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  (H  L1  …  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 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)| c  HU(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