Reductions to LIA*

Example formula

See (Piskac and Kuncak, 2008)

Step 1 - rewrite with summations

To

Step 2 - Introduce auxiliary variables

Step 3 - Combine summations

• Cardinality constraints added to LIA formula
• Set constraints are under summations

Step 4 - Boolean reduction

Boolean Reduction - General form

Solving B*

Claim: for set cardinalities we are creating sets based on semi-linear combinations of vectors based on Boolean assignments. Thus, the vectors are based on satisfying implicants of the formula under .

General form:

Define

Solving B* with Constrained Horn Clauses (CHC)

If this is satisfiable with solution , then is UNSAT.

Justification:

• if is a solution to CHC, then .

Solving B* with CHC

Conversely, if there is some set of assignments , such that , then it shows that .

State reachability - example

Solutions to are .

Set .

it is unsat

B* represents a special case vector addition system

• Pretend it as a state-machine where the initial state is .

• tells us the space of possible increments.

• The increments are independent of the current state.

• Each vector fits within a cube (each coordinate is either 0 or 1).

State reachability is decidable

• Z3 contains an engine for Horn clauses (PDR/SPACER).

• Simple algorithm: Enumerate all solutions to .

• Check satisfiability of .

From PDR to conflicts

• Assume we have a set of literals over set constraints and cardinalities of sets.

• We want to check satisfiability of them.

• Extract , solve for .

  • If the system is unsat: it means there is some solution for sets that satisfy the cardinality constraints.
  • if the system has a solution: it means the conjunction of set constraints cannot satisfy the cardinality constraints.
    • we could look for a subset of set constraints that are sufficient using the unsat core from cardinality constraints.

State reachability refinements

• Start with initial state ,
  

• While is unsat and is sat:

  • Check sat of .
    • Add new solution .
    • If there are no new solutions is completed.
  • Dual process for strengthening .

Over-approximations

Solving Over-approximations

(set-logic HORN)
(declare-fun F2Star (Int Int Int) Bool)
(define-fun F2 ((bs Int) (bt Int) (bu Int)) Bool 
    (and (<= 0 bs) (<= bs 1)
         (<= 0 bt) (<= bt 1)
         (<= 0 bu) (<= bu 1)
         (or (= bs 0) (= bt 1))
         (= (= bu 1) (and (= bt 1) (= bs 0)))))
(define-fun F1 ((ks Int) (kt Int) (ku Int)) Bool
    (and (= ks 1) (not (= ku (- kt 1)))))

(assert (F2Star 0 0 0))
(assert (forall ((ks Int) (kt Int) (ku Int) (bs Int) (bt Int) (bu Int))
             (=> (and (F2Star ks kt ku) (F2 bs bt bu)) (F2Star (+ ks bs) (+ kt bt) (+ ku bu)))))


(assert (forall ((ks Int) (kt Int) (ku Int))
(=> (F2Star ks kt ku) (not (F1 ks kt ku)))))

(check-sat)
(get-model)

Theory combination

Not all set constraints are pure.

• means the set is a singleton set and if with , then .
• Abstract singleton constraints as sets of size 1 for variables that are distinct. Track distinctness assumptions.

Claim: we can eliminate singleton sets and just consider set variables with cardinality 1 from the point of view of the set solver.

• Can anything reasonable be done with ranges, subsets, map?

Integration - Outline

• Set of literals over set constraints.
• All sub-expressions with cardinality operators
• Sub-terms over sets .
• Introduce fresh variables for sub-term definitions.
• Define for equality constraints.
• Define for cardinality operators, arithmetic constraints and set disequalities.
• Check reachability.

The extra mile

What if we want to solve CHC over finite sets (not just Boolean programs)?

• Is there a (non-trivial) class of CHC over finite sets that is solvable?
• Trivial := finite sets over finite base sorts

Bibliography