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A connection between Similarity Logic Programming and Gödel Modal Logic

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Luciano BlandiDept. Matematica & InformaticaUniversity of Salerno 84084 Fisciano (Salerno) ITALY

lblandi@unisa.it

A connection between Similarity Logic Programming and Gödel Modal Logic

Lluis GodoInstitut 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

- Two approaches to Similarity-based Approximate Reasoning:
- Extended Logic Programming
- Extended Modal Logic

- Correspondencesbetween the twoapproaches

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.

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.

MATCH

SLD Resolution: an example

Goal:meeting(‘Data Mining’, date(D, M,Y))

X = ‘Data Mining’

- Program: meeting(X, date(D, M,Y)) 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))

R(‘Data Mining’,‘Information Retrieval’) = 0.6 > 0

R(‘Data Mining’,‘Data Mining’) = 1 > 0

R(‘Data Mining’, ‘Data Discovery’) = 0.4 > 0

Similarity-based SLD Resolution: an example

Goal:

conference(‘Data Mining’, date(D, M,Y))

Program:

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: Approximation degree:

conference(‘Data Mining’,date(22, 6, 2000)) = 0.6

conference(‘Data Mining’, date(12, 5, 2001)) = 1 exact solution

conference(‘Data Mining’,date(22, 9, 2001)) = 0.4

Similarity-based SLD Resolution

To determine the computed answer substitutions by means similarity-based SLD derivation

G0C1,1,U1 G1 … Cm,m,Um Gm …

Approximation degree associated to the computed answer

= min{U1, U2, …, Uk}

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.

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)).

check(D,M,Y,Agenda) :- …..

…

Weak Solutions:

meeting('Data Mining',date(12, 5,01)) (=0.4)

meeting('Data Mining', date(22, 9, 01)) (=0.8)

Exact Solutions:

meeting('Data Mining',date(12,5,01))

Weak Solutions:

meeting('Data Mining',date(12, 5, 01)) ( =0.4)

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.

Similarity interface of Silog

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: for each r [0, 1]

Axioms:

Axioms(BL) + {G} + {Bookkeeping axioms} + {Approximation axioms}

Inference rules:

From and infer

From infer

BL: The basic many-valued propositional logic

Language:

Set of the symbols of propositional constant: Const = {p1, p2, …}

Logic connectives: (strong conjunction), (implication)

Truth-constant:

Axioms:

(A1) () (()()) (A2) ()

(A3) () () (A4) (()) (())

(A5a) (()) (()) (A5b) (()) (())

(A6) (()) ((()))(A7)

Inference rule

Modus ponens:From and infer

Extended BL with rationals

Language: Language(BL) + { for each r [0, 1] }

Axioms:Axioms(BL) + {Book-keeping axioms}

Book-keeping axioms:

Inference rule:

Modus ponens:From and infer

Notation: (, r) is ( )

Derived inference rule:

From (, r) and ( , s) infer (, r * s)

RGL: Rational Gödel Logic

Language: Language(BL) + { 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}

RGS5◊: The rational Gödel similarity-based S5 modal logic

Language: Language(RGL) + { (approximation modal operator) }

Axioms:Axioms(RGL) + {Approximation axioms}

Approximation axioms:

D◊: ◊( ) (◊ ◊) F◊: ¬◊0

T◊: ◊ Z◊+ : ◊¬¬ ¬¬◊

4◊: ◊◊ ◊ B◊: ¬◊¬◊

R1: ◊ R2: ◊ ◊( )

Inference rules:

From and infer

From infer

Many-valued similarity-based Kripke modelsM=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( , w) = r for all r [0, 1]

e(◊, w) = supw’W min{S(w, w’), e(, w’)}

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 LGby

(q p1, . . . , pn)* = p1pnq

(p1, . . . , pn)* = p1 pn

Then, we can define

Rules* = {* | Rules}

Crisp = {p ¬p | p Const}

Sim = { ((p q) (q p)) | p, q Const}

and the following theory in the language LG

P= Facts Rules* Crisp Sim

Sim = { ((p q) (q p)) | p, q Const}

(*)

In RGS5◊,

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.

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 = G0C1,1 G1 · · · Ck−1,k−1 Gk−1Ck,k⊥

where G0 = q’ and = min{1, . . . , k}, then

P◊⊢S◊q’

We have shown and put into relation two approaches to

Similarity-based Approximate Reasoning:

- Framework: Extended Logic Programming
Tool: Similarity-based SLD Resolution

(Similarity relation defined on symbols of the language)

- 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