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4/30/2010
1
Reduction of
inductive predicates
for shape analysis
of circular lists
Daniel Stutzman
April 27, 2010
circular lists
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shape analysis
of circular lists
inductive predicates
for shape analysis
of circular lists
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Reduction of
inductive predicates
for shape analysis
of circular lists
Let’s make a sorted linked list
struct ListNode {
int data;
struct ListNode* next;
struct ListNode* prev;
}
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Let’s make a sorted linked list
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0
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0
x
Run-time “checker” functions
boolean is_sll(ListNode* x) {
return (x->prev == NULL) && is_sll2(x);
}
boolean is_sll2(ListNode* x, int min_data) {
return (x->data >= min_data) &&
(x->next->prev == x) &&
is_sll2(x->next, x->data);
}
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Run-time “checker” functions
boolean is_sll(ListNode* x) {
return (x->prev == NULL) && is_sll2(x);
}
boolean is_sll2(ListNode* x) {
return (x->next->data >= x->data) &&
(x->next->prev == x) &&
is_sll2(x->next);
}
Run-time “checker” functions
boolean is_sll(ListNode* x) {
return (x->prev == NULL) && is_sll2(x);
}
boolean is_sll2(ListNode* x) {
return (x->next == NULL) ||
((x->next->data >= x->data) &&
(x->next->prev == x) &&
is_sll2(x->next));
}
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Static inductive predicates
is_sll(x) = null(x->prev) /\ is_sll2(x)
is_sll2(x) = null(x->next) V
((x->next->data >= x->data) /\
(x->next->prev == x) /\
is_sll2(x->next))
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is_sll2
Example in a shape domain
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Unfolding
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x
is_sll2
?
Unfolding
?
0
x
is_sll2
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Unfolding
?
0
0
x
Unfolding
?
0
x
is_sll2
?
?
0
0
x
V
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Let’s make a circular list
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Let’s make a circular list
3
0
4 7
x
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Let’s make a circular list
3 4 7
x
Walking backwards
?
?
?
x
is_circular_list
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Walking backwards
?
?
?
x
is_circular_list
Walking backwards
?
?
?
x
is_circular_list
Error: Unable to find an
appropriate edge to unfold
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Reduction
? ?
x
is_list
Reduction
? ?
x
is_list
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Reduction
? ?
is_list
?
x
Two equivalent summaries
is_bounded_list
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Two equivalent summaries
is_bounded_list
Two equivalent summaries
is_bounded_list
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Two equivalent summaries
is_bounded_
list
Two equivalent summaries
is_
bound
ed_list
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Two equivalent summaries
Two equivalent summaries
is_list
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Two equivalent summaries
is_list
Two equivalent summaries
is_list
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Two equivalent summaries
is_list
Two equivalent summaries
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Other equivalent summaries
dll1NULL
Other equivalent summaries
dll1NULL
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Other equivalent summaries
dll1NULL
Other equivalent summaries
dll1NULL
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Other equivalent summaries
dll1NULL
Other equivalent summaries
NULLNULL
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Other equivalent summaries
dll2
NULLNULL
NULLNULL
Other equivalent summaries
dll2
NULLNULL
NULL
NULL
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Other equivalent summaries
dll2
NULLNULL
NULL
NULL
Other equivalent summaries
dll2
NULLNULL
NULL
NULL
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Other equivalent summaries
dll2
NULLNULL
NULL
NULL
Other equivalent summaries
NULLNULL
NULLNULL
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