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alcq-system-fm.maude
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alcq-system-fm.maude
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(fmod SYNTAX is
inc NAT .
sorts AConcept Concept ARole Role .
subsort AConcept < Concept .
subsort ARole < Role .
ops ALL EXIST : Role Concept -> Concept .
ops CTRUE CFALSE : -> AConcept .
op _&_ : Concept Concept -> Concept [gather (e E) prec 31] .
op _|_ : Concept Concept -> Concept [gather (e E) prec 32] .
op ~_ : Concept -> Concept [prec 30] .
eq ~ CTRUE = CFALSE .
eq ~ CFALSE = CTRUE .
ops AT-MOST AT-LEAST : Nat Role Concept -> Concept .
op INV : Role -> Role .
var R : Role .
var C : Concept .
--- eq AT-LEAST(1, R, C) = EXIST(R, C) .
endfm)
(fmod LABEL is
inc SYNTAX .
sorts Label ELabel ALabel QLabel .
subsorts ELabel ALabel QLabel < Label .
ops gt lt : Nat Role -> QLabel .
op ex : Role -> ELabel .
op al : Role -> ALabel .
endfm)
(view Label from TRIV to LABEL is
sort Elt to Label .
endv)
(fmod LALC-SYNTAX is
inc LABEL .
inc LIST{Label} .
vars L1 L2 L : List{Label} .
vars R S : Role .
var C : Concept .
var T : Label .
var n : Nat .
sorts Expression LConcept FzConcept .
subsorts LConcept FzConcept < Expression .
op <_|_> : List{Label} Concept -> LConcept [ctor] .
op `[_`,_`] : Nat LConcept -> FzConcept .
ops has-quant has-lt has-gt : List{Label} -> Bool .
ops has-al has-ex : List{Label} -> Bool .
ceq has-quant(T L) = true if (T :: QLabel) .
eq has-quant(T L) = has-quant(L) [owise] .
eq has-quant(nil) = false .
ceq has-lt(T L) = true if (T :: QLabel) /\ lt(n, R) := T .
eq has-lt(T L) = has-lt(L) [owise] .
eq has-lt(nil) = false .
ceq has-gt(T L) = true if (T :: QLabel) /\ gt(n, R) := T .
eq has-gt(T L) = has-gt(L) [owise] .
eq has-gt(nil) = false .
ceq has-ex(T L) = true if (T :: ELabel) .
eq has-ex(T L) = has-ex(L) [owise] .
eq has-ex(nil) = false .
ceq has-al(T L) = true if (T :: ALabel) .
eq has-al(T L) = has-al(L) [owise] .
eq has-al(nil) = false .
op neg : List{Label} -> List{Label} .
op neg-aux : List{Label} List{Label} -> List{Label} .
eq neg(L1) = neg-aux(L1, nil) .
eq neg-aux(L1 al(R), L2) = neg-aux(L1, ex(R) L2) .
eq neg-aux(L1 ex(R), L2) = neg-aux(L1, al(R) L2) .
eq neg-aux(nil, L2) = L2 .
endfm)
(view Expression from TRIV to LALC-SYNTAX is
sort Elt to Expression .
endv)
(fmod SEQUENT-CALCULUS is
inc LALC-SYNTAX .
inc SET{Expression} .
inc SET{Label} .
inc SET{Nat} .
inc NAT .
inc QID .
sort Sequent .
sorts Goal State Proof .
subsort Goal State < Proof .
op next : Nat -> State .
op goals : Set{Nat} -> State .
op `[_from_by_is_`] : Nat Nat Qid Sequent -> Goal [format (n d d d d d d d d d)] .
op nil : -> Proof [ctor] .
op __ : Proof Proof -> Proof [ctor comm assoc] .
op _|-_ : Set{Expression} Set{Expression} -> Sequent [ctor prec 122 gather(e e)] .
op _:_|-_:_ : Set{Expression} Set{Expression} Set{Expression} Set{Expression} -> Sequent [ctor prec 122 gather(e e e e)] .
endfm)
(mod LALCQ-SYSTEM is
inc SEQUENT-CALCULUS .
inc INT .
vars ALFA GAMMA : Set{Expression} .
vars X Y N M : Nat .
var XS : Set{Nat} .
vars A B C D : Concept .
vars R S : Role .
var E : Expression .
var AT : AConcept .
var Q : Qid .
vars L L1 L2 L3 L4 : List{Label} .
--- como lidar com regras nao deterministicas? m > n sss inc em um, controle de aplicacoes
rl [initial] :
[ X from Y by Q is ALFA, < nil | AT > |- < nil | AT >, GAMMA ] goals((X, XS)) =>
[ X from Y by Q is ALFA, < nil | AT > |- < nil | AT >, GAMMA ] goals((XS)) .
crl [initial] :
[ X from Y by Q is ALFA, < lt(n,R) L | A > |- < lt(m,R) L | A >, GAMMA ] goals((X, XS)) =>
[ X from Y by Q is ALFA, < lt(n,R) L | A > |- < lt(m,R) L | A >, GAMMA ] goals((XS))
if n <= m .
crl [initial] :
[ X from Y by Q is ALFA, < gt(n,R) L | A > |- < gt(m,R) L | A >, GAMMA ] goals((X, XS)) =>
[ X from Y by Q is ALFA, < gt(n,R) L | A > |- < gt(m,R) L | A >, GAMMA ] goals((XS))
if m <= n .
rl [weak-l] :
[ X from Y by Q is ALFA, E |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, E |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'weak-l is ALFA |- GAMMA ] .
rl [weak-r] :
[ X from Y by Q is ALFA |- GAMMA, E ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- GAMMA, E ] next(N + 1) goals((XS, N))
[ N from X by 'weak-r is ALFA |- GAMMA ] .
rl [forall-r] :
[ X from Y by Q is ALFA |- GAMMA, < L1 | ALL(R, A) > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- GAMMA, < L1 | ALL(R, A) > ] next(N + 1) goals((XS, N))
[ N from X by 'forall-r is ALFA |- GAMMA, < L1 al(R) | A > ] .
rl [forall-l] :
[ X from Y by Q is ALFA, < L1 | ALL(R, A) > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L1 | ALL(R, A) > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'forall-l is ALFA, < L1 al(R) | A > |- GAMMA ] .
rl [exist-r] :
[ X from Y by Q is ALFA |- GAMMA, < L1 | EXIST(R, A) > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- GAMMA, < L1 | EXIST(R, A) > ] next(N + 1) goals((XS, N))
[ N from X by 'exist-r is ALFA |- GAMMA, < L1 ex(R) | A > ] .
rl [exist-l] :
[ X from Y by Q is ALFA, < L1 | EXIST(R, A) > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L1 | EXIST(R, A) > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'exist-l is ALFA, < L1 ex(R) | A > |- GAMMA ] .
rl [least-l] :
[ X from Y by Q is ALFA, < L1 | AT-LEAST(M, R, A) > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L1 | AT-LEAST(M, R, A) > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'least-l is ALFA, < L1 gt(M,R) | A > |- GAMMA ] .
rl [least-r] :
[ X from Y by Q is ALFA |- < L1 | AT-LEAST(M, R, A) >, GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < L1 | AT-LEAST(M, R, A) >, GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'least-r is ALFA |- < L1 gt(M,R) | A >, GAMMA ] .
rl [most-l] :
[ X from Y by Q is ALFA, < L1 | AT-MOST(M, R, A) > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L1 | AT-MOST(M, R, A) > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'most-l is ALFA, < L1 lt(M,R) | A > |- GAMMA ] .
rl [most-r] :
[ X from Y by Q is ALFA |- < L1 | AT-MOST(M, R, A) >, GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < L1 | AT-MOST(M, R, A) >, GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'most-r is ALFA |- < L1 lt(M,R) | A >, GAMMA ] .
crl [neg-l] :
[ X from Y by Q is ALFA, < L1 | ~ A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L1 | ~ A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'neg-l is ALFA |- GAMMA, < neg(L1) | A > ]
if not has-quant(L1) .
crl [neg-r] :
[ X from Y by Q is ALFA |- GAMMA, < L1 | ~ A > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- GAMMA, < L1 | ~ A > ] next(N + 1) goals((XS, N))
[ N from X by 'neg-r is ALFA, < neg(L1) | A > |- GAMMA ]
if not has-quant(L1) .
vars m n : Nat .
rl [quant-exist-l] :
[ X from Y by Q is ALFA, < ex(R) L1 | A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < ex(R) L1 | A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'quant-exist-l is ALFA, < gt(1,R) L1 | A > |- GAMMA ] .
--- or gt(n,R) where n >= 1
rl [quant-exist-r] :
[ X from Y by Q is ALFA |- < ex(R) L1 | A >, GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < ex(R) L1 | A >, GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'quant-exist-r is ALFA |- < gt(1,R) L1 | A >, GAMMA ] .
crl [exist-quant-l] :
[ X from Y by Q is ALFA, < gt(n,R) L1 | A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < gt(n,R) L1 | A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'exist-quant-l is ALFA, < ex(R) L1 | A > |- GAMMA ]
if n >= 1 .
rl [exist-quant-r] :
[ X from Y by Q is ALFA |- < gt(1,R) L1 | A >, GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < gt(1,R) L1 | A >, GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'exist-quant-r is ALFA |- < ex(R) L1 | A >, GAMMA ] .
crl [quant-gt-l] :
[ X from Y by Q is ALFA, < gt(m,R) L1 | A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < gt(m,R) L1 | A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'quant-qt-r is ALFA, < gt((m - 1), R) L1 | A > |- GAMMA ]
if m >= 1 .
rl [quant-gt-r] :
[ X from Y by Q is ALFA |- < gt(m,R) L1 | A >, GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < gt(m,R) L1 | A >, GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'quant-qt-r is ALFA |- < gt((m + 1), R) L1 | A >, GAMMA ] .
rl [quant-lt-l] :
[ X from Y by Q is ALFA, < lt(m,R) L1 | A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < lt(m,R) L1 | A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'quant-lt-r is ALFA, < lt((m + 1), R) L1 | A > |- GAMMA ] .
crl [quant-lt-r] :
[ X from Y by Q is ALFA |- < lt(m,R) L1 | A >, GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < lt(m,R) L1 | A >, GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'quant-lt-r is ALFA |- < lt((m - 1), R) L1 | A >, GAMMA ]
if m >= 1 .
crl [and-r] :
[ X from Y by Q is ALFA |- GAMMA, < L | A & B > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- GAMMA, < L | A & B > ] next(N + 2) goals((XS, N, N + 1))
[ N from X by 'and-r is ALFA |- GAMMA, < L | A > ]
[ N + 1 from X by 'and-r is ALFA |- GAMMA, < L | B > ]
if not has-ex(L) /\ not has-gt(L) .
crl [and-l] :
[ X from Y by Q is ALFA, < L | A & B > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L | A & B > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'and-l is ALFA, < L | A >, < L | B > |- GAMMA ]
if not has-ex(L) /\ not has-lt(L) .
crl [or-l] :
[ X from Y by Q is ALFA, < L | (A | B) > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA, < L | (A | B) > |- GAMMA ] next(N + 2) goals((XS, N, N + 1))
[ N from X by 'or-l is ALFA, < L | A > |- GAMMA ]
[ N + 1 from X by 'or-l is ALFA, < L | B > |- GAMMA ]
if not has-al(L) /\ not has-gt(L) .
crl [or-r] :
[ X from Y by Q is ALFA |- GAMMA, < L | (A | B) > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- GAMMA, < L | (A | B) > ] next(N + 1) goals((XS, N))
[ N from X by 'or-r is ALFA |- GAMMA, < L | A >, < L | B > ]
if not has-al(L) /\ not has-lt(L) .
op all-label : Set{Expression} Label -> Bool .
op remove-label : Set{Expression} Label Set{Expression} -> Set{Expression} .
vars ALFA' GAMMA' : Set{Expression} .
var LBL : Label .
eq all-label((< LBL L | A >, GAMMA), LBL) =
all-label(GAMMA, LBL) .
eq all-label(empty, LBL) = true .
eq all-label(GAMMA, LBL) = false [owise] .
eq remove-label((< LBL L | A >, GAMMA), LBL, GAMMA') =
remove-label(GAMMA, LBL, (GAMMA', < L | A >)) .
eq remove-label(empty, LBL, GAMMA) = GAMMA .
crl [prom-exist] :
[ X from Y by Q is < ex(R) L | A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is < ex(R) L | A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'prom-exist is < L | A > |- GAMMA' ]
if all-label(GAMMA, ex(R)) = true /\
GAMMA' := remove-label(GAMMA, ex(R), empty) .
crl [prom-all] :
[ X from Y by Q is ALFA |- < al(R) L | A > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < al(R) L | A > ] next(N + 1) goals((XS, N))
[ N from X by 'prom-all is ALFA' |- < L | A > ]
if all-label(ALFA, al(R)) = true /\
ALFA' := remove-label(ALFA, al(R), empty) .
crl [prom-gt-l] :
[ X from Y by Q is < gt(n,R) L | A > |- GAMMA ] next(N) goals((X, XS)) =>
[ X from Y by Q is < gt(n,R) L | A > |- GAMMA ] next(N + 1) goals((XS, N))
[ N from X by 'prom-gt-l is < L | A > |- GAMMA' ]
if all-label(GAMMA, gt(n,R)) = true /\
GAMMA' := remove-label(GAMMA, gt(n,R), empty) .
crl [prom-gt-r] :
[ X from Y by Q is ALFA |- < gt(n,R) L | A > ] next(N) goals((X, XS)) =>
[ X from Y by Q is ALFA |- < gt(n,R) L | A > ] next(N + 1) goals((XS, N))
[ N from X by 'prom-gt-r is ALFA' |- < L | A > ]
if all-label(ALFA, gt(n,R)) = true /\
ALFA' := remove-label(ALFA, gt(n,R), empty) .
endm)