Automated Theorem Proving

Automated Theorem Proving Lecture 3 Satisfiability modulo theories

Arithmetic programs • In addition, integer-valued variables with affine operations •  Formula := A |   |    A  Atom := b | t = 0 | t > 0 | t  0 t  Term := c | x | t + t | t – t | ct b  SymBoolConst x  SymIntConst c  {…,-1,0,1,…}

Satisfiability modulo arithmetic • A formula is a boolean combination of literals • Each literal is a positive or negative atom • Each atom is either a boolean variable or a linear constraint over integer variables

x  y  (a  z > 0)  (a  x > y)  y + z  x b  x  y c  z > 0 d  x > y e  y + z  x b  (a  c)  (a  d)  e

x  y  (a  z > 0)  (a  x > y)  y + z  x b  x  y c  z > 0 d  x > y e  y + z  x b  (a  c)  (a  d)  e Arithmetic Solver

x  y  (a  z > 0)  (a  x > y)  y + z  x b  x  y c  z > 0 d  x > y e  y + z  x b  (a  c)  (a  d)  e b = T, e = T Arithmetic Solver Satisfiable

x  y  (a  z > 0)  (a  x > y)  y + z  x b  x  y c  z > 0 d  x > y e  y + z  x b  (a  c)  (a  d)  e b = T, e = T Arithmetic Solver a = F Unsatisfiable b = T, c = T, e = T

x  y  (a  z > 0)  (a  x > y)  y + z  x b  x  y c  z > 0 d  x > y e  y + z  x b  (a  c)  (a  d)  e b = T, e = T Arithmetic Solver a = T Unsatisfiable b = T, d = T, e = T

Affine constraints A collection of m constraints over n variables: a11 x1 + a12 x2 + … + a1n xn + c1  0 a21 x1 + a22 x2 + … + a2n xn + c2  0 … am1 x1 + am2 x2 + … + amn xn + cm  0 a1 x1 + a2 x2 + … + an xn + c> 0 a1 x1 + a2 x2 + … + an xn + c-1 0 a1 x1 + a2 x2 + … + an xn + c 0 (-a1)x1 + (-a2)x2 + … + (-an xn) + (-c) 0 a1 x1 + a2 x2 + … + an xn + c= 0

Satisfiability problem for affine constraints A collection of m constraints over n variables: a11 x1 + a12 x2 + … + a1n xn + c1  0 a21 x1 + a22 x2 + … + a2n xn + c2  0 … am1 x1 + am2 x2 + … + amn xn + cm  0 Does there exist an assignment of x1,x2, …,xn over the integers such that each constraint is satisfied ?

Solving affine constraints • Integer linear programming • NP-complete • Approximate integers by rationals/reals • Linear programming • Polynomial time (Khachian 1978, Karmarkar 1984) • Simplex algorithm (Dantzig 63) • exponential worst-case time • polynomial behavior in practice

Simplex Algorithm for Affine Constraints

Tableau x1 x2 …xn y1 a11 a12 … a1n c1 y2 a21 a22 … a2n c2 … ym am1 am2 … amn cm Row variables Column variables Read it as: y1 = a11 x1 + a12 x2 + … + a1n xn + c1 y2 = a21 x1 + a22 x2 + … + a2n xn + c2 … ym = am1 x1 + am2 x2 + … + amn xn + cm y1  0 y2  0 … ym  0

x – y + 1  0 x + y + 3  0 -x + -4  0 x y a 1 -1 1 b 1 1 3 c -1 0 -4

c = 0 x = 0 a = 0 y = 0 b = 0

Sample point x1 x2 …xn y1 a11 a12 … a1n c1 y2 a21 a22 … a2n c2 … ym am1 am2 … amn cm x1 = 0x2 = 0…xn = 0 y1 = c1 y2 = c2 … ym = cm

A tableau is feasible if the sample point satisfies • all sign constraints. • Otherwise, drop a subset of sign constraints to • get a feasible tableau. • For each unsatisfied sign constraint: • Look for a different point satisfying the constraint • while preserving existing constraints • If such a point is found, add the constraint • Otherwise, declare unsatisfiable • Declare satisfiable

Pivot operation Exchange row i and column j: 1. Solve for xj yi = ai1 x1 + … + aij xj + … + ain xn + ci xj = (-1/aij) (ai1 x1 + … + (-1)yi + … + ain xn + ci) 2. Substitute in row k  i yk = ak1 x1 + … + akj xj + … + akn xn + ck yk = (ak1 – akjai1/aij) x1 + … + (akj/aij)yi + … + (akn – akjain/aij) xn + (ck – akjci/aij)

x1 …xj …xn y1 a11 … a1j … a1n c1 … yi ai1 … aij … ain ci … ym am1 … amj … amn cm x1 …yi…xn y1 (a11 – a1jai1/aij)… (a1j/aij) … (a1n – a1jain/aij)(c1 – a1jci/aij) … xj (- ai1/aij) … (1/aij) … (- ain/aij)(-ci/aij) … ym (am1 – amjai1/aij) … (amj/aij) … (amn – amjain/aij)(cm – amjci/aij)

Observation A pivot operation preserves the solution set of any tableau.

x y a 1 -1 1 b 1 1 3 c -1 0 -4 x y a 1 -1 1 b 1 1 3 c -1 0 -4 Drop sign constraint for c Pivot a and x a b x 1/2 1/2 -2 y -1/2 1/2 -1 c -1/2 -1/2 -2 a y x 1 1 -1 b 1 2 2 c -1 -1 -3 Pivot b and y

c = 0 x = 0 a = 0 y = 0 b = 0

Manifestly maximized row variable A row variable is manifestly maximized if every non-zero entry, other than the entry in the constant column, in its row is negative and lies in a column owned by a restricted variable. m n x y 1 -1 2 0 l -1 -3 0 -1 • - l is manifestly maximized in the above tableau. • l is constrained to be at most -1. • y is not manifestly maximized in the above tableau.

Manifestly unbounded column variable A column variable is manifestly unbounded if every negative entry in its column is in a row owned by an unrestricted variable. x u l 1 -1 0 y -1 -1 1 z -1 -2 -1 m 0 1 2 • x is manifestly unbounded in the above tableau. • x can take arbitrarily large values. • u is not manifestly unbounded in the above tableau.

Observation • Given a feasible tableau T and a variable v, there • is a sequence of pivot operations on T leading to a • tableau T’ such that either • v is manifestly maximized in T’, or • 2. v is manifestly unbounded in T’

Algorithm • Create initial tableau T with only those sign constraints that are • satisfied by the sample point of T • 2. If every row variable satisfies its sign constraint, return satisfiable • 3. Pick a row k owned by variable y such that the sign constraint is • not satisfied by the sample point of T • 4. If y is manifestly maximized in T, return unsatisfiable • 5. Pick a column j such that akj is positive • 6. If every restricted row has a non-negative entry in column j, • perform Pivot(k,j). y becomes manifestly unbounded in T. • Therefore, add the sign constraint for y. Go to 2. • 7. (i, j) = ComputePivot(k) • 8. Perform Pivot(T,i,j) • 9. If the sample point of T satisfies the sign constraint for y, then • add the sign constraint for y. Go to 2. • 9. Go to 4

Observation • If a row variable y is not manifestly maximized • either there is a positive entry in some column • or there is a negative entry in a column owned by an unrestricted variable

Algorithm • Create initial tableau T with only those sign constraints that are • satisfied by the sample point of T • 2. If every row variable satisfies its sign constraint, return satisfiable • 3. Pick a row k owned by variable y such that the sign constraint is • not satisfied by the sample point of T • 4. If y is manifestly maximized in T, return unsatisfiable • 5’. Pick a column j such that akj is negative and the variable in column j • is unrestricted. • 6. If every restricted row has a non-positive entry in column j, • perform Pivot(k,j). y becomes manifestly unbounded in T. • Therefore, add the sign constraint for y. Go to 2. • 7. (i, j) = ComputePivot(k) • 8. Perform Pivot(T,i,j) • 9. If the sample point of T satisfies the sign constraint for y, then • add the sign constraint for y. Go to 2. • 9. Go to 4

Pratt’s Algorithm for Difference Constraints

Difference constraints Three different kinds of constraints: x – y  c x  c -y  c • - very common in program verification • satisfiability procedure more efficient than • for general affine constraints • - satisfiability procedure complete for integers

Variable x Vertex x Constraint x – y  c Edge from y to x with weight c Reduction to a graph problem Introduce a new variable z to denote the value 0 x - z  c x  c z - y  c -y  c - Add a new vertex s. - Add an edge with weight 0 from s to every other vertex v.

Theorem The set of constraints is satisfiable iff there is no negative cycle in the graph.

Soundness If there is a negative cycle in the graph, the set of constraints is unsatisfiable. x1 - x2  c1 x2 - x3  c2 … xn - x1  cn 0  c1 + c2 + … + cn < 0

Completeness If there is no negative cycle in the graph, the set of constraints is satisfiable.

Bellman-Ford algorithm d(s) := 0 for each vertex v  s: d(v) :=  for each vertex: for each edge (u,v): if d(v) > d(u) + weight(u,v) d(v) := d(u) + weight(u,v) for each edge (u,v): if d(v) > d(u) + weight(u,v) Graph contains a negative-weight cycle

Completeness If there is no negative cycle in the graph, then d(v) - d(u)  weight(u,v) for each edge (u,v). Model: Assign to variable x the value d(x) –d(z).

Automated Theorem Proving