From F-logic to CHORD

F-Logic Syntax

• Atoms– Focus on only two kinds of atoms

• s :: c (subclass relationship)• s[m->>v] (inheritable multi-valued method)• s,c,m,v are logic terms

• Literal (A or ¬A)• Rules (HL1 … Ln)• Program

– A set of rules

F-Logic Semantics

Herbrand Universe (HU)

• The set of all ground (variable-free) terms constructed from the function symbols and constants in program

Ex: P = { a[m(X)->>v] p(X) :: z, p(2) :: z, p(3) :: z,

p(4) :: w }

HU (P) = { a, v, z, w, 2, 3, 4, m(a), m(v), …, p(a), p(v)… }

Herbrand Instantiation (ground)

• The set of rules obtained by consistently substituting all terms in HU(P) for all variables in every rule of P

Ex: P = { a[m(X)->>v] p(X) :: z, p(2) :: z, p(3) :: z,

p(4) :: w }

ground(P) = { a[m(a)->>v] p(a) :: z, a[m(v)->>v] p(v) :: z, … }

Herbrand Base (HB)

• The set of atoms of the form s::c and s[m->>v]/c where s,m,c,v are terms in HU(P)

Ex: Ex: P = { a[m(X)->>v] p(X) :: z, p(2) :: z, p(3) :: z,

p(4) :: w }

HB(P) = { p(a)::z, p(v)::z, … a[m(a)->>v]/a, … }

Three Valued Interpretation (I)

• Is a pair <T;U> where T and U are disjoint subsets of the Herbrand base HB(P).– T contains all atoms that are true in I– U contains all atoms that are undefined in I

• Truth values: f < u < t, ¬f = t, ¬t = f, ¬u = u– I(A) = t, if A T– I(A) = u, if A U– I(A) = f, otherwise– If Ai HB(P), I(A1 … An) = min{ I(Ai) }

Truth Valuation Funcitons (Vh, Vb)

• Vh(A,I) – truth value of A if it appears in the head of some rule– Vh(s::c, I) = I(s::c)– Vh(c[m->>v], I) = I(c[m->>v]/c)

• Vb(A,I) – truth value of A if it appears in the body of some rule– Vb(s::c, I) = I(s::c)– Vb(s[m->>v], I) = max{ I(s[m->>v]/c)| cHU(P) }– Vb(¬L, I) = ¬V(L, I)– Vb(L1 … Ln, I) = min{ Vb(Li, I) }

Three valued interpretation for Rules

• For rules– I(H B) = t if Vh(H,I) ≥ Vb(B,I),– I(H B) = f, otherwise.

• For facts– I(H) = t if Vh(H,I) = t),– I(H) = f, otherwise.

Program Satisfaction

• I satisfies P if for every R in ground(P), I(R) = t

Local Inheritance Contexts

• s[m->>v] is a strong local context for s in I if I(s[m->>v]/s) = t

• s[m->>v] is a weak local context for s in I if I(s[m->>v]/s) = u

Inheritance Context

• c[m->>v] is a strong inheritance context for s in I, if:– s is a proper subclass of c– m ->> v is locally defined in c– There’s no other weak of strong local inheritance

context of m in s

• c[m->>v] is a weak inheritance context for s in I, if:– s is a proper subclass of c– m ->> v is locally defined in c– All local inheritance contexts of m in s are weak

Overriding

• o strongly overrides c[m->>v] for s in I, if– o is a proper subclass of c x o[m->>x] is a strong inheritance context

for s

• o weakly overrides c[m->>v] for s in I, if x o[m->>x] is a weak inheritance context for

s or o::c is undefined

Inheritance Candidates

• c[m->>v] is a strongly inheritance candidate for s in I, if:– c[m->>v] is a strong inheritance context for s– Theres no o such that o weakly or strongly overrides

c[m->>v] for s

• c[m->>v] is a weak inheritance candidate for s in I, if:– c[m->>v] is a weak inheritance context for s or– c[m->>v] is a strong inheritance context for s and

there is o such that o weakly overrides c[m->>v] for s

CHORD

Differences Between CHR and Prolog

• Prolog / F-Logic– Closed World

Assumption• True / False /

Undefined

– Logical Rules– Semantics deals with

True/False/Undefined parts of HB(P)

• CHR / CHORD– Open World

Aassumption• True / False / Unknown

– Non-Logical Rules– Constraint Store

contains only part of the True facts (Remaining True/False/Unknown are inacessible)

My Proposal

• Add F-Atoms as CHR constraints– They would be able to be used both as BIC

and as UDC• BIC: allows part of the class structure to be defined

by the built-in solver• UDC: allows the user to defined the “user defined”

part of the class structure

• At each step, the Optimistic Object Model of the constraint store would be computer