Add examples
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10
AST.py
10
AST.py
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@ -31,9 +31,9 @@ class LRule:
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nbody: tuple[atom, ...]
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def text(self):
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head = ",".join(self.head)
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head = ", ".join(self.head)
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nbody = tuple(f"not {atom}" for atom in self.nbody)
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body = ",".join(self.pbody + nbody)
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body = ", ".join(self.pbody + nbody)
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return f"{head} :- {body}."
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@ -62,7 +62,7 @@ class OBinary(OFormula):
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right: OFormula
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def text(self):
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return f"{self.left.text()} {self.operator} {self.right.text()}"
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return f"({self.left.text()} {self.operator} {self.right.text()})"
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@dataclass
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@ -100,8 +100,8 @@ class HMKNFVisitor(ParseTreeVisitor):
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self.oatoms = set()
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def visitKb(self, ctx: HMKNFParser.KbContext):
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orules = (self.visit(orule) for orule in ctx.orule()) if ctx.orule() else ()
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ont = reduce(partial(OBinary, "&"), orules, OConst.TRUE)
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orules = (self.visit(orule) for orule in ctx.orule()) if ctx.orule() else (OConst.TRUE,)
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ont = reduce(partial(OBinary, "&"), orules)
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lrules = tuple(self.visit(lrule) for lrule in ctx.lrule())
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return KB(ont, lrules, Set(self.katoms), Set(self.oatoms))
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24
aft.py
24
aft.py
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@ -10,7 +10,7 @@ usage: python aft.py < knowledge_bases/simple.hmknf
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from sys import stdin, flags, argv
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from AST import KB, Set, atom, loadProgram
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from hmknf import objective_knowledge
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from util import printp
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from util import format_set, printp, textp
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Kinterp = tuple[Set[atom], Set[atom]]
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@ -21,14 +21,14 @@ def add_immediate_XY(kb: KB, X: Set[atom], Y: Set[atom]) -> Set[atom]:
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When they are flipped, i.e. X = P, and Y = T, then it computes what is "possibly true"
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This function is monotone w.r.t. the precision ordering
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"""
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_, X, _ = objective_knowledge(kb, X, Set())
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_, newX, _ = objective_knowledge(kb, X, Set())
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for rule in kb.rules:
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if not X.issuperset(rule.pbody):
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continue
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if Y.intersection(rule.nbody):
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continue
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X = X.union({rule.head[0]})
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return X
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newX = newX.union({rule.head[0]})
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return newX
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def extract_OBT_entails_false(kb: KB, T: Set[atom]):
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@ -59,11 +59,15 @@ def fixpoint(op, initial):
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return prev
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def stable_revision(kb: KB, T: Set[atom], P: Set[atom]):
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def stable_revision(kb: KB, T: Set[atom], P: Set[atom], debug=False):
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def left(T):
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if debug:
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print("left:", format_set(T))
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return add_immediate_XY(kb, T, P)
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def right(P):
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if debug:
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print("right:", format_set(P))
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return add_immediate_XY(kb, P, T) - extract_OBT_entails_false(kb, T)
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return (
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@ -73,9 +77,14 @@ def stable_revision(kb: KB, T: Set[atom], P: Set[atom]):
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def stable_revision_extend(kb: KB, initialT: Set[atom], initialP: Set[atom]):
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i = -1
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def stable_revision_tuple(TP: Kinterp) -> Kinterp:
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nonlocal i
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i += 1
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T, P = TP
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return stable_revision(kb, T, P)
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print(f"{i}:", textp(T, P))
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return stable_revision(kb, T, P, True)
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return fixpoint(stable_revision_tuple, (initialT, initialP))
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@ -83,7 +92,6 @@ def stable_revision_extend(kb: KB, initialT: Set[atom], initialP: Set[atom]):
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def least_stable_fixedpoint(kb: KB):
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return stable_revision_extend(kb, Set(), kb.katoms)
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def main():
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""""""
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if len(argv) > 1:
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@ -92,7 +100,7 @@ def main():
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in_file = stdin
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kb = loadProgram(in_file.read())
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T, P = least_stable_fixedpoint(kb)
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printp(T, P)
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print("fixpoint:", textp(T, P))
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if __name__ == "__main__" and not flags.interactive:
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@ -1,4 +1,4 @@
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This is BibTeX, Version 0.99d (TeX Live 2022/dev/Debian)
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This is BibTeX, Version 0.99d (TeX Live 2022/Debian)
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Capacity: max_strings=200000, hash_size=200000, hash_prime=170003
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The top-level auxiliary file: document.aux
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The style file: plain.bst
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@ -0,0 +1,7 @@
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\BOOKMARK [1][-]{section.1}{\376\377\000A\000n\000\040\000A\000p\000p\000r\000o\000x\000i\000m\000a\000t\000o\000r\000\040\000H\000y\000b\000r\000i\000d\000\040\000M\000K\000N\000F\000\040\000k\000n\000o\000w\000l\000e\000d\000g\000e\000\040\000b\000a\000s\000e\000s}{}% 1
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\BOOKMARK [1][-]{appendix.A}{\376\377\000L\000a\000t\000t\000i\000c\000e\000\040\000T\000h\000e\000o\000r\000y}{}% 2
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\BOOKMARK [1][-]{appendix.B}{\376\377\000P\000a\000r\000t\000i\000a\000l\000\040\000S\000t\000a\000b\000l\000e\000\040\000M\000o\000d\000e\000l\000\040\000S\000e\000m\000a\000n\000t\000i\000c\000s}{}% 3
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\BOOKMARK [2][-]{subsection.B.1}{\376\377\000F\000i\000x\000p\000o\000i\000n\000t\000\040\000O\000p\000e\000r\000a\000t\000o\000r\000s}{appendix.B}% 4
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\BOOKMARK [1][-]{appendix.C}{\376\377\000A\000p\000p\000r\000o\000x\000i\000m\000a\000t\000i\000o\000n\000\040\000F\000i\000x\000p\000o\000i\000n\000t\000\040\000T\000h\000e\000o\000r\000y}{}% 5
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\BOOKMARK [1][-]{appendix.D}{\376\377\000H\000y\000b\000r\000i\000d\000\040\000M\000K\000N\000F\000\040\000K\000n\000o\000w\000l\000e\000d\000g\000e\000\040\000B\000a\000s\000e\000s}{}% 6
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\BOOKMARK [2][-]{subsection.D.1}{\376\377\000A\000n\000\040\000E\000x\000a\000m\000p\000l\000e\000\040\000K\000n\000o\000w\000l\000e\000d\000g\000e\000\040\000B\000a\000s\000e}{appendix.D}% 7
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@ -1,6 +1,7 @@
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\documentclass{article}
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\usepackage{natbib}
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\usepackage{hyperref}
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\input{common}
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@ -14,6 +15,20 @@
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\section{An Approximator Hybrid MKNF knowledge bases}
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A program $\P$ is a set of (normal) rules. An ontology $\OO$ is a first-order formula.
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\begin{definition}
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Given a set of atoms $S$ and an atom $a$. We use $\OBO{S}$ to denote the first-order formula obtained by extending $\OO$ with every atom from $S$ fixed to be true. The relation
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\begin{align*}
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\OBO{S} \models a
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\end{align*}
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Holds if either
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\begin{itemize}
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\item there is no satisfying assignment for $\OBO{S}$ (The principle of explosion)
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\item for every satisfying assignment for $\OBO{S}$, $a$ is true, $\OBO{S}$ is logically equivalent to $\OBO{S \union \{ a\}}$
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\end{itemize}
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Holds if when we fix every atom in $S$ to be true with
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\end{definition}
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The lattice $\langle \L^2, \preceq_p \rangle$
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\begin{align*}
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@ -26,6 +41,16 @@ The lattice $\langle \L^2, \preceq_p \rangle$
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S(\Phi)(T, P) \define \Bigl( \lfp~\Gamma(\cdot, P),~ \lfp\,\Bigl( \Gamma(\cdot, T) \setminus extract(T) \Bigr) \Bigr)
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\end{align*}
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\begin{definition}
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A (hybrid MKNF) knowledge base $\KB = (\OO, \P)$ is a program $\P$ and an ontology $\OO$\footnote{See \Section{section-prelim-hmknf} for a formal definition}
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An interpretation $(T, P)$\footnote{See \Section{psms} for definition of interpretations} is an MKNF model of $\KB$ if
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\begin{itemize}
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\item $T \subseteq P$
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\item $\OBO{P}$ is consistent, i.e., $\OBO{P} \not\models \bot$ (It follows that $\OBO{T} \not\models \bot$)
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\item All the rules are satisfied, $\lfp~{\Gamma(\cdot, T)} = P$.
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\end{itemize}
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\end{definition}
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\appendix
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\section{Lattice Theory}
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@ -59,7 +84,7 @@ The lattice $\langle \L^2, \preceq_p \rangle$
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\end{definition}
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\section{Partial Stable Model Semantics}
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\section{Partial Stable Model Semantics}\label{psms}
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\begin{definition}
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A (ground and normal) {\em answer set program} $\P$ is a set of rules where each rule $r$ is of the form
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@ -1,3 +0,0 @@
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-b.
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a :- not b.
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b :- not a.
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@ -1,4 +0,0 @@
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:- c.
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:- b.
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:- a.
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a.
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@ -0,0 +1,8 @@
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# Simple two valued model
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# oatoms vs katoms
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(-a | c) & d.
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a :- not b.
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c :- c.
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@ -0,0 +1,4 @@
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# KB has a WFM and 2, 2-valued models
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a :- not b.
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b :- not a.
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@ -0,0 +1,6 @@
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# KB has no WFM
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-a | -b.
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a :- not b.
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b :- not a.
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@ -0,0 +1,8 @@
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# Interleaved positive inferences with ontology and program
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(a -> b) & (d -> e).
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a.
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c :- b.
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d :- c.
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@ -0,0 +1,10 @@
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# Generate some undefined atoms
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a :- not a'.
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a' :- not a.
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:- a'.
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b.
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b :- a.
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@ -0,0 +1,5 @@
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# Ontology collapses "choice"
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-b.
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a :- not b.
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b :- not a.
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@ -14,6 +14,7 @@ from more_itertools import peekable
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from AST import loadProgram, atom, Set, KB
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from aft import Kinterp, add_immediate_XY, fixpoint, stable_revision
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from hmknf import oformula_sat
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from util import printp, powerset
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@ -34,7 +35,7 @@ def verify(kb: KB, T: Set[atom], P: Set[atom]) -> bool:
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def check_rules(P: Set[atom]):
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return add_immediate_XY(kb, P, T)
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return fixpoint(check_rules, Set()) == P
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return oformula_sat(kb.ont, P) and fixpoint(check_rules, Set()) == P
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def main():
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9
util.py
9
util.py
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@ -2,6 +2,7 @@ from functools import wraps
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from AST import Set, atom
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import more_itertools
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def powerset(items):
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return map(Set, more_itertools.powerset(items))
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@ -10,12 +11,16 @@ def format_set(s: Set[atom]):
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return "{" + ", ".join(sorted(s)) + "}"
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def printp(*args):
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def textp(*args):
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whole = "("
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sets = (format_set(arg) for arg in args)
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whole += ", ".join(sets)
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whole += ")"
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print(whole)
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return whole
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def printp(*args):
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print(textp(*args))
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def prints_input(*pos):
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