
conference('Neural Network', date(17, 9, 00)).
conference('Artificial Intelligence', date(10, 1, 02)).
conference('Data Mining', date(12, 5, 01)).
conference('Data Discovery', date(22, 9, 01)).
Weak Solutions:
meeting('Data Mining',date(12, 5,01)) (=0.4)
meeting('Data Mining', date(22, 9, 01)) (=0.8)
check(D,M,Y,Agenda) :- …..
…
Exact Solutions:
Weak Solutions:
meeting('Data Mining',date(12, 5, 01)) ( =0.4)
meeting('Data Mining',date(12,5,01))
meeting('Data Mining', date(22, 9, 01)) (=0.8)
Exact Solution:
meeting('Data Mining',date(12,5,2001))
(*) V. Loia, S. Senatore, M.I. Sessa, Similarity-based SLD Resolution and its implementation in an Extended
Prolog System, Proc. 10th IEEE International Conference on Fuzzy Systems, 2001, Melbourne, Australia.
I. Approximate reasoning by extension of the Logic Programming paradigm
Similarity interface of Silog
II. Approximate reasoning by extension of the Modal Logic
RGS5◊: The rational Gödel similarity-based S5 modal logic
Language :
Set of the symbols of propositional constant:
Const = {p1, p2, …}
Logic connectives: (conjunction),  (implication)
Modal operator: (approximation)
Truth-constants: r for each r  [0, 1]
Axioms:
Axioms(BL) + {G} + {Bookkeeping axioms} + {Approximation axioms}
Inference rules:
From  and    infer 
From    infer   
II. Approximate reasoning by extension of the Modal Logic
BL: The basic many-valued propositional logic
Language:
Set of the symbols of propositional constant: Const = {p1, p2,
…}
Logic connectives:
 (strong conjunction),  (implication)
0
Truth-constant:
Axioms:
(A1) ()  (()())
(A2) ()  
(A3) ()  ()
(A4) (())  (())
(A5a) (())  (())
(A5b) (())  (())
(A6) (())  ((()))
(A7) 0 
Inference rule
Modus ponens:
From  and    infer 
II. Approximate reasoning by extension of the Modal Logic
Extended BL with rationals
Language: Language(BL) + { r for each r 
Axioms:
[0, 1] }
Axioms(BL) + {Book-keeping axioms}
Book-keeping axioms:
r & s  r *s
r  s  r * s
Inference rule:
Modus ponens:
From  and    infer 
Notation: (, r) is ( r )
Derived inference rule:
From (, r) and (  , s) infer (, r * s)
II. Approximate reasoning by extension of the Modal Logic
RGL: Rational Gödel Logic
Language: Language(BL) + { r for each r 
[0, 1] }
Axioms: Axioms(BL) + {Book-keeping axioms} + {G}
Axiom G:
  ( & )
Inference rule:
Modus ponens:
From  and    infer 
Remark:
Axioms(G) = Axioms (BL) + {G}
II. Approximate reasoning by extension of the Modal Logic
RGS5◊: The rational Gödel similarity-based S5 modal logic
Language: Language(RGL) + {  (approximation modal operator) }
Axioms:
Axioms(RGL) + {Approximation axioms}
Approximation axioms:
D◊: ◊(  )  (◊  ◊)
T◊:   ◊
4◊: ◊◊  ◊
R1: r  ◊r
Inference rules:
From  and    infer 
From    infer   
F◊: ¬◊0
Z◊+ : ◊¬¬  ¬¬◊
B◊:   ¬◊¬◊
R2: r  ◊  ◊( r  )
II. Approximate reasoning by extension of the Modal Logic
Many-valued similarity-based Kripke models
M=W, S, e
W   is a set of possible worlds
S: W × W  [0, 1 ] is a similarity relation on W
e: Const × W  [0, 1 ] represents an evaluation assigning to each atomic
formula pi  Const and each interpretation w  W
a truth value e(pi, w)  [0, 1] of pi in w.
e is extended to formulas by defining
e(  , w) = min{e(, w), e(, w)}
e(  , w) = e(, w) G e(, w)
e( r , w) = r for all r   [0, 1]
e(◊, w) = supw’W min{S(w, w’), e(, w’)}
Correspondences between the two approaches (1)
Let R : Const × Const  [0, 1] be a similarity relation on Const.
Let P = Facts  Rules  L be a logic program
Define a mapping * : L  LG by
(q  p1, . . . , pn)* = p1    pn  q
(p1, . . . , pn)* = p1    pn
Then, we can define
Rules* = {* |   Rules}
Crisp = {p  ¬p | p  Const}
Sim = { R(p, q)  ((p  q)  (q  p)) | p, q  Const}
and the following theory in the language LG
P = Facts  Rules*  Crisp  Sim
Correspondences between the two approaches (2)
Sim = { R(p, q)  ((p  q)  (q  p)) | p, q  Const}
IS (p | q)  sup
w W:
e(q, w)1
sup S(w, w' )
(*)
w' W:
e(p, w') 1
In RGS5◊,
IS (p | q)  q  p
We note that
IS(p | p) = 1
(reflexivity)
IS(r | q)  min{IS(r | p), IS(p | q)}
(min-transitivity)
RS(p, q) = min{IS(p |q), IS(q | p)}
is a min-similarity
(*) E.H. Ruspini, “On the semantics of fuzzy logic”, in: Int. Journal of Approximate Reasoning, 5, 1991,
pp. 45-88.
Correspondences between the two approaches (3)
Proposition
Let R : Const × Const  [0, 1] be a similarity relation,
P = Rules  Facts a definite program on a propositional language L
and q’ a goal..
If there exists a similarity-based SLD refutation with approximation
degree  for P  { q’}
D = G0 C1,1 G1  · · · Ck−1,k−1 Gk−1 Ck,k ⊥
where G0 =  q’ and  = min{1, . . . , k}, then
P◊ ⊢S α ◊q’
Conclusions and related works
We have shown and put into relation two approaches to
Similarity-based Approximate Reasoning:
I.
Framework: Extended Logic Programming
Tool: Similarity-based SLD Resolution
(Similarity relation defined on symbols of the language)
II.
Framework: Many-valued Rational Gödel Modal Logic
Tool: Similarity relation as accessibility relation
(Similarity relation defined on possible worlds)
Future tasks:
 To prove the full relationship among the two approaches
 To prove the relative counterpart for predicate languages.
 To study possible links with respect to other approaches to
similarity-based reasoning