On the Use of Automata Techniques to Decide Satisfiability

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On the Use of Automata Techniques to Decide
Satisfiability

• Mia Minnes
• May 3, 2005

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Motivation

• Weve seen how useful ILP formulations can be.
• Often, the main question is whether such a system
  (of linear equations and inequalities) has a
  solution, and if so, what it is.
• Omega is nice, but can we do better?

• Moreover, can we have a general framework for
  solving such systems which is amenable to
  extension?
• E.g. to Mixed Integer Linear Programming
• E.g. to more general systems (with quantifiers).

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Motivation

• One possible such framework is Finite Automata.
• Why are we interested in it?
• It worked beforeMONA (model checking for LTL).
• Automata are amenable to algorithmic
  manipulation.
• It often has a compact representation.

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Goal

• Given a system of linear equations and
  inequalities, construct an automaton which
  accepts exactly those vectors which are solutions
  to the system.
• In order to get there, well need to
• Define the systems of equations / inequalities
• Define automata
• Formulate algorithm.

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1. The Systems

• Presburger Arithmetic is the first order theory
  (sentences involving ?, ?, Boolean combinations)
  of

• Atomic (basic) formulas are of the form

or
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1. The Systems

• A formula ?(x1,, xn) in Presburger Arithmetic is
  SATISFIABLE if there are integers c1,, cn such
  that ?(c1,, cn) is true.

• Examples

?
?
?
ILP Problem
?

• Presburger (1927) gave decision method.

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2. Automata

• A finite state AUTOMATON is a machine which
  reads input bit-by-bit and changes state
  according to the current state and input.
• It has designated initial and accepting states,
  and a transition function which gives the rule
  for moving between states.

• Example

Sample Input
00011 ?
10011 ?
01100 ?
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3. Algorithm

• Given ?, a formula in Presburger Arithmetic
• Construct A?, the automaton accepting exactly
  those words satisfying ?
• Check if the set of words accepted by A? is empty
• If yes, ? is not satisfiable if no, it is.

Observe The hardest part of this algorithm is
(2). Why?
Cf. (Wolper, Boigelot 02, extending Boudet,
Comon 96)
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3. Algorithm
To tackle the translation of formulas to
automata

• First construct A? for atomic ?
• Then define automata corresponding to Boolean
  opns
• Finally, define automata for quantified formulas.

(Note that the third step is not important for
ILP problems.)
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3. Algorithm Equations
Some implementation details

• Integers encoded in 2s complement binary
  notation.
• Read numbers MSB first.
• States represent current value of sum. Hence,

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3. Algorithm Forward Construction
x-y2 a (1,-1)


Inputs (b) ----- 01 ----- 10 ----- 00,11
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3. Algorithm Pruning Construction
x-y2
----- 01 ----- 10 ----- 00,11 -----
00,01,10,11
Note this is not yet minimal!
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3. Algorithm Backward Construction
x-y2
?(?0a.b)/2 ? Z? ?-a.b? ?0 ? ? via b
Hsi,2 L2
Hsi,2,1,0 L0
Hsi,2,1,0 L empty
Hsi,2,1 L1
----- 01 ----- 10 ----- 00,11 -----
00,01,10,11
Note this is minimal!
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3. Algorithm Inequalities

• For forward construction include all states
  representing numbers less than RHS as accepting
  (plus modify bad states).
• For backward construction, add transitions for
  all

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3. Algorithm Inequalities
x-y lt 2
----- 01 ----- 10 ----- 00,11 -----
00,01,10,11
Also need to add transitions!
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3. Algorithm General Formulas

• Weve taken care of any atomic formula .
• For negation, conjunction, and disjunction of
  formulas, use the standard techniques for
  complementation, intersection, and union of
  finite automata.
• Complementation switch accepting / non-accepting
  states
• Intersection run both automata simultaneously
• Union run the automata side by side
• To eliminate quantifiers, use the projection
  operation on automata.

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Complexity Results

• If ? is a quantifier free formula, then the
  minimal automaton for ? has at most 1(??)?
  states, where ? is the number of terms in ?,
  ? and ? depend on the absolute values of the
  coefficients in the linear terms.
• If ? is an arbitrary formula of length ?, the
  minimal automaton for ? has at most
• states. Moreover, there is an example requiring
  this many states, so the bound is tight.

Cf. (Klaedtke 03)
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Evaluation of Aut. Decision Proc.

• Relatively simple algorithm.
• Handles general case well (Pres. Arithmetic).
• Uses graph theoretic simplifications.
• Exhibits solution if one exists.
• Doesnt use native arithmetic so will spend more
  time on computation.
• Calculates a lot of data that may not be
  necessary for specific question.

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Experimental Results Quant. Free

• Satisfiability problem for quantifier free
  Presburger Arithmetic is NP complete, so
  experiments were designed to determine which
  classes of formulas were handled well by each
  technique (LASH vs. ILP vs. Omega)
• 25 runs with parameters
• atomic formulas
• variables
• LASH was able to complete nearly all test runs,
  and do so correctly. ILP, Omega were faster when
  they worked but were prone to error / failure.

Cf. (Ganesh, Berezin, Dill 02)
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Extension Mixed Integers and Reals

• Hybrid systems and timed systems require systems
  of equations in which some of the variables have
  real values and some have integer values.
• To use the automata techniques in this case we
  have two challenges
• How to encode real numbers as binary input to
  automata?
• How to design automata which accept particular
  sets of reals?
• A nice feature of the design which follows is
  that the set of integers is automata recognizable.

Cf. (Boigelot, Bronne, Rassart 97 Boigelot,
Jodogne, Wolper 01 Boigelot, Wolper 02)
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Encoding Real Numbers

• Given ? a real number, we encode it in binary
  with MSB first, using 2s complement for negative
  numbers.
• Result is an infinite word
• ? ?I ?F
• where ?I encodes the integer part of ? as a
  finite binary word,
• and ?F encodes the fractional part of ? as an
  infinite binary word.

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Automata Accepting Real Numbers

• Decompose the problem as follows
• Using a finite automaton (as before) to accept
  the integer part of the real number
• Concatenating with a special automaton which only
  accepts
• And then appending a new kind of automaton which
  will have to be able to accept the fractional
  (infinite) part of the real number.

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Automata Accepting Real Numbers

• What is a Buchi Automaton?
• A finite state automaton where the notion of
  accepting a word is different The automaton
  accepts a word ? if, on reading ?, the automaton
  enters an accepting state infinitely many times.
• By the Buchi characterization theorem, the set of
  words accepted by a Buchi automaton is nonempty
  exactly if there is a lasso path in the
  underlying directed graph from an initial state
  to an accepting state and then looping back to
  the accepting state.

Accepts e.g. 011111
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Automata Accepting Real Numbers

• Now the satisfiability question is
• Does the integer part automaton accept a
  non-empty set of words? And,
• Does the fractional part automaton accept a
  non-empty set of words?
• Since both of these are answerable (in linear
  time) by graph algorithms, were in good shape.

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Future Directions Open Questions

• Further, more extensive, comparative studies
• For general formulas (with quantifiers)
• In the mixed case (compare to Fourier-Motzkin
  approach)
• Explore different encoding schemes
• LSB vs. MSB first
• Expand/ optimize existing toolsets.

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References

• Boigelot, Bronne, Rassart (CAV 97) An Improved
  Reachability Analysis for Strongly Linear Hybrid
  Systems.
• Wolper, Boigelot (TACAS 00) On the Construction
  of Automata from Linear Arithmetic Constraints.
• Boigelot, Jodogne, Wolper (IJCAR 01) On the Use
  of Weak Automata for Deciding Linear Arithmetic
  with Integer and Real Variables.
• Boigelot, Wolper (ICLP 02) Representing
  Arithmetic Constraints with Automata An
  Overview.
• Ganesh, Berezin, Dill (FMCAD 02) Deciding
  Presburger Arithmetic by Model Checking and
  Comparison with Other Methods.
• Klaedtke (Technical Report, 03 LICS 04) On
  the Automata Size for Presburger Arithmetic.