Assertion Answer Set Logic
Program and its Application
Mingyi Zhang
Guizhou Academy of Science
2012-6-18
Motivation
Motivation
(1) Lack of intuition
Noncumulativity
• (Brewka,1991; Giorddano and Martelli,1994)
Cause :
Inconsistency between justification of defaults
( does not commit to assumptions)
Example 1 Broken-arm (Pool, 1988)
:Useable(x)¬Broken(x)/Useable(x) (1)
Broken(Leftarm)Broken(Rithtarm)
E=Th(Broken({Leftarm)Broken(Rithtarm), Useable(Leftarm),
Useable(Leftarm)})
Motivation
Adding the information
x.BROKEN(x) USABLE(x) , (3)
will generate two extensions as desired. This is a type of "hard"
exceptions to a default, i.e. exceptions for which the negation of
the default's consequent can be proven. In this case (1) can be
replaced by
:Useable(x)/Useable(x)
But we hope to consider a weaker type of exception to a
default: we want to block a default but we don't assert the
negation of its consequent. Nonnormal defaults are needed
precisely to express this weaker type of exceptions.
Motivation
Example 2 (Brewka, 1991)
DOG  BIRD PET
DOG¬BIRD
PET:DOG/DOG
SINGS:BIRD/BIRD
S1NGS .
E=Th(W{BIRD)}) which contains PET
Adding PET to the DT, we get another extension which
contains DOG.
Motivation
(2) Destroy part of the additional expressiveness of nonnormal
defaults
CDL is cumulative and commits to assumptions. However, CDL
is in the style of Lukaszewicz's default logic. It happens to be
semimonotonic, which destroys part of the additional
expressiveness of nonnormal defaults, for instance, it makes it
impossible to represent priorities among defaults.
Motivation
Example 3 (Giorddano and Martelli,1994)
student :¬ married/¬married (1)
adult : married/married
(2)
living-in-college : student/student (3)
beard : adult’adult
(4)
living-in-college
beard
We want default (1) to have a higher priority than default (2).
To achieve this purpose, default (2) can be replaced with the
seminormal default:
adult : married¬student/) married (2’)
Motivation
The resulting theory has a single DL extension, Th({living-in-college,
beard, student, adult, ¬married}), as wanted, since applying default
(3) blocks (2'). This same theory, however, has two CDL extensions:
F 1 containing (adult : (adult}), (married : (adult, married¬student} )
and:
F 2 containing (student: {student}), (adult : {adult}),
(¬married :{student,¬married}),
since in CDL (3) blocks (2'), but (2') blocks (3) in turn. Hence, in CDL,
using seminormal defaults does not allow to enforce priorities among
default rules (1) and (2), as in Reiter's default logic.
Motivation
To do this, Giorddano and Martelli introduced CADL (Commitment
to Assumption Default Logic) and QDL(Quasi-Default Logic).
Guaranteeing cumulativity of QDL, they imposed a condition on
definition of QDL extension. Zhang (1999) gave the definition similar
to Reiter’s one, algorithms for reasoning task for CADL and QDL, and
a sufficient and necessary condition for cumulitiviy of QDL. (2010)
Motivation
(3) Keep trace of an inference supporting a conclusion
Abduction
AB, B
--------------A
I. Proof in Geometry
II. Diagnosis based on consistency (Reiter 1980)
Motivation
A system is a pair (SD, COMPS), where
SD, the system description: a set of first-order sentences
COMPS, the system components: a finite set of constants.
An observation of a system is a finite set of first-order sentences
(SD, COMPS, OBS) ---- a system (SD, COMPS) with observation OBS.
A diagnosis for (SD, COMPS, OBS) is a minimal set COMPS such
that
SD  OBS  {AB(c)  c }  {AB(c)  c  COMPS}
is consistent.
Motivation
Proposition 1 (Zhang mingyi et al, 2004)
Let (SD,Comps) be a system and OBS an observation. Given
COMPS), if SDOBS{AB(c) cCOPMPS-} is consist, and for
any ci , is a diagnosis for any ci, SD  OBS  {AB(c)c
COMPS}╞ AB(ci), then  is a diagnosis for (SD, COMPS, OBS).
Proposition 2 (Reiter, 1987) ---- Relation between Diagnosis and
premise-free normal DL
Consider a system (SD, COMPONENTS) under observation OBS,
where so and OBS are sets of first-order sentences. Then E is an
extension for the default theor yDT =( {:¬ AB(c)/ AB(c) c COMPS},
SD  OBS) iff for some diagnosis  for (SD, COMPONENTS, OBS), E =
{  predicts II}.
Assertion ASP
Assertion ASP
(1) ASP--– a special case of DL
Assertion ASP---a special case of QDL in Reiter’s sense
Definition 1 An assertion is an expression of the form <L:S>,
where L and S are a literal and a set of literals respectively. For
any set of assertions W, we denote:
Form(W)={L| <L:S> }, the set of literals of W.
Supp(W)= , the support of W.
In particular, Form(A)={L} and Supp(A)=S are the literal and
support of A, for an assertion A=<L:S>.
Assertion ASP
Definition 2 A rule is an expression of the form L0←L1,L2, ...,Lm,
not Lm+1, …, not Ln(m+n>0), where L0, Li (1≤i≤n) are literals
respectively. For a set of rules R, we use the following notations:
Head(R)={L0|L0←L1,L2, ...,Lm, not Lm+1, …, not LnR(m+n>0)}
Pos(R)= {Li|L0←L1,L2, ...,Lm, not Lm+1, …, not LnR,1≤i≤m,(m+n>0)}
Neg(R)= {Li|L0←L1,L2, ...,Lm, not Lm+1, …, not LnR,m+1≤i≤n,(m+n>0)}
Write a rule as HedPos, not Neg. A rule r is basic if Neg(r)=∅ .
Definition 3 Π=W∪R is an assertion answer set program (AASP)
if W is a set of assertions and R is a set of rules. Π=W∪R is basic if
R is a set of basic rules. Π is finite iff W and R both are finite.
Assertion ASP
Definition 4 Let Π=W∪R be an AALP. A set of assertions E is an
answer set of if E=∪0≤iEi , Where
E0=W if Form(W) is consistent, otherwise {<Lit:>},
Ei+1=Ei∪Fi if Form( ) is consistent, otherwise {<Lit:>}.
where
Fi={<L0:∪1≤j≤mSj>|L0←L1, L2, …, Lm∈R and <Li:Sj>∈Ei for 1≤ j≤m}.
Definition 5 Let =WR be an AASP and F a set of assertions . The
F-reduct F is an AASP obtained by removing each rR if Neg(r)
Form(F) .
Assertion ASP
Definition 6 Let E is an answer set of Π=W∪R . The set of
generating rules for E, GD(E,Π), is the set
{L0←L1,L2, ...,Lm, not Lm+1, …, not Ln∈R,(m+n>0)|for i:1≤i ≤
m,<Li:Si>∈E and for j:m+1≤j≤n,Lj∉Form(E)}.
Definition 7 Let =W∪R be an AASP. For any R,. we define
an operator : (R,)=∪n0 n(R,), as follows:
{ Head Pos, not Neg R  Form(W)∪{Head} is consistent},
0(R,)= {<Lit:>}, otherwise
{HeadPos, not NegR  Pos Form(W)∪Head(n(R,) )
and Pos Form(W)∪Head(n(R,)) is consistent
Tn+1(R,)= {<L: >LLit} otherwise
Assertion ASP
Definition 8 Let =WR be an AASP. RR is compatible (w.r.t. ),
if , for any LNeg(R), LForm(W)Head(R). R’ is strongly
compatible if R’ is compatible and (R , )=R. R’ is -robust (or
just robust, if A is understood), if R’ is strongly compatible and, for
any R R, if R is strongly compatible, then R=R.
Corollary 1  has an inconsistent answer set iff
Form(W)Head((R*,)) is inconsistent, where
R*={rRNeg(r)=.
Corollary 2 E={<Lit,>} is an answer set of  iff
Form(W)Head((R*,)) is inconsistent.
Corollary 3 Suppose E is an answer set of . Then
Form(E) =Form(W)Head(GR(E,)) ,
Supp(E)=Supp(W)Neg(GR(E, ))
Assertion ASP
Theorem 1. (Characterization of assertional answer sets) Let
=WR be an AASP.  has an answer set iff there is a strongly
compatible subset R of R such that, for any rRR, Pos(r)
⊈Form(W)Head(R) or (Form(W)Head()) Neg(r) .
Corollary 4 If R is compatible then =W∪R has a unique answer
set.
Assertion ASP
Based on the characterization we can give an algorithm for
computing answer sets and its complexity. Now we deal with the
following problem, which we call the membership problem: Given
an AASP , a robust set ' and an assertion <p:S>, determine
whether <p:S> is an element of the answer set generated by
'.The main reasoning tasks in AASP are skeptical reasoning (SR)
and credulous reasoning.
• SR: Decide whether a given assertion <p :S> occurs in all
assertional answer sets of .
• CR: Decide whether a given assertion <p:S> occurs in some
assertional answer set of .
We derive some formal results that will allow us to deal properly
with supports and to solve the membership problem.
Assertion ASP
Definition 6 Let S be a support,  an AASP, and F a set of
assertions. Then
[S] = {r Neg(r)S}, F[S] = {<L:V>F VS}
Lemma 1 Let  be an AASP. If R is compatible, then it has a unique
answer set E and GD(E, ) = (R,).
Lemma 2 Let E be an answer set of a finite AASP  and < L:S> an
assertion in E. Then the assertion <L:Supp(E)> is also an element of
E.
From the definition of AASP extension, one may be tempted to
think that reasoning in AASP requires exponential space, due to
the combinatorial explosion of the supports in the intermediately
generated assertions. However, in this paper we show that AASP
has the same complexity as ASP.
Assertion ASP
Theorem 2 Let  be a finite AASP, ' a -robust set of rules, and
<L:S> an assertion . Let S='[S]. Then <L:S> is an element of the
answer set E' of  generated by ' iff the following two conditions
(1) and (2) are satisfied:
(1) LHead(('[S],S));
(2) Neg(('[S],S)) = S.
Theorem 3 Let  be a finite AASP and let <p:S> be an assertion and
S.
(1) If Head((B)) is inconsistent, then <p:S> is not an element of
any assertional answer set of ..
(2) If Head((B)) is consistent, the AASP S has a unique answer
set E with set of generating rules GD(E) =(S). The assertion
<p:S> is an element of some extension of  iff <p:S>E.
Assertion ASP
Lemma 3. Let  be a finite AASP and <p:S> an assertion such that
Head((B)) is consistent. Then, <p:S> belongs to some answer set
of  iff for every , (S)S , it holds that
(1) <p:S>Head(), and
(2) Neg() = R, where S='[S].
Theorem 4. In propositional AASP the reasoning tasks SR is NPomplete, while the reasoning task CR is DP-complete.
Theorem 5 AASP is cumulative.
Assertion ASP
Splitting (Wu Maonian and Zhang Mingyi)
Based on the Finest Splitting of Belief Set and the
characterization of AASP answer sets we explored a method, which
constructs a reduce without guessing a set of assertions.
Applications
Application
(1) Alias analysis (Yang Bo and Zhang Ming)
Find a well tradeoff between precision and computation cost of
alias analysis for some programming languages. Alias in Java
program is one of the main causes for program errors.
Applying AASP can determine the propagation path of points-to
information more accurately, and hence improve the precision of
points-to analysis.
Application
ASP Rules for Points-to Analysis
ref(x,o,cs,s')←pred(s,s'), ref(x,o,cs,s), not ¬ref(x,o,cs,s')
(r1)
reff(o,f,o',cs,s')←pred(s,s'), reff(o,f,o',cs,s), not ¬reff(o,f,o',cs,s')
(r2)
ref(x,os,cs,s)←new(s,os,x), call(m,c,cs,cs'), in(s,c,m), not inbranch(s)
(r3)
ref(x,os,cs,s)←new(s,os,x), call(m,c,cs,cs'), in(s,c,m), pred(s1,s2), if(s1),
branch(s2,s1), inbranch(s), same(s2,s) (r4)
¬ref(x,o',cs,s')←pred(s,s'), new(s',o,x), ref(x,o',cs,s)
(r5)
ref(x,oy,cs,s')←pred(s,s'), ref(y,oy,cs,s), asn(s',x,y)
(r6)
¬ref(x,ox,cs,s')← pred(s,s'), ref(x,ox,cs,s), asn(s',x,y), not sp(s')
(r7)
reff(ox,f,oy,cs,s')←pred(s,s'), ref(y,oy,cs,s), ref(x,ox,cs,s), store(s',x,f,y)
(r8)
¬reff(ox,f,o,cs,s')←pred(s,s'), ref(x,ox,cs,s), reff(ox,f,o,cs,s),
store(s',x,f,y), not sp(s')
(r9)
ref(y,o,cs,s')←pred(s,s'), ref(x,ox,cs,s), reff(ox,f,o,cs,s), load(s',y,x,f)
(r10)
¬ref(y,oy,cs,s')←pred(s,s'), ref(y,oy,cs,s), load(s',y,x,f), not sp(s')
(r11)
Application
sp(s')← pred(s,s'), ref(x,ox,cs,s), ref(y,oy,cs,s), ox=oy , asn(s',x,y)
sp(s')←pred(s,s'), ref(x,ox,cs,s), reff(ox,f,o,cs,s), ref(y,oy,cs,s),
o=oy, store(s',x,f,y)
sp(s')←pred(s,s'), ref(y,oy,cs,s), ref(x,ox,cs,s), reff(ox,f,o,cs,s),
oy=o, load(s',y,x,f)
1{pred(s,s'):branch(s',s)}1←if(s)
(r12)
(r13)
(r14)
(r15)
Applications
Definition 9 Let P is a program and R a set of rules. R is sound w.r.t.
a point—to relation of P if, for any statement StP with
corresponding node s in a control flowchart such that <σ, σ′>
∈PT[St],
1) for any reference x occurring in P, if σ′ x=ox，then ref(x,ox,cs,s)
∈Rule(s)；
2) for any reference-type variable with a form x.f, if σ′ x=ox and σ′
ox.f=o，then ref(x,ox,cs,s)∈Rule(s) and reff(ox,f,o,cs,s)∈Rule(s),
where PT：Stm→(PS↪PS) is a point-to semantic relation, PS is the
set of all point-to states and Stm is a set of statements formed by
Assignment statement , Skip, Load, Store, Sequential Compound
statement, Conditional statement and While statement.
Applications
Theorem 3 The set R of rules is sound w.r.t. inference of points-to
relation on a program.
For a diagnosis system (SD, COMPONENTS,OBS)，we call =*
∪ C∏ the program corresponding to the diagnosis system, where
*={ L0←L1,…, Lm L0L1…Lm CLOBS, m0}R and C∏=
{←L1,…, Lm L1…LmCLOBS, m1}.
Definition 10 Given a diagnosis system DIS=(SD, COMPONENTS,
OBS) and its corresponding standard program =* ∪ C∏ ,  a
diagnosis for DIS if there is an answer set S of  such that
  COMPONENT  N(GD(S ,  )
Applications
From Theorem 1 we have
Theorem 4  is a diagnosis for DIS= (SD, COMPONENTS, OBS) iff
 is a minimal subset of COMPONENTS such that
I. ()=, where ={r*H(r)COMPONENTS-  } and  is
a standard program corresponding to DIS,
II. CLOBS N( ' ) ) is consistent.
Applications
Deal (Chen Wu, Zhang Dongmo and Zhang Mingyi)
Definition 11 A bargaining game is a pair ((1, ≿ 1),(2, ≿ 2)),
where i (i=1,2) is an AASP and ≿i is a complete transitive and
reflexive order (total preorder or weak order) over i that
satisfies the following condition: for r, r1,…rn i,
If Pos(r) and Pos(r)Head({r1,…rn}) then there is k (1kn)
such that rk ≿ir.
The pair (i, ≿i i) is called the prioritized demand of player i. For
any r, ti, r≻i t denotes that r ≿i t and t≵i r. r≈it denotes r≿i t
and t≿i r.
Clearly, ≈i is an equivalent relation on i, which uniquely
determines a partition of i. We use ≈i to give a hierarchy of i,
that is, each equivalent class is viewed a level of the hierarchy.
Applications
Definition 12 The sets of assertions E1 and E2 are co-consistent if
both Form(Ei)Form(E-i) and Form(Ei)SuppE-i) are consistent,
where –i=3-i.
Definition 13 Given a prioritized demand set (,≿) for an agent,
its hierarchy {(j)}j1 is defined as follows:
1={rt (r≿t)}; 1=1. for k1
k+1={rktk (r≿t)}; k+1=k k+1.
It is easy to see that 1={ rPos(r)=}. In the sequent, we
write k to denote {(j)}1jk..
Applications
Definition 14 Let G=((1, ≿ 1),(2, ≿ 2)) be a bargaining game, where i= WiRi
(i=1,2). A deal of G is a pair (D1,D2) satisfying the following conditions: for each
i=1,2
1.W1W2  Di.
2.Di =Form(Ei), where Ei is an answer set of i=(Vi,Ti) such that Ti =GD(Ei, i).
3.Vi is a maximal subset of Wi s.t. Vi and E-I is co-consistent.
4.for each k(k=1,2…), Ti Rik is a maximal subset of Rik in the following
sense:
(1) Vi  1jk {<L:S>Ei LHead(Ti Rij ), Neg((Ti Rij))} and E-i are coconsistent .
(2) There is no ri Rik- Ti such that
I. ik=(Vi, 1jkTi {ri})) has an answer set F(I,k) with GD(F(I,k), ik)= 1jk
(TiRik ){ri}.
II. Vi  1jk {<L:S>Ei LHead((Ti Rij){r}), Neg((Ti Rij){r}))} and E-i are
co- consistent.
where {Rij}j1 is the hierarchy of Ri. The set of all deals of G is denoted by (G),
called the feasible set of the game.
Applications
Definition 15 i, Gi as before (i=1,2). A pair of sets of literals
Oi=( Di, Pi) implements gi* Gi, where
DiB+(i){ gi }, PiB+(-i){ g-i }, giGi , g-iG-i (B+
represents the positive body of a program)
if
1. SiANS(i{l l DiPi} s.t. gi* Si, i=1,2
2.gi* g-i* is consistent, i=1,2
3.For i=1,2, gi* are maximal goals satisfying conditions 1 and 2,
i.e. there are no gi’ Gi s.t. conditions 1 and 2 hold for gi’ and s.t.
gi’ Si and gi*i gi’.
(3) Approach for Handling Conflicts in Authorization (Zhang
Mingsheng and Zhang Mingyi)