Transcript My CAV'06 talk

Heuristic Theorem Prover
Kenneth Roe
8/20/2006
www.fordocsys.com
Key contributions
• Preprocessor before DPLL(T)
– Unate detection algorithm
– Rewriting
– Introduction of Symmetry Breaking to SMT
Modulo Theorem
– Incremental difference logic encoding
SMT-COMP’06 configuration
• BIconnected Graph Meta System for Encoding
to Sat or Smt (BIG MESS)
HTP
YICES
BCLT
MiniSat
Rewriting
• Algebraic simplification
• Rules for arithmetic and boolean logic
• Examples
a+1+1+1+1 rewrites to a+4
a+2<b+3 rewrites to a<b+1
a and a rewrites to a
• Some context rewriting
a<b and b<a rewrites to False
a<b and not(a=b) rewrites to a<b
Symmetry breaking
• Currently only works for QF_UF division
• Step 1: Detect symmetric variables:
– Example: P(a,b) ^ P(b,a)
– The variables a and b can be swapped and form a
symmetric pair.
• Step 2: Detect symmetric predicates:
– In the example above P(a,b) and P(b,a) are
symmetric
• Step 3: Generate Symmetry breaking tuples
– P(a,b) or not(P(b,a))
Unate detection algorithm
• Detect atomic predicates that when asserted or denied
make a complex boolean expression true or false
• Example problem:
– (a<b) and (if b=c then a+1=b else a<b+1)
• For each subterm we compute four sets
–
–
–
–
Assert_makes_true
Deny_makes_true
Assert_makes_false
Deny_makes_false
• Also compute pair wise implications between atomic
predicates
Unate detection algorithm example
(a<b) and (if b=c then a+1=b else a<b+1)
b=c
assert_makes_true={b=c}
deny_makes_false={b=c}
a<b
assert_makes_true={a<b,a+1=b}
deny_makes_false={a<b,a<b+1}
a+1=b
assert_makes_true={a+1=b}
deny_makes_false={a+1=b,a<b,a<b+1}
a<b+1
assert_makes_true={a<b,a<b+1,a+1=b}
deny_makes_false={a<b+1}
Unate detection algorithm example
(a<b) and (if b=c then a+1=b else a<b+1)
if b=c then a+1=b else a<b+1
assert_makes_true={a+1=b}
deny_makes_false={a<b+1}
(a<b) and (if b=c then a+1=b else a<b+1)
assert_makes_true={a+1=b}
deny_makes_false={a<b,a<b+1}
Difference logic encoding
• Example
– not(a < b) or ((a<b) and (b < c) and (c < a))
• Introduce boolean variables
– (a < b) => A
– (b < c) => B
– (c < a) => C
• Encode and add predicates for illegal
combinations
– (not(A) or (A and B and C)) and
(not(A) or not(B) or not(C))
Results
• Difference logic encoding useful on most
cases in QF_LIA, QF_LRA and QF_IDL—
quite effective in gaining performance
• Symmetry breaking effective in improving
performance
Results
• Rewriting useful on select problems (WISA
for example)
• Unate detection not useful on its own but
is a critical building block for the difference
logic encoding