5. Delarative Diagnosis in the CLP sheme
Alexandre Tessier and Gerard Ferrand
LIFO, BP 6759, F-45067 Orleans Cedex 2, Frane
: fAlexandre.Tessier, Gerard.Ferrandginria.fr
email
When a result is omputed but it is onsidered as inorret beause it is not
expeted, we onsider that we have a symptom (of error). The symptom may
be a wrong answer or a missing answer. The role of diagnosis is to loate
an error, that is a limited program fragment responsible for the symptom.
The notions of symptom and error have a meaning only w.r.t. some notion of
expeted semantis. We onsider only delarative semantis. The user does not
need to understand the operational behaviour of the CLP system. Symptom
and error are onneted via some kind of tree and the diagnosis amounts to
searh for a kind of minimal symptom in this tree. Several searh strategies are
possible. The priniples of an implementation are desribed, with a diagnosis
session.
5.1 Introdution
To intuitively introdue the basi notions of symptom and error let us onsider
a toy program in the paradigm lp(FD) :
fa(N,F) :- B#=1, aux(N,B,F).
aux(N,B,P) :- N#=0, B#=P.
aux(N,B,P) :- N#=M+1, C#=N, aux(M,C,P).
with an expeted semantis suh as:
{ f a is the fatorial funtion, that is f a(N; F ) , F = N !
{ and aux(N; B; P ) , P = N ! B .
So the last lause should be:
aux(N,B,P) :- N#=M+1, C#=N*B, aux(M,C,P).
For the goal: N#=<2, fa(N,F) we have three omputed answer onstraints whih are
F = 1, N = 0.
F = 1, N = 1.
F = 1, N = 2.
A goal is the desription of a relation. Here the expeted relation is:
= 1; N = 0 or F = 1; N = 1 or F = 2; N = 2. So this omputed result
is onsidered as inorret beause it is not expeted. Let us onsider that we
have a symptom (of error). But there are two ways to understand that there
is a symptom:
F
132
A. Tessier and G. Ferrand
Firstly we have a wrong answer, F = 1, N = 2.
To be more formal, a way to express that it is a wrong answer is to say that,
with respet to the expeted semantis of the program, the following logi
formula (of the form: omputed answer onstraint ) goal ) is false :
F
=1^N =2)N
62^
(
f a N; F
)
It is a rst kind of symptom.
Seondly we have a missing answer, F = 2, N = 2.
Unlike the rst kind of symptom, to be able to say that it is a missing
answer we have to take all the omputed answer onstraints into aount: It
is missing beause among these omputed answer onstraints we do not see
F = 2, N = 2.
To be more formal, a way to express that it is a missing answer is to say
that, with respet to the expeted semantis of the program, the following
logi formula is false :
N
62^
(
f a N; F
) ) (F = 1 ^ N = 0) _ (F = 1 ^ N = 1) _
(F = 1 ^ N = 2)
(sine with respet to the expeted semantis, for F = 2; N = 2, the left hand
side is true but the right hand side is false). It is a seond kind of symptom.
So we have two kinds of symptoms: for a question (that is a goal) it is
possible to have several omputed answers C1 ; ; Ci ; so we may be
interested either in a single answer or in the sequene of all the answers. We
onsider that there are two levels of omputation and two kinds of results. At
eah level the omputation is nite : At a rst level a result is a single Ci . If
Ci is a wrong answer we have a symptom of the rst kind. At a seond level
of omputation it is the sequene C1 ; ; Ci ; whih is the result. To be
more preise, sine it is a nite omputation, it is a sequene: C1 ; ; Cn
(terminated by \no more" answer). Finite failure is the partiular ase where
n = 0. If an expeted answer C is missing among C1 ; ; Ci ; we have a
symptom of the seond kind. With these intuitive motivations we all positive
the rst level, where there is the rst kind of symptom, and we all negative
the seond level, where there is the seond kind of symptom ([5.8℄). It is
beause we are in a relational paradigm that these two levels are shown to
be so dierent.
From an intuitive viewpoint symptoms are \aused" by errors. Roughly
speaking an error is a limited program fragment responsible for the symptom
and the role of diagnosis is to loate the error.
For the positive level, in our example, the lause
aux(N,B,P) :- N#=M+1, C#=N, aux(M,C,P)
is erroneous and is responsible for the wrong answer F = 1, N = 2, that is to
say for the positive symptom whih is formalised by the (unexpeted) formula
5. Delarative Diagnosis
F
=1^N =2)N
62^
(
f a N; F
133
)
but it is possible to give more information about the \ause" of a symptom: In
this small example it is easy to understand that the symptom, that is f a(2; 1)
whih is false, omes from aux(2; 1; 1), whih omes from aux(1; 2; 1), whih
omes from aux(0; 1; 1). But aux(0; 1; 1) is true whereas aux(1; 2; 1) is false.
This transition between true and false is through the erroneous lause and
the onstraint N = 1; B = 2; P = 1; M = 0; C = 1, sine it is for these values
that the body of the erroneous lause is true and its head is false. So the
onstraint gives more information about the \ause" of the symptom. We
onsider that it is the pair made of this erroneous lause and this onstraint
whih is an error (inorretness), alled positive error (positive inorretness)
beause it is responsible for the positive symptom.
For the negative level the program fragment responsible for the symptom
is a \paket of lauses" (all the lauses beginning with a same prediate
symbol): the intuitive reason is that some answers are missing beause some
lause instanes are missing. In our example, the \paket"
aux(N,B,P) :- N#=0, B#=P.
aux(N,B,P) :- N#=M+1, C#=N, aux(M,C,P).
is erroneous and is responsible for the missing answer F = 2, N = 2., that
is to say for the negative symptom whih is formalised by the (unexpeted)
formula
N
62^
(
f a N; F
) ) (F = 1 ^ N = 0) _ (F = 1 ^ N = 1) _
(F = 1 ^ N = 2)
but it is possible to give more information about the \ause" of a negative
symptom, like for the positive level. In this small example it is easy to understand that there is again a transition between true and false through the
erroneous \paket" and a onstraint: With respet to the expeted semantis,
aux(N; B; P ) is true for N = 1; B = 2; P = 2 but it is not possible to have one
of the two bodies: either N = 0; B = P or N = M + 1; C = N; aux(M; C; P )
true for some values of M; C . A more formal way to express this is to say
that for N = 1; B = 2; P = 2 the formula
9 9
M
C
(N = 0 ^ B = P ) _ (N = M + 1 ^ C = N ^ aux(M; C; P ))
is false w.r.t. the expeted semantis. We onsider again that it is the pair
made of the erroneous \paket" and the onstraint N = 1; B = 2; P = 2
whih is an error (inorretness), alled negative error (negative inorretness) beause it is responsible for the negative symptom.
The notions of symptom and error have a meaning only w.r.t. some notion of expeted semantis. We onsider only delarative semantis and we
presuppose that, during a diagnosis session, the user is able to deide, for
some omputed answers, if they are wrong, or if some expeted answer is
134
A. Tessier and G. Ferrand
missing. In pratie a omputed answer may be intriate, it is the origin of
the presentation problem (stressed by Lloyd [5.5℄) whih may be a diÆulty
of delarative diagnosis. However this presupposition is neessary to give a
meaning to delarative debugging questions. From a oneptual viewpoint
the user behaves like an orale whih is able to deide if something is wrong
or missing (it is the same abstrat notion of orale and the same theoretial
framework if we an partially replae the user by a system using some form
of speiation for the expeted semantis of the program).
The notion of positive symptom and positive error omes from Shapiro's
seminal work (wrong answer and inorretness, [5.7℄).
For the negative side, the notion of negative symptom and negative error
does not orrespond to Shapiro's notions (missing answer and insuÆieny)
whih need more omplex interation with the orale. Theses notions ome
from W. Drabent, S. Nadjm-Tehrani, J. Maluszynski (non ompletely overed
atom, [5.2℄).
The rest of the hapter is organised as follows. Setion 5.2 denes the
theoretial notions of symptom and error. Setion 5.3 denes the proof-trees
and the diagnosis sheme. Setion 5.4 desribes the diagnosis algorithms. Setion 5.5 links the omputation of a result whih is a symptom with the prooftree used in the diagnosis sheme. Setion 5.6 desribes an implementation.
Setion 5.7 shows a diagnosis session. Setion 5.8 presents a onlusion and
future work.
5.2 Basi Notions of Symptom and Error
We use the basi theoretial notions of the CLP Sheme ([5.4℄). D is a onstraint domain. We onsider only denite programs, that is to say without
negation (thanks to disequations and global onstraints, negation as failure
is less useful than in lassial Prolog). A program P is supposed to be normalised in suh a way that only distint variables are allowed as prediate
arguments (all the links between variables are expressed by the onstraints).
It is a faility to simplify the explanation but it is not a loss of generality of
the diagnosis method.
An interpretation I for the language of the program is supposed to be
given. It is a formalisation of the expeted semantis of the program. I must
be an expansion of D, that is to say that it adds to D an interpretation for
the new prediate symbols.
The following denitions are intuitively motivated by the previous introdution.
Denition 5.2.1.
symptom (of
false in I .
P
C
!
G
w.r.t. I ) if
(C onstraint, G goal) is a omputed positive
is a omputed answer for G, but C ! G is
C
5. Delarative Diagnosis
135
Denition 5.2.2. A positive error (positive inorretness) (of P w.r.t. I ) is
a pair made up of a lause p(X ) B in P and a onstraint C suh that, for
some solution of C , B is true in I but p(X ) is false in I .
In some logi formulas we use the following notation : 9 A F means quantiation over the variables of F whih have no ourrene in the expression
A.
In the program P a \paket" of lauses
( )
p X
B1
:::
( )
p X
B
m
is the set of all the lauses of P beginning with a same p. The \paket" is
also alled the denition of the prediate p.
The ompleted denition of the prediate p is the formula
( ) $ 9 p(X ) (B1 _ _ Bm )
It an be rewritten as
p X
[p(X )
9
X (B1 _ _ Bm )℄ ^ [p(X ) ! 9 X (B1 _ _ Bm )℄
( ) 9 X (B1 _ _ Bm ) is merely equivalent to p(X ) (B1 _ _ Bm ),
whih is merely equivalent to the \paket" of p.
The other diretion: p(X ) ! 9 X (B1 _ _ Bm ) annot be simplied so
muh but it is going to be a useful notation. F I (P ) (only-if(P)) is the set of
the formulas
p X
( ) ! 9 X (B1 _ _ Bm )
Remark that we ould also mention I F (P ), the set of the formulas
p X
( )
p X
9
X (B1 _ _ Bm )
but it is merely equivalent to P . Usually the set of the ompleted denitions
is denoted by P , it is equivalent to I F (P ) ^ F I (P ).
Denition 5.2.3. G ! 9 G (C1 _ _ Cn ) (Ci onstraints, G goal) is a
omputed negative symptom (of P w.r.t. I ) if for the goal G there exists a
nite SLD-tree whose omputed answer onstraints are C1 ; ; Cn , but
G ! 9 G (C1 _ _ Cn ) is false in I .
Note that in the partiular ase n = 0 there is a nite failure and in suh
a ase (C1 _ _ Cn ) is merely false so G ! 9 G (C1 _ _ Cn ) is :G.
Denition 5.2.4. A negative error (negative inorretness) (of P w.r.t. I )
is a pair made up of p(X ) ! 9 X (B1 _ _ Bm ) in F I (P ) and a onstraint
C suh that, for some solution of C , p(X ) is true in I but 9
X (B1 _ _ Bm )
is false in I .
136
A. Tessier and G. Ferrand
In the rest of this setion we explain why, if there is a omputed positive
(resp. negative) symptom then there is a positive (resp. negative) error. It is a
short and easy veriation but this result is purely logial and non onstrutive. In the next setion we set out a renement of this result giving more
information about the onnetion between symptom and error.
Positive level: At rst we reall the basi soundness result ([5.4℄): If C is a
omputed answer onstraint for the goal G then P j=D C ! G, that is to say
C ! G is true in all the expansions of D whih are models of P . With regard
to the question of the appliability to CLP systems with inomplete solvers,
it is interesting to remark that, for this kind of soundness, neither orretness
nor ompleteness is required for the solver (intuitively: if an inomplete solver
does not rejet C , if C is unsatisable then C ! G is true. If an inorret
solver rejets C , C is not a omputed answer, even if it is satisable).
Let C ! G be a omputed positive symptom. It is false in I so, thanks
to the previous soundness result, I is not a model of P so there is in P some
lause p(X ) B whih is false in I , so for some assignment a in D, (namely
for some values in D), B is true in I but p(X ) is false in I . Suh a ould give
some information about the \ause" of symptoms. To have a positive error
it is suÆient to take a C suh as a is solution of C . There is always suh a
C , e.g. true. However the more preise C is, the more informative it is.
Negative level: Similar explanation, using another basi soundness result: If
for the goal G there is a nite SLD-tree whose omputed answer onstraints
are C1 ; ; Cn , then F I (P ) j=D G ! 9 G (C1 _ _ Cn ).
It is interesting to remark that now, for this kind of soundness, orretness
(but not ompleteness) is required for the solver (intuitively: a satisable Ci
annot be removed from the onlusion of the impliation, but an unsatisable Ci an be added).
5.3 Connetion between Symptom and Error via
Proof-Trees
Now we onsider a notion of symptom whih is more general than a omputed symptom. Intuitively in these symptoms the onstraint may be more
informative than in a omputed symptom.
Denition 5.3.1. C 0
symptom (of
is false in I .
P
^
C
w.r.t. I ) if
(C; C 0 onstraints, G goal) is a positive
is a omputed answer for G, but C 0 ^ C ! G
!
C
G
So if C 0 = true the symptom is a omputed symptom. Likewise:
Denition 5.3.2. C 0 ^ G ! 9 G(C1 _ _ Cn ) (C 0 ; Ci onstraints, G goal)
is a negative symptom (of
P
w.r.t. I ) if for the goal
G
there exists a nite
5. Delarative Diagnosis
SLD-tree whose omputed answer onstraints are C1 ;
9 G (C1 _ _ Cn ) is false in I .
;
C
137
n , but C 0 ^ G !
At eah level of omputation, we an get a logial representation of the
omputation as a tree. This tree is a proof-tree aording to some rules as
usual in logial formalisms. The rules and the proof-trees are not the same
for the two levels of omputation but the diagnosis sheme is the same and
it is easy to explain it beause at eah level the diagnosis amounts to searh
for a kind of minimal symptom in this tree.
To learly explain what are the rules and the proof-trees we use a new
toy theoretial example: The program is
( )
( )
q (X )
r (X )
( ) ( )
0:
X < 0:
X < 1:
p X
q X ;r X :
q X
X >
and the goal is p(X ).
At the positive level a omputation is a SLD-derivation. In our example
for the goal p(X ) there is a SLD-derivation giving the omputed answer X >
0 ^ X < 1. We an onsider this omputation as a proof of the formula X >
0 ^ X < 1 ! p(X ) and to be more preise we an onsider the omputation
as the onstrution of the following tree:
1!X <1
1 ! r(X )
X
X > 0 ^ X < 1 ! X > 0 ^ r (X )
X > 0 ^ X < 1 ! q (X ) ^ r (X )
X > 0 ^ X < 1 ! p(X )
Suh a tree is made of rules and is alled a proof-tree aording to these
rules. There are two kinds of rules:
0!X >0
> 0 ! q (X )
X >
X <
X <
{ For eah lause p(X ) B in P , a \program" rule
0
C ! B
0
C ^ C ! C ^ p(X )
{ and the \logial" rules. In our framework it is onvenient to onsider all
the rules
(C1 ! G1 ); ; (Cn ! Gn )
C
!
G
where C ! G is a logial onsequene of (C1 ! G1 ); ; (Cn ! Gn ).
These rules are alled logial rules. For example we an get the previous
proof-tree with the two following rules:
138
A. Tessier and G. Ferrand
C
C
0
0
C ! C ^ G
0
0
C ! G ^ G
!
G
!
C
But from the same omputation giving the same answer we an also extrat another proof-tree:
X > 0^ X < 1 ! X > 0
X > 0^ X < 1 ! X < 1
X > 0 ^ X < 1 ! q (X )
X > 0 ^ X < 1 ! r (X )
X > 0 ^ X < 1 ! q (X ) ^ r (X )
X > 0 ^ X < 1 ! p(X )
We an get this seond proof-tree with the \program rule" and the following \logial" rules:
C
! !
! ^ ^
G1
C
C
^
C
G1
C1
^ ^
C
n
G
G
n
n!C
So for eah omputed answer onstraint C for a goal G, the formula C ! G
is the root of various proof-trees, alled (positive) proof-trees, aording to
these various rules, and eah of these proof-trees an be easily obtained from
the omputation namely from the orresponding SLD-derivation.
In partiular if C ! G is a omputed (positive) symptom then it is the
root of various (positive) proof-trees. Let us onsider suh a proof-tree. Eah
node of this proof-tree is labelled by a C 0 ! G0 . It may be a symptom node
(the root is a symptom node). Let us onsider the notion of minimal symptom
node where \minimal" is dened w.r.t the binary relation: x hild of y . So a
node is a minimal symptom node if it is a symptom node but no hild of it
is a symptom node.
To eah node of the proof-tree there is a rule whih is assoiated (the label
C ! G of the node is the onlusion of the rule). Clearly in a \logial" rule,
if the hypotheses are not symptoms then the onlusion is not a symptom.
So the rule whih is assoiated to a minimal symptom node annot be a
\logial" rule so neessarily it is a \program" rule
C
^
C
C
0!B
0 ! C ^ p(X )
Moreover, in this program rule, for some solution of C ^ C 0 , p(X ) is false in
0
0
I (sine C ^ C ! C ^ p(X ) is a symptom), but B is true in I (sine C ! B
0
is not a symptom), so the lause p(X ) B and the onstraint C ^ C of this
program rule give a positive error.
5. Delarative Diagnosis
139
Clearly there are always minimal symptoms (sine proof-trees are nite)
and any way to nd a minimal symptom in a positive proof-tree gives the
loalisation of a positive error. Moreover, to nd a minimal symptom the
following method is suÆient: To onsider only onlusions of program rules
(nodes of the form C ^ C 0 ! C ^ p(X )), and to searh for minimal symptoms
with respet to these nodes. So we have to test formulas C ^ C 0 ! C ^ p(X )
for symptoms. But suh a test amounts to querying (the orale) whether
0
C ^ C ! p(X ) is expeted, namely is true in I .
At the negative level a omputation is a nite SLD-tree. In our example
for the goal p(X ) there is a nite SLD-tree giving the omputed answers
(X > 0 ^ X < 1) and (X < 0 ^ X < 1). We an onsider this omputation as
a proof of the formula p(X ) ! (X > 0 ^ X < 1) _ (X < 0 ^ X < 1) and to
be more preise we an onsider the omputation as the onstrution of the
following tree:
X > 0 !X > 0 X < 0 !X < 0
X <1!X <1
X<1!X<1
q(X ) ! X > 0 _ X < 0
X > 0 ^r(X ) !X> 0 ^X < 1 X < 0 ^r(X ) !X< 0 ^X < 1
q(X ) ^ r(X ) ! (X > 0 ^ X < 1) _ (X < 0 ^ X < 1)
p(X ) ! (X > 0 ^ X < 1) _ (X < 0 ^ X < 1)
Suh a tree is made of rules and is alled a proof-tree aording to these
rules. There are two kinds of rules (with, as usual, appropriate onditions on
the free variables, whih are not detailed here):
{ For eah p(X ) ! 9 X (B1 _ _ Bm ) in F I (P ), a \program" rule
W
for i : Bi ! 9 Bi j Cji
W W
i
C ^ p(X ) ! 9
Cp(X ) i j C ^ Cj
{ and \logial" rules, for example
G
!9
W
G i Ci
G
C
!
^
0 ! 9 Ci G
for i : CiW^ G
W
0 ! 9 GG
G
0
i
i j Cj
0
W
i
j Cj
C
So if for a goal G there is a nite SLD-tree whose omputed answer onstraints are C1 ; ; Cn , then the formula G ! 9 G (C1 _ _ Cn ) is the root
of various proof-trees, alled (negative) proof-trees, aording to these rules,
and eah of these proof-trees an be easily obtained from the omputation
namely from the orresponding SLD-tree ([5.3℄, [5.6℄).
In partiular if G ! 9 G (C1 _ _ Cn ) is a omputed (negative) symptom
then it is the root of various (negative) proof-trees. Let us onsider suh a
proof-tree, in whih eah node is labelled by a G0 ! 9 G (C10 _ _ Cn0 ). A
node may be a symptom node (the root is a symptom node). Let us onsider
the notion of minimal symptom node where \minimal" is dened w.r.t the
0
140
A. Tessier and G. Ferrand
binary relation: x hild of y . So a node is a minimal symptom node if it is a
symptom node but no hild of it is a symptom node.
To eah node of the proof-tree there is a rule whih is assoiated (the label
G ! 9 G (C1 _ _ Cn ) of the node is the onlusion of the rule). Clearly in
a \logial" rule, if the hypotheses are not symptoms then the onlusion is
not a symptom. So the rule whih is assoiated to a minimal symptom node
annot be a \logial" rule so neessarily it is a \program" rule
for i : Bi ! 9 Bi j Cji
W W
i
C ^ p(X ) ! 9
Cp(X ) i j C ^ Cj
W
MoreoverWinWthis program rule, for some solution of C , sine C ^Wp(WX ) !
i
Cp(X ) i j C ^ Cj is a symptom,Wp(X ) is true in I but 9 Cp(X ) i j C ^
i
i
Cj is false in I . But the Bi ! 9 Bi
j Cj are not symptoms. Let us suppose
W
9 X (B1 _ _ Bm )Wis true
in I , then, for some i, 9 X Bi is true, so 9 Bi j Cji
W
is true, so 9 Cp(X ) i j C ^ Cji is true whih is a ontradition. So 9 X (B1 _
_ Bm ) is false in I . So p(X ) ! 9 X (B1 _ _ Bm ) and the onstraint C
of this program rule give a negative error.
Clearly there are always minimal symptoms (sine proof-trees are nite)
and any way to nd a minimal symptom in a negative proof-tree gives the loalisation of a negative error. Moreover, to nd a minimal symptom the following method is suÆient: To onsider
only onlusions of program rules (nodes
W W
of the form C ^ p(X ) ! 9 Cp(X ) i j C ^ Cji ), and to searh for minimal
symptoms with respet to theses nodes. Suh
a minimal symptom an be found
W W
by testing if some C ^ p(X ) ! 9 Cp(X ) i j C ^ Cji are symptoms. But this
W W
test amounts to querying (the orale) whether C ^ p(X ) ! 9 Cp(X ) i j Cji
is expeted, namely is true in I .
9
5.4 Diagnosis Algorithm
So for eah level of omputation we have a notion of proof-tree. And any
way to nd a minimal symptom in a proof-tree gives the loalisation of an
error. Let us onsider a proof-tree rooted by a (omputed) symptom. The
diagnoser queries the orale about the labels of the nodes of the proof-tree,
it is a test: \is this a symptom?". The orale does not need to understand
the operational behaviour of the system.
The objetive of the diagnoser is to loate a minimal symptom (a minimal
symptom node). With this end in view the diagnoser uses a strategy in order
to hoose the node whih orresponds to the next query to the orale.
Let us assume that eah node of the proof-tree an be either expeted (it
is not a symptom) or unexpeted (it is a symptom) or unknown (it is not
yet determined). The diagnosis algorithm is the following:
5. Delarative Diagnosis
141
while no minimal symptom appears in the proof tree
hoose an unknown node aording to some strategy
query the orale about the hosen node in order to determine if it is
expeted or unexpeted
show the error assoiated with a minimal symptom
It is easy to show that this algorithm is orret and that it is omplete in the
sense that it always founds an error (the proof-tree is nite).
In general the orale is the user and we have to take into aount the fat
that the user annot answer some queries of the diagnoser. Thus we have a
fourth ategory of nodes: the dontask nodes (the orale annot determine
if it is a symptom). It is trivial to see that this possibility involves that
the diagnoser is not always able to loate a minimal symptom. Moreover it
involves that the strategies used to hoose a node are more intriate and
sometimes annot hoose a node.
As an example, we desribe here the strategies Top-Down and Divide&Query whih are the most useful in pratie.
The Top-Down strategy follows a path of unexpeted nodes from the root
until it reahes an unknown node. This unknown node is the hosen node.
Beause of the dontask nodes, this strategy does not always nd a node to
query (see Fig 5.7). It looks like a prex traversal of the proof-tree where subproof-trees rooted by an expeted or dontask node are removed (see Fig. 5.1).
The relation topdown(N ode; C hosenN ode) provides the node hosen
(C hosenN ode) by the Top-Down strategy in the tree rooted by N ode. If
the strategy annot onlude C hosenN ode = notf ound.
(
)
if N ode is unknown
then N ode = C hosenN ode
else if N ode is unexpeted
then topdown0 (hildren(N ode); C hosenN ode)
else C hosenN ode = notf ound
topdown N ode; C hosenN ode
0 ([℄; notf ound)
0
topdown ([N odejC hildren℄; C hosenN ode)
topdown(N ode; C hosenN ode0 )
if C hosenN ode0 = notf ound
then topdown0 (C hildren; C hosenN ode)
else C hosenN ode = C hosenN ode0
topdown
Fig. 5.1 shows an example of proof-tree and the node hosen by the TopDown strategy. The proof-tree is drawn as usual: the root of the proof-tree is
at the top of the drawing, the parent relation of the proof-tree links a node
whih is the onlusion of a rule with its hypotheses whih are its hildren
on the drawing. In the drawing the following notation are used: unexpeted
142
A. Tessier and G. Ferrand
nodes are labelled by \N", expeted nodes by \Y", dontask nodes by \X" and
unknown nodes by \?").
N
N
Y
X
N
X
?
?
Y
N
N
?
choosen node
?
?
?
?
N
?
?
?
?
?
Y
Y
?
X
Y
?
?
Fig. 5.1.
Top-Down Strategy
The basi priniple of the Divide&Query strategy is to hoose a node in
order to divide the searh spae in two parts:
{ if the node is
expeted, the subtree rooted by this node an be removed
from the searh spae,
{ if the node is unexpeted, eah node whih is not a desendant of this node
an be removed from the searh spae.
The point is to hoose a node suh that there is as muh unknown nodes in
its subtree as out of its subtree (onsidering that all subtrees rooted by an
expeted node are removed from the searh spae). Note that it is not always
possible to have as muh unknown nodes in the subtree as out of it, but the
strategy hooses the nodes whih is the most near this ondition.
The Divide&Query strategy here is an improvement of the Divide-andquery strategy of [5.7℄. It really hooses the best node (that is the node whih
better divides the searh spae in two parts) and it takes into aount dontask
nodes and the fat that the strategy may be hanged during a diagnosis
session.
The tree onsidered is the tree rooted by an unexpeted node with the
least number of unknown and dontask nodes, but where a minimal symptom
is always possible (remember the dontask nodes).
In Fig. 5.2 we use the same notations as in the previous example. In addition, the integer at the left of the nodes is the number of unknown or dontask
nodes in their subtrees. The 1st subtree and the 4th subtree are not onsidered beause of the dontask nodes: if the unknown nodes are expeted no
minimal symptom an be deteted. The 3rd subtree is onsidered beause the
number of unknown nodes is lower than in the 2nd subtree. The 3rd subtree
has 5 unknown nodes so the Divide&Query strategy hooses the root of the
5. Delarative Diagnosis
143
subtree with 3 unknown nodes: when the user answer the query the subtree
will have 2 unknown nodes inside and 2 unknown nodes outside ((5 1)=2).
15 N
11 N
2 N
2 N
0 Y
2 X
6 N
choosen node
1 X
1 ?
6 ?
3 ?
0 Y
1 ?
3 ?
1 ?
1 ?
1 ?
5 N
1 X
2 ?
1 ?
1 ?
0 Y
0 Y
1 ?
1 ?
0 Y
1 ?
1 ?
Fig.
5.2.
Strategy
Divide&Query
It is possible to build pathologial ases (where whatever node you hoose
there is few nodes in one side and a lot of nodes in the other side) but in
pratie this strategy queries about log2 (N ) nodes (where N is the number
of unknown nodes). For instane, in a proof-tree with 1,000,000 nodes there
is less than 20 queries (try to nd an error without delarative diagnosis!).
5.5 Abstrat Proof-Trees
As said previously, only the nodes whih orresponds to onlusion of program rules an be minimal symptom nodes. So, in order to derease the size
(number of nodes) of proof-trees, we dene abstrat proof-trees. Let us assume
that a proof tree is given, the abstrat proof tree whih orresponds to the
proof tree is dened as follow:
1. the root of the abstrat proof-tree is the root of the proof-tree,
2. y is a hild of a node x in the abstrat proof-tree if
{ y is the onlusion of a program rule in the proof tree,
{ y is a desendant of x in the proof-tree,
{ and there is no other onlusion of program rules between x and
the proof-tree.
y
in
The abstrat proof trees A whih orresponds to a proof tree B an be also
dened as an abstration of the proof tree B in the sense of Chapter 8: all
the onlusion of program rules are seleted (in addition of the root).
In the following we say proof-tree instead of abstrat proof-tree.
144
A. Tessier and G. Ferrand
The user invokes the diagnoser when a (positive or negative) symptom
appears at the end of a (positive or negative) omputation. The omputation
is either a SLD-derivation (a branh of a SLD-tree) or a SLD-tree. But the
diagnoser uses a proof-tree!
The point is that it is possible to ompute diretly a (positive or negative)
proof-tree rooted by the (positive or negative) omputed symptom from a
(positive or negative) omputation, that is from (a branh of) a searh-tree.
In order to simplify, the omputation rule is assumed to be without oroutining (for instane the standard omputation rule of Prolog), then prooftrees an be dedued from searh-tree using the notion of erasing dened
below. [5.6℄ shows an extension to any omputation rule.
The main useful notion is: let x and y be two nodes of the searh-tree, y
is a node where x is erased if
{ Ai is the atom seleted at the node x in its goal (the goal on x is
A1 ; : : : ; Ai ; : : : ; An and the store is C ),
{ y is a desendants of x where Ai is fully solved (the goal on y is
0
0
A1 ; : : : ; Ai 1 ; Ai+1 ; : : : ; An and the store is C ^ C where C is a omputed answer to Ai ).
Let x be a node of the searh-tree, we denote by
{
{
{
{
( ) the set of onstraints aumulated from the root until the node x,
( ) the atom seleted at the node x in its goal,
erased(x) the set of nodes where x is erased,
sthildren(x) the set of hildren of x in the searh-tree.
store x
selet x
Let C be a omputed answer onstraint for the goal G suh that C ! G
is a positive symptom. C is made of the onstraints aumulated along a
suess branh of the searh-tree. The set of nodes of this branh is denoted
by branh.
In order to dene the positive proof tree whih orresponds to the positive
symptom omputed, we have to determine:
1. the root of the proof-tree;
2. the binary relation: x hild of y in the proof-tree;
3. the formula (query) whih labels a node of the proof-tree.
We onsider, in order to simplify, that the nodes of the proof-tree are a subset
of the nodes of the searh-tree, but the labels of a node is dierent depending
on whether it is onsidered as a node of the proof-tree or a node of the
searh-tree.
First, the root of the positive proof-tree is the root of the searh-tree.
Seondly, the list L of hildren of a node x of the positive proof-tree is
given by the relation +hildren:
5. Delarative Diagnosis
145
+hildren
(x; L)
fy g = sthildren(x) \ branh
if y 2 erased(x)
then L = [℄
else + hildren0 (x; y; L1); L = [y jL1℄
0(x; y; L)
+hildren
fz g = erased(y ) \ branh
if z 2 erased(x)
then L = [℄
else + hildren0 (x; z; L1);
L
= [z jL1℄
Finally, the formula assoiated with a node x of the positive proof-tree
is store(y ) ! selet(x) where y is the suess leaf of branh. We use the
onstraint store(y ) in the formula beause it is the most preise we an know,
but as said in Setion 5.3 there exists various proof-trees.
For example, let us onsider the small program (without onstraints and
variables to be more onise, in other words the onstraints are always true):
p
q; r:
p
q; a:
r
a:
q
w:
w:
a
z:
a:
The positive proof-tree whih orresponds to the rst answer to the goal p
is given by Fig. 5.3. The nodes of the searh-tree used by the proof-tree have
been dupliated for the legibility of the drawing. The seleted atom in the
goals is underlined in the searh-tree. For example, p has two hildren in the
positive proof-tree: its hild q in the searh-tree, the node r where q is erased.
Note that the node where r is erased is also the node where p is erased.
On Fig. 5.3 you an reognise a more lassial notion of proof-tree in logi
programming (a node orresponds to the head of a lause of the program and
its hildren orrespond to the body of the lause).
Let C1 ; : : : ; Cn be the omputed answer onstraints for the goal G suh
that G ! 9 G (C1 _ _ Cn ) is a negative symptom.
In order to dene the negative proof tree whih orresponds to the negative symptom omputed, we have to determine:
1. the root of the proof-tree;
2. the binary relation: x hild of y in the proof-tree;
3. the formula (query) whih labels a node of the proof-tree.
146
A. Tessier and G. Ferrand
Search Tree
Positive Proof Tree
p
p
q,r
q,a
q
r
w,r
w,a
w
a
r
a
a
z
First Positive Answer
z
Fig. 5.3. From a branh
of the searh-tree to the
positive proof-tree.
First, the root of the negative proof-tree is the root of the searh-tree.
Seondly, the list L of hildren of a node x of the negative proof-tree is
given by the relation hildren:
hildren(x; L)
let S be the list of nodes of (sthildren(x) n erased(x))
hildren0 (x; S; L)
0(x; [℄; [℄)
0
hildren (x; [y jS ℄; [y jL℄)
hildren00 (x; y; L1)
hildren0 (x; S; L2)
hildren
append L ; L ; L
( 1 2 )
hildren
00 (x; y; L)
let S be the list of nodes of (erased(y ) n erased(x))
0
hildren (x; S; L)
Finally, the formula assoiated
with a node x of the negative proof-tree
W
is store(x) ^ selet(x) ! y2erased(x) store(y ).
For example, let us onsider again the previous program:
p
q; r:
p
q; a:
r
a:
q
w:
w:
a
a:
z:
5. Delarative Diagnosis
147
The negative proof-tree whih orresponds to the searh-tree for the goal p
is given by Fig. 5.4. The nodes of the searh-tree used by the proof-tree have
been dupliated for the legibility of the drawing. The seleted atom in the
goals is underlined in the searh-tree. For example, p has four hildren in the
negative proof-tree: its hildren q and q in the searh-tree, the node r where
the rst q is erased, the node a where the seond q is erased. Note that the
nodes where r and a are erased are also the nodes where p is erased.
Search Tree
Negative Proof Tree
p
p
q,r
q,a
q
r
q
a
w,r
w,a
w
a
w
z
r
a
a
z
z
z
Fig.
5.4.
From the
searh-tree to the negative proof-tree.
5.6 Implementation
The positive part of the delarative diagnoser has been implemented and
tested on the INRIA platform: TkCalypso, developed in the DiSCiPl projet.
At the time of writing this book, implementation of the negative part is in
progress.
TkCalypso is an extension of GNU-Prolog [5.1℄. It inludes a graphial interfae (Fig. 5.5) and some debugger modules. Eah module an be plugged or
unplugged. Fig. 5.6 shows the struture of TkCalypso with the three modules
that are atually implemented: searh-tree visualisation, stati debugger and
delarative diagnoser. Communiations between GNU-Prolog and the modules and the graphial interfae are handled by the \Core/Gestionnary" pakages. This setion desribes the main features of the module alled \Delarative Diagnoser".
When a goal is given to TkCalypso, it stores informations on the searhtree in order to reompute it eÆiently and make post-mortem analysis of
the searh-tree.
148
A. Tessier and G. Ferrand
Fig. 5.5.
Graphial interfae: a positive symptom
GNU-Prolog
GUI (Tcl/Tk)
Core
Search Tree
Gestionnary
Static Debugger
Declarative Diagnoser
Fig. 5.6. Arhiteture
of the TkCalypso platform.
5. Delarative Diagnosis
149
If the user noties a omputed symptom, then the delarative diagnoser
is alled and the parent relation of the (positive or negative) proof-tree is
omputed dynamially from the searh-tree.
One we have a (positive or negative) proof-tree, the diagnosis priniple is
always the same: hoose a node of the proof-tree (aording to some strategy),
hek if it is expeted or not expeted, until a minimal symptom is founded.
Several strategies have been implemented in order to hoose the node
to query: Top-Down, Bottom-Up, Divide&Query, Nearby-Error and UserGuided (Top-Down and Divide&Query are desribed in Setion 5.4).
Built-in prediates are known as orret prediates, but also user prediates ould be known as orret. For example, when the stati analysis (see
Chapter 2) has proved their orretness. Another example is when the user
is onvined that some prediates are orret. So the user an set a list of
orret prediates, whih will be used by the diagnoser.
It is possible that the user does not want to be questioned on some prediates. For example, the semantis of the prediate is very intriate and the
user want to suspend the queries on that prediate as long as possible. So
the user an set a list of prediates whih must not be questioned.
The status of a node of the proof-tree is more preise that the ones desribe
in Setion 5.4, beause we want to know the origin of this status, it an be:
{
{
{
{
{
{
{
when the prediate assoiated to the node is not a built-in and
the node has not been queried,
expeted(user ) when the user said that the node is expeted,
expeted(list) when the prediates assoiated to the node is in the list of
orret prediates (when the user is onvined that the prediate is orret),
expeted(system) when the prediate assoiated to the node is a built-in
prediate (the are not suspeted!),
unexpeted(user ) when the user said that the node is not expeted,
dontask (user ) when the user does not want to answer the query assoiated
to the node (for example the query is very intriate), note that the user
an ome bak later to the query assoiated with this node,
dontask (list) when the prediates assoiated to the node is in the list of
prediates that the user does not want to answer (for example the user
does not know the semantis of the prediates or the user wants to delay
the queries about the prediate).
unknown
The user has the possibility to dynamially add some prediates to the
list of orret prediates or remove some prediates from it. If the user adds
a prediate, eah unknown or dontask ( ) node onerning the prediate is
labelled by expeted(list); if a node is unexpeted(user) then the problem
is given to the user: either remove the prediate from the orret prediate
list or label the node as expeted(user). When the user removes a prediate
of the list, eah expeted(list) node onerning the prediate are labelled by
unknown.
150
A. Tessier and G. Ferrand
The user an also hange dynamially the list of prediates whih must
not be questioned. If the user adds a prediate to the list, eah unknown node
onerning the prediate beomes a dontask (list) node. If the user removes
a prediate of the list, eah dontask (list) node onerning the prediate beomes an unknown node.
Fig. 5.7.
Delarative Diagnoser Interfae
The graphial interfae (see Fig.5.7) of the diagnoser works like an hypertext navigator: the user an navigate between queries, beause its answers are
seen as hyper-link between queries (it is possible to go bak, go forward, see
an history...). The proof-tree is displayed with informations about the nodes.
For example, the user an see the status of a node
{ with olours: red = unexpeted, grey = unknown, blue = dontask, green =
expeted,
{ and with shades: light = system, medium = user, dark = list.
The user an also hoose the query with the mouse on the proof tree: it is
the \User Guided" strategy.
5. Delarative Diagnosis
151
The query C ! A is displayed as A :- 9 A C . The diagnoser uses a trivial
store simpliation in order to simplify the onstraint 9 A C in the query.
But this point has to be improved (this is disussed later in Setion 5.8).
5.7 A Diagnosis Session
Let us onsider the following QuikSort program whih is intended to sort a
list of nite domain terms:
qs([℄, [℄).
qs([Pivot | List℄, SortList) :partition(Pivot, List, MinList, MaxList),
qs(MinList, SortMinList),
qs(MaxList, SortMaxList),
append([Pivot|SortMinList℄, SortMaxList, SortList).
partition(_, [℄, [℄, [℄).
partition(Pivot, [X | List℄, [X | MinList℄, MaxList) :X #< Pivot,
partition(Pivot, List, MinList, MaxList).
partition(Pivot, [X | List℄, MinList, [X | MaxList℄) :X #> Pivot,
partition(Pivot, List, MinList, MaxList).
In the expeted model of qs(A,B) A is a list of distint nite domain terms,
B is a permutation of A and B is an inreasing list (for example X < 5 !
qs([5; X; 7℄; [X; 5; 7℄) is expeted). In the expeted model of partition(A,
B, C, D) A is a nite domain term, B is a list of nite domain terms, C is
the list of members of B whih are lower than A, D is the list of members of
B whih are greater than A, B is obtained by a fusion of C and D.
Fig. 5.5 shows that for the goal qs([12,X,Y,6,Z℄,L) the rst answer is:
L
X
Y
Z
=
=
=
=
[12, #2(8..11), #23(7..10), 6, #44(0..5)℄
#2(8..11)
#23(7..10)
#44(0..5)
(X = #2(8..11) means that X is a nite domain variable whose domain
is 8::11). It is a positive symptom: L is not an inreasing list. So we all
the positive delarative diagnoser module of TkCalypso and a new window
appears on the sreen, with a drawing of the positive proof tree.
The diagnoser queries the user on some nodes of the proof-tree. For example in Fig. 5.8, the user will answer (lik on) \Not Expeted" beause the
last onstraint in the store is #44(0..5) #< 6, so [6, #44(0..5)℄ is not an
inreasing list. The store simpliation has been disabled on the gure and
we see that without simpliation the store quikly beomes unreadable.
152
A. Tessier and G. Ferrand
Fig. 5.8.
A query
5. Delarative Diagnosis
153
After several queries the diagnoser nds a minimal symptom and shows
the orresponding error. Fig 5.9 shows an error, rst it displays the inorret
lause and next the error: a lause instane and a onstraint store. The problem is that in append([A|F℄, G, C) the pivot A should be between the list F
and the list G: append(F, [A|G℄, C).
Fig. 5.9.
An error
Sometimes it is not easy to x the error provided by the delarative diagnoser, but the user is sure that the lause is inorret, so the user does
not need to searh elsewhere in the program. Thus delarative diagnosis is
very eÆient espeially for large programs (with a lot of lauses) or for large
omputations (searh-tree with a lot of nodes). It is also a good tool for
eduational purpose.
5.8 Conlusion
The main remaining task onerns the interation with the orale.
Of ourse, answers to previous queries may be used by the delarative
diagnoser in order to automatially answer to some other queries: Let us
onsider a query \C ! A expeted?" If it is stored that C 0 ! A is expeted
and if C ! C 0 is true then C ! A is expeted. If it is stored that C 0 ! A
is unexpeted and if C 0 ! C is true then C ! A is unexpeted. Likewise for
the negative side.
154
Referenes
The diÆulty is to deide if C ! C 0 is true (or in general C1 _ _ Cm !
0
0
C1 _ _ Cn ), that is the entailment problem.
It is interesting to study how assertions dened in Chapter 1 ould be used
to answer to queries of the delarative diagnoser. Then some assertions are
viewed as a partial speiation of the expeted semantis of the program.
Despite these tehniques it is not possible to ompletely avoid interation
with the user. So, works in progress onerns the presentation problem, that is
to show queries in an understandable form. Variable elimination, redundant
onstraint elimination, onstraint simpliation and approximation may be
useful methods to present queries to the user.
Referenes
5.1 D. Diaz.
A Native Prolog Compiler with Constraint Solving over
Finite Domains Edition 1.0, for GNU Prolog version 1.0.0, 1999.
http://www.gnu.org/software/prolog/
5.2 W. Drabent, S. Nadjm-Tehrani, and J. Maluszynski. Algorithmi Debugging with Assertions. In Harvey Abramson and M. H. Rogers, editors, MetaProgramming in Logi Programming, pages 501{522. The MIT Press, 1989.
5.3 G. Ferrand and A. Tessier. Clariation of the bases of Delarative Diagnosers
for CLP. Deliverable D.WP2.1.M1.1-1. Debugging Systems for Constraint
Programming (ESPRIT 22532), 1997. http://disipl.inria.fr/
5.4 J. Jaar, M. J. Maher, K. Marriott, and P. J. Stukey. Semantis of Constraint
Logi Programs. Journal of Logi Programming, 37(1-3):1{46, 1998.
5.5 J. W. Lloyd. Delarative Programming in Esher. Tehnial Report CSTR-95013, Department of Computer Siene, University of Bristol, 1995.
5.6 B. Malfon and A. Tessier. An Adaptation of Negative Delarative Error Diagnosis to any Computation Rule. Deliverable D.WP2.1.M2.1-1.
Debugging Systems for Constraint Programming (ESPRIT 22532), 1998.
http://disipl.inria.fr/
5.7 E. Y. Shapiro. Algorithmi Program Debugging. ACM Distinguished Dissertation. The MIT Press, 1982.
5.8 A. Tessier. Corretness and Completeness of CLP Semantis revisited with (Co)Indution. Deliverable D.WP2.1.M2.1-2. Debugging Systems for Constraint
Programming (ESPRIT 22532), 1998. http://disipl.inria.fr/