Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DAG Shortest Path (n=3, weight 2)


Label-increasing edges guarantee acyclicity by construction.

the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

the defining fixpoint admits exactly one value

Answer: 1

the unique verified answer

import json
import sys

data = json.load(sys.stdin)
n = data["num_vertices"]
edges = data["edges"]
target_vertex = data["target_vertex"]

successors = [[] for _ in range(n)]
predecessors = [[] for _ in range(n)]
for u, v in edges:
    successors[u].append(v)
    predecessors[v].append(u)

indegree = [len(preds) for preds in predecessors]
order = []
ready = [vertex for vertex in range(n) if indegree[vertex] < 1]
while ready:
    ready.sort()
    current = ready.pop(0)
    order.append(current)
    if current == target_vertex:
        break
    for nxt in successors[current]:
        indegree[nxt] -= 1
        if indegree[nxt] < 1:
            ready.append(nxt)

print(order.index(target_vertex))

You are given a directed acyclic graph with 4 vertices labelled 0 to
3 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 1 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=4, vertex 1)

Explanation: Canonical verified instance: Z3 proved satisfiability and blocked 1 model(s) to establish uniqueness.


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_1 member_2)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

import json
import sys

data = json.load(sys.stdin)
n = data["num_vertices"]
adjacency = data["adjacency"]
source = data["source"]

seen = [0] * n
seen[source] = 1
stack = [source]
while stack:
    vertex = stack.pop()
    for nxt in adjacency[vertex]:
        if not seen[nxt]:
            seen[nxt] = 1
            stack.append(nxt)
total = 0
for flag in seen:
    if flag:
        total += 1
print(total)

You are given a directed acyclic graph with 4 vertices labelled 0 to 3,
described by adjacency lists. Starting from the source vertex s = 0, a
vertex is reachable when some directed path leads to it; the source itself is
always reachable.

Print how many vertices are reachable from s. The exact count is unique and
its uniqueness has been formally proven for this instance with the Z3 SMT
solver.

DAG Reachability Count (n=4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_2 member_3)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_2))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Reachability Count (n=4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

import json
import sys

data = json.load(sys.stdin)
n = data["num_vertices"]
edges = data["edges"]
source = data["source"]
target = data["target"]

infinity = float("inf")
dist = [infinity] * n
dist[source] = 0
for u, v, w in edges:
    if dist[u] != infinity and dist[u] + w < dist[v]:
        dist[v] = dist[u] + w
print(dist[target])

You are given a directed acyclic graph with 6 vertices labelled 0 to
5. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 5. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=6, weight 2)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

You are given a directed acyclic graph with 4 vertices labelled 0 to
3. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 3. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=4, weight 7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a directed acyclic graph with 5 vertices labelled 0 to
4 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 2 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=5, vertex 2)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_0 member_4)
     (=> member_1 member_2)
     (=> member_2 member_5)
     (=> member_4 member_5)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0))
     (=> member_4 (or member_0))
     (=> member_5 (or member_2 member_4))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0)
           (ite member_5 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

You are given a directed acyclic graph with 6 vertices labelled 0 to 5,
described by adjacency lists. Starting from the source vertex s = 0, a
vertex is reachable when some directed path leads to it; the source itself is
always reachable.

Print how many vertices are reachable from s. The exact count is unique and
its uniqueness has been formally proven for this instance with the Z3 SMT
solver.

DAG Reachability Count (n=6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 15

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to
6. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 6. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=7, weight 15)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

DAG Topological Position (n=4, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_4)
     (=> member_2 member_3)
     (=> member_2 member_5)
     (=> member_3 member_5)
     (=> member_4 member_5)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_2))
     (=> member_4 (or member_0))
     (=> member_5 (or member_2 member_3 member_4))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0)
           (ite member_5 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Reachability Count (n=6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to
6 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 1 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=7, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_1 member_2)
     (=> member_1 member_4)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_3 member_4)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_2))
     (=> member_4 (or member_1 member_2 member_3))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given a directed acyclic graph with 5 vertices labelled 0 to 4,
described by adjacency lists. Starting from the source vertex s = 0, a
vertex is reachable when some directed path leads to it; the source itself is
always reachable.

Print how many vertices are reachable from s. The exact count is unique and
its uniqueness has been formally proven for this instance with the Z3 SMT
solver.

DAG Reachability Count (n=5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_1 member_2)
     (=> member_2 member_3)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0 member_1))
     (=> member_3 (or member_2))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Reachability Count (n=4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to
6 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 5 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=7, vertex 5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a directed acyclic graph with 6 vertices labelled 0 to
5 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 2 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=6, vertex 2)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_4)
     (=> member_1 member_4)
     (=> member_2 member_3)
     (=> member_3 member_4)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_2))
     (=> member_4 (or member_0 member_1 member_3))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

DAG Reachability Count (n=5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_1 member_3)
     (=> member_2 member_3)
     (=> member_3 member_4)
     (=> member_4 member_5)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_0 member_1 member_2))
     (=> member_4 (or member_3))
     (=> member_5 (or member_4))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0)
           (ite member_5 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Reachability Count (n=6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0 member_1))
     (=> member_3 (or member_1))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Reachability Count (n=4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

You are given a directed acyclic graph with 5 vertices labelled 0 to
4 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 1 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=5, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

You are given a directed acyclic graph with 6 vertices labelled 0 to
5 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 1 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=6, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_4)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_2 member_4)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_1))
     (=> member_4 (or member_0 member_2))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

DAG Reachability Count (n=5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 15

the unique verified answer

DAG Shortest Path (n=4, weight 15)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

DAG Topological Position (n=7, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 16

the unique verified answer

You are given a directed acyclic graph with 5 vertices labelled 0 to
4. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 4. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=5, weight 16)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

DAG Topological Position (n=4, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

DAG Topological Position (n=4, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_3 member_4)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0 member_1))
     (=> member_4 (or member_3))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_4 1 0))))

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

DAG Reachability Count (n=5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to
6 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 3 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=7, vertex 3)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_0 member_5)
     (=> member_1 member_3)
     (=> member_1 member_5)
     (=> member_2 member_4)
     (=> member_2 member_5)
     (=> member_2 member_6)
     (=> member_4 member_5)
     (=> member_5 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_0 member_1))
     (=> member_4 (or member_2))
     (=> member_5 (or member_0 member_1 member_2 member_4))
     (=> member_6 (or member_2 member_5))
     (= reachable_count
        ...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to 6,
described by adjacency lists. Starting from the source vertex s = 0, a
vertex is reachable when some directed path leads to it; the source itself is
always reachable.

Print how many vertices are reachable from s. The exact count is unique and
its uniqueness has been formally proven for this instance with the Z3 SMT
solver.

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 13

the unique verified answer

DAG Shortest Path (n=7, weight 13)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 16

the unique verified answer

DAG Shortest Path (n=6, weight 16)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_4)
     (=> member_0 member_5)
     (=> member_0 member_6)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 member_5)
     (=> member_1 member_6)
     (=> member_2 member_7)
     (=> member_3 member_4)
     (=> member_3 member_6)
     (=> member_3 member_7)
     (=> member_4 member_5)
     (=> member_4 member_7)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_1))
     (=> member_4 (or member_0 member_3))
     (=> member_5 (or member_0 member_1 member_4))...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to 7,
described by adjacency lists. Starting from the source vertex s = 0, a
vertex is reachable when some directed path leads to it; the source itself is
always reachable.

Print how many vertices are reachable from s. The exact count is unique and
its uniqueness has been formally proven for this instance with the Z3 SMT
solver.

DAG Reachability Count (n=8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_4)
     (=> member_0 member_5)
     (=> member_1 member_3)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_2 member_5)
     (=> member_4 member_5)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_1 member_2))
     (=> member_4 (or member_0 member_2))
     (=> member_5 (or member_0 member_2 member_4))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite ...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Reachability Count (n=6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given a directed acyclic graph with 6 vertices labelled 0 to
5 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 3 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=6, vertex 3)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_0 member_5)
     (=> member_0 member_7)
     (=> member_1 member_2)
     (=> member_1 member_4)
     (=> member_2 member_3)
     (=> member_2 member_5)
     (=> member_2 member_6)
     (=> member_3 member_5)
     (=> member_3 member_7)
     (=> member_5 member_7)
     (=> member_6 member_7)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0 member_2))
     (=> member_4 (or member_1))
     (=> member_5 (or member_0 member_2 member_3))
     (=> member_6 (or membe...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

DAG Reachability Count (n=8)


Label-increasing edges guarantee acyclicity by construction.

the input is a DAG: every edge (u, v) satisfies u < v, weights lie in [1, 9]

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

the defining fixpoint admits exactly one value

Answer: 4

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to
6 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 4 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=7, vertex 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to
7. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 7. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=8, weight 8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Shortest Path (n=8, weight 6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

DAG Topological Position (n=6, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to
7 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 2 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=8, vertex 2)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Topological Position (n=7, vertex 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 10

the unique verified answer

DAG Shortest Path (n=7, weight 10)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a directed acyclic graph with 6 vertices labelled 0 to
5. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 5. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=6, weight 2)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Shortest Path (n=7, weight 6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

DAG Topological Position (n=7, vertex 5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to
7 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 4 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=8, vertex 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_1 member_3)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_3 member_5)
     (=> member_3 member_6)
     (=> member_4 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_0 member_1 member_2))
     (=> member_4 (or member_2))
     (=> member_5 (or member_3))
     (=> member_6 (or member_3 member_4))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given a directed acyclic graph with 7 vertices labelled 0 to
6. Every edge [u, v, w] goes from a lower to a higher label and has
integer weight 1 <= w <= 9; edges are listed in ascending (u, v) order.

Print the minimum possible total weight of a path from source s = 0 to
target t = 6. The target is guaranteed reachable, the optimum is
unique, and both facts have been formally proven for this instance with the
Z3 SMT solver.

DAG Shortest Path (n=7, weight 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_4)
     (=> member_1 member_2)
     (=> member_1 member_4)
     (=> member_1 member_5)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_3 member_4)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_2))
     (=> member_4 (or member_0 member_1 member_2 member_3))
     (=> member_5 (or member_1))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member_2 1 0)
           (ite member_3 1 0)
           (ite member_...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

You are given a directed acyclic graph with 6 vertices labelled 0 to 5,
described by adjacency lists. Starting from the source vertex s = 0, a
vertex is reachable when some directed path leads to it; the source itself is
always reachable.

Print how many vertices are reachable from s. The exact count is unique and
its uniqueness has been formally proven for this instance with the Z3 SMT
solver.

DAG Reachability Count (n=6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_1 member_3)
     (=> member_2 member_4)
     (=> member_2 member_5)
     (=> member_2 member_6)
     (=> member_3 member_4)
     (=> member_4 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_0 member_1))
     (=> member_4 (or member_2 member_3))
     (=> member_5 (or member_2))
     (=> member_6 (or member_2 member_4))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0)
           (ite member...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 9

the unique verified answer

DAG Shortest Path (n=6, weight 9)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_0 member_5)
     (=> member_1 member_2)
     (=> member_1 member_4)
     (=> member_1 member_6)
     (=> member_2 member_5)
     (=> member_2 member_6)
     (=> member_3 member_4)
     (=> member_5 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0))
     (=> member_4 (or member_1 member_3))
     (=> member_5 (or member_0 member_2))
     (=> member_6 (or member_1 member_2 member_5))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

DAG Topological Position (n=6, vertex 3)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_4)
     (=> member_0 member_5)
     (=> member_0 member_6)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 member_4)
     (=> member_2 member_4)
     (=> member_2 member_5)
     (=> member_3 member_4)
     (=> member_3 member_5)
     (=> member_3 member_6)
     (=> member_4 member_5)
     (=> member_4 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_1))
     (=> member_4 (or member_0 member_1 member_2 member_3))
     (=> member_5 (or member_0 m...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_0 member_4)
     (=> member_0 member_6)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 member_4)
     (=> member_1 member_6)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_2 member_5)
     (=> member_2 member_6)
     (=> member_3 member_4)
     (=> member_3 member_5)
     (=> member_4 member_5)
     (=> member_4 member_6)
     (=> member_5 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0 member_1))
    ...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_7)
     (=> member_1 member_3)
     (=> member_1 member_4)
     (=> member_1 member_5)
     (=> member_1 member_6)
     (=> member_2 member_4)
     (=> member_2 member_6)
     (=> member_2 member_7)
     (=> member_3 member_4)
     (=> member_3 member_7)
     (=> member_4 member_5)
     (=> member_4 member_6)
     (=> member_4 member_7)
     (=> member_5 member_7)
     (=> member_6 member_7)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_1))
     (=>...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

DAG Reachability Count (n=8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_6)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 member_5)
     (=> member_1 member_6)
     (=> member_2 member_3)
     (=> member_3 member_4)
     (=> member_3 member_6)
     (=> member_4 member_5)
     (=> member_4 member_6)
     (=> member_5 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0 member_1))
     (=> member_3 (or member_1 member_2))
     (=> member_4 (or member_3))
     (=> member_5 (or member_1 member_4))
     (=> member_6 (or membe...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_0 member_5)
     (=> member_0 member_6)
     (=> member_1 member_2)
     (=> member_1 member_6)
     (=> member_2 member_4)
     (=> member_2 member_5)
     (=> member_2 member_6)
     (=> member_3 member_4)
     (=> member_3 member_5)
     (=> member_3 member_6)
     (=> member_4 member_5)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0))
     (=> member_4 (or member_2 member_3))
     (=> member_5 (or member_0 member_2 member_3 member_4))
     (=> member_6 ...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_0 member_4)
     (=> member_0 member_6)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 member_4)
     (=> member_1 member_6)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_2 member_6)
     (=> member_3 member_6)
     (=> member_4 member_5)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0 member_1))
     (=> member_3 (or member_0 member_1 member_2))
     (=> member_4 (or member_0 member_1 member_2))
     (=> membe...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

DAG Shortest Path (n=8, weight 3)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Shortest Path (n=7, weight 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_0 member_5)
     (=> member_1 member_2)
     (=> member_1 member_3)
     (=> member_1 member_4)
     (=> member_2 member_3)
     (=> member_2 member_4)
     (=> member_2 member_6)
     (=> member_2 member_7)
     (=> member_3 member_4)
     (=> member_3 member_5)
     (=> member_3 member_6)
     (=> member_3 member_7)
     (=> member_4 member_6)
     (=> member_4 member_7)
     (=> member_5 member_6)
     (=> member_5 member_7)
     (=> member_1 (or member_0))
     (=> member_2 ...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

DAG Reachability Count (n=8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Shortest Path (n=7, weight 6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

DAG Topological Position (n=7, vertex 5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Shortest Path (n=7, weight 7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

DAG Shortest Path (n=8, weight 6)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Shortest Path (n=7, weight 7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Topological Position (n=8, vertex 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Topological Position (n=7, vertex 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_0 member_4)
     (=> member_1 member_6)
     (=> member_2 member_4)
     (=> member_3 member_4)
     (=> member_3 member_5)
     (=> member_3 member_6)
     (=> member_4 member_5)
     (=> member_4 member_6)
     (=> member_5 member_6)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0))
     (=> member_3 (or member_0))
     (=> member_4 (or member_0 member_2 member_3))
     (=> member_5 (or member_3 member_4))
     (=> member_6 (or member_1 member_3 member_4 member...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

DAG Reachability Count (n=7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_2)
     (=> member_0 member_3)
     (=> member_0 member_4)
     (=> member_1 member_2)
     (=> member_1 member_5)
     (=> member_1 member_6)
     (=> member_2 member_3)
     (=> member_2 member_6)
     (=> member_3 member_4)
     (=> member_3 member_6)
     (=> member_3 member_7)
     (=> member_4 member_7)
     (=> member_5 member_6)
     (=> member_6 member_7)
     (=> member_1 (or member_0))
     (=> member_2 (or member_0 member_1))
     (=> member_3 (or member_0 member_2))
     (=> member_4 (or member_0 member_3))
    ...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

DAG Reachability Count (n=8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

DAG Shortest Path (n=7, weight 8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

(and member_0
     (=> member_0 member_1)
     (=> member_0 member_3)
     (=> member_1 member_2)
     (=> member_1 member_4)
     (=> member_1 member_5)
     (=> member_1 member_7)
     (=> member_2 member_6)
     (=> member_3 member_5)
     (=> member_5 member_7)
     (=> member_1 (or member_0))
     (=> member_2 (or member_1))
     (=> member_3 (or member_0))
     (=> member_4 (or member_1))
     (=> member_5 (or member_1 member_3))
     (=> member_6 (or member_2))
     (=> member_7 (or member_1 member_5))
     (= reachable_count
        (+ (ite member_0 1 0)
           (ite member_1 1 0...

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

DAG Reachability Count (n=8)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to
7 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 5 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=8, vertex 5)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DAG Shortest Path (n=8, weight 4)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to
7 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 7 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=8, vertex 7)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

You are given a directed acyclic graph with 8 vertices labelled 0 to
7 and edge list [u, v].

Run Kahn's algorithm with lexicographic tie-breaking: repeatedly remove the
smallest-labelled vertex whose remaining in-degree is zero, appending it to
the removal order (rounds are numbered from 0). Print the round at which the
designated vertex 1 is removed. The resulting removal order is unique;
existence and uniqueness of the answer have been formally proven with the
Z3 SMT solver.

DAG Topological Position (n=8, vertex 1)


Label-increasing edges guarantee acyclicity by construction.

The declarative semantics of the task over the concrete graph; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

DAG Shortest Path (n=8, weight 5)


Transition rows fully define the next-state function.

the automaton is complete and deterministic over {{0, 1}}: each row lists exactly two successor states

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

import json
import sys

data = json.load(sys.stdin)
transitions = data["transitions"]
start = data["start"]
accepting = data["accepting"]
length = data["length"]

size = len(accepting)
counts = [0] * size
counts[start] = 1
steps = 0
while steps < length:
    nxt = [0] * size
    for state in range(size):
        amount = counts[state]
        if amount != 0:
            for nxt_state in transitions[state]:
                nxt[nxt_state] += amount
    counts = nxt
    steps += 1
total = 0
for state in range(size):
    if accepting[state]:
        total += counts[state]
print(total)

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 2 states numbered 0 to 1. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

import json
import sys

data = json.load(sys.stdin)
first = data["first"]
second = data["second"]

rows = len(first["accepting"])
cols = len(second["accepting"])
seen = [[0] * cols for _ in range(rows)]
start_left = first["start"]
start_right = second["start"]
seen[start_left][start_right] = 1
stack = [[start_left, start_right]]
total = 0
while stack:
    left, right = stack.pop()
    total += 1
    for symbol in range(2):
        nxt_left = first["transitions"][left][symbol]
        nxt_right = second["transitions"][right][symbol]
        if not seen[nxt_left][nxt_right]:
            seen[nxt_left][nxt_right] = 1
            stack.append([nxt_left, nxt_right])
print(total)

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(0, 1) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (3x3 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 0) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (2x3 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 3 states numbered 0 to 2. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 2.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (2x4 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 2 states numbered 0 to 1. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 1.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(2, 0) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (3x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (2x3 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (2x3 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (2x2 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 3 states numbered 0 to 2. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 3 states numbered 0 to 2. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 1.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (2x3 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 1) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (2x3 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA String Acceptance Count (states=2, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA String Acceptance Count (states=3, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 2.

Print how many distinct binary strings of length exactly 4 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=4)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 16

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 3 states numbered 0 to 2. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 2.

Print how many distinct binary strings of length exactly 4 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=3, L=4)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(3, 0) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (4x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 2) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (3x4 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(2, 3) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (4x4 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(0, 2) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x7 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

DFA Product Reachability (4x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (4x4 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(2, 2) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x7 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 2.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(4, 1) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x7 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

DFA Product Reachability (3x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 4.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 3) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x6 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(3, 4) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x6 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(3, 2) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x6 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 1.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 3 states numbered 0 to 2. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=3, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

DFA Product Reachability (4x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 16

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 3 states numbered 0 to 2. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 4 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=3, L=4)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 4.

Print how many distinct binary strings of length exactly 2 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=2)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 4) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 7

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 3.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 1.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 6 states numbered 0 to 5. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 1.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=6, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 27

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 6 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=6)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(0, 1) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (4x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 13

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 4 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=4)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 5

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(2, 3) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 4

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 2) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (4x4 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 22

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 3.

Print how many distinct binary strings of length exactly 5 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=5)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

DFA Product Reachability (4x6 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 6

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 5 states numbered 0 to 4. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 4.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=5, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 2

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(0, 4) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (5x6 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 8

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 0.

Print how many distinct binary strings of length exactly 3 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=3)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 24

the unique verified answer

You are given a deterministic finite automaton over the binary alphabet
{0, 1} with 4 states numbered 0 to 3. The row transitions[s]
contains exactly two entries: the successor state after reading a 0 and the
successor state after reading a 1, respectively. The automaton starts in
state 2.

Print how many distinct binary strings of length exactly 5 move the
automaton from its start state to an accepting state. The count is unique;
it has been formally proven for this instance with the Z3 SMT solver.

DFA String Acceptance Count (states=4, L=5)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 3

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(1, 1) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (4x5 states)


Transition rows fully define the next-state function.

The declarative semantics of the task over the concrete machine; Z3 reports this system satisfiable.

Proven by Z3 model exclusion: blocking the witness model makes the system unsatisfiable.

Answer: 1

the unique verified answer

You are given two deterministic finite automata over the binary alphabet
{0, 1} that recognize the same language (equivalence holds by construction).
Consider their synchronous product: execution starts from the state pair
(2, 1) and every input symbol advances both machines
simultaneously.

Print how many distinct state pairs are reachable in this product. The count
is unique; it has been formally proven for this instance with the Z3 SMT
solver.

DFA Product Reachability (4x6 states)