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A connection between Similarity Logic Programming and GÃ¶del Modal Logic

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### A connection between Similarity Logic Programming and Gödel Modal Logic

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

lblandi@unisa.it

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

Overview

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

- Correspondencesbetween the twoapproaches

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))

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

I. Approximate reasoning by extension of the Logic Programming paradigmSimilarity-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

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

G0C1,1,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)).

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.

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: 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)

Truth-constant:

Axioms:

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

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

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

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

Inference rule

Modus ponens:From and infer

II. Approximate reasoning by extension of the Modal Logic

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)

II. Approximate reasoning by extension of the Modal Logic

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}

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◊: ◊( ) (◊ ◊) F◊: ¬◊0

T◊: ◊ Z◊+ : ◊¬¬ ¬¬◊

4◊: ◊◊ ◊ B◊: ¬◊¬◊

R1: ◊ R2: ◊ ◊( )

Inference rules:

From and infer

From infer

II. Approximate reasoning by extension of the Modal Logic

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’)}

Correspondencesbetween the twoapproaches (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 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

Correspondencesbetween the twoapproaches (2)

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.

Correspondencesbetween the twoapproaches (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 = G0C1,1 G1 · · · Ck−1,k−1 Gk−1Ck,k⊥

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

P◊⊢S◊q’

Conclusions and related works

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

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