diff --git a/src/problems/MultipleSource/algo/matrix_bfs/intersection.py b/src/problems/MultipleSource/algo/matrix_bfs/intersection.py
new file mode 100644
index 0000000..85dfb20
--- /dev/null
+++ b/src/problems/MultipleSource/algo/matrix_bfs/intersection.py
@@ -0,0 +1,200 @@
+from itertools import product
+from typing import Dict
+
+from pyformlang.finite_automaton.state import State
+from pyformlang.finite_automaton import EpsilonNFA
+from pyformlang.finite_automaton.symbol import Symbol
+
+from pygraphblas.types import BOOL
+from pygraphblas.matrix import Matrix
+from pygraphblas.vector import Vector
+from pygraphblas import descriptor
+from pygraphblas import Accum, binaryop
+
+from src.graph.graph import Graph
+from src.problems.MultipleSource.algo.matrix_bfs.reg_automaton import RegAutomaton
+
+
+class Intersection:
+    """
+    Implementations of graph and regular grammar intersection algorithm
+    """
+
+    def __init__(self, graph: Graph, regular_automaton: RegAutomaton):
+        self.graph = graph
+        self.graph.load_bool_graph()
+        self.regular_automaton = regular_automaton
+        self.intersection_matrices = dict()
+        self.__create_intersection_matrices__()
+
+    def __create_intersection_matrices__(self):
+        num_vert_graph = self.graph.get_number_of_vertices()
+        num_vert_regex = self.regular_automaton.num_states
+        num_verts_inter = num_vert_graph * num_vert_regex
+
+        for symbol in self.regular_automaton.matrices:
+            if symbol in self.graph:
+                self.intersection_matrices[symbol] = Matrix.sparse(
+                    BOOL, num_verts_inter, num_verts_inter
+                )
+
+    def __to_automaton__(self) -> EpsilonNFA:
+        """
+        Build automata from matrices
+        """
+        enfa = EpsilonNFA()
+        graph_vertices_num = self.graph.get_number_of_vertices()
+
+        start_states = [
+            self.to_inter_coord(x, y)
+            for x, y in product(
+                range(graph_vertices_num), self.regular_automaton.start_states
+            )
+        ]
+
+        final_states = [
+            self.to_inter_coord(x, y)
+            for x, y in product(
+                range(graph_vertices_num), self.regular_automaton.final_states
+            )
+        ]
+
+        for start_state in start_states:
+            enfa.add_start_state(State(start_state))
+
+        for final_state in final_states:
+            enfa.add_final_state(State(final_state))
+
+        for symbol in self.intersection_matrices:
+            matrix = self.intersection_matrices[symbol]
+
+            for row, col in zip(matrix.rows, matrix.cols):
+                enfa.add_transition(State(row), Symbol(symbol), State(col))
+
+        return enfa
+
+    def to_inter_coord(self, graph_vert, reg_vert) -> int:
+        """
+        Converts coordinates of graph vertice and regex vertice
+        to intersection coordinates vertice
+        """
+        return reg_vert * self.graph.get_number_of_vertices() + graph_vert
+
+    def create_diag_matrices(self) -> Dict[str, Matrix]:
+        """
+        Create a block diagonal matrices from graph and regex matrices for each symbol
+        """
+        num_vert_graph = self.graph.get_number_of_vertices()
+        num_vert_regex = self.regular_automaton.num_states
+        diag_num_verts = num_vert_graph + num_vert_regex
+
+        diag_matrices = dict()
+        for symbol in self.regular_automaton.matrices:
+            if symbol in self.graph:
+                diag_matrix = Matrix.sparse(BOOL, diag_num_verts, diag_num_verts)
+                diag_matrix.assign_matrix(
+                    self.regular_automaton.matrices[symbol],
+                    slice(0, num_vert_regex - 1),
+                    slice(0, num_vert_regex - 1),
+                )
+                diag_matrix.assign_matrix(
+                    self.graph[symbol],
+                    slice(num_vert_regex, diag_num_verts - 1),
+                    slice(num_vert_regex, diag_num_verts - 1),
+                )
+
+                diag_matrices[symbol] = diag_matrix
+
+        return diag_matrices
+
+    def create_masks_matrix(self) -> Matrix:
+        num_vert_graph = self.graph.get_number_of_vertices()
+        num_vert_regex = self.regular_automaton.num_states
+        num_verts_diag = num_vert_graph + num_vert_regex
+
+        mask_matrix = Matrix.identity(BOOL, num_vert_regex, value=True)
+        mask_matrix.resize(num_vert_regex, num_verts_diag)
+
+        return mask_matrix
+
+    def intersect_bfs(self, src_verts) -> EpsilonNFA:
+        """
+        Intersection implementation with synchronous breadth first traversal
+        of a graph and regular grammar represented in automata
+        """
+        num_vert_graph = self.graph.get_number_of_vertices()
+        num_vert_regex = self.regular_automaton.num_states
+
+        num_verts_inter = num_vert_graph * num_vert_regex
+        num_verts_diag = num_vert_graph + num_vert_regex
+
+        graph = self.graph
+        regex = self.regular_automaton.matrices
+
+        regex_start_states = self.regular_automaton.start_states
+
+        diag_matrices = self.create_diag_matrices()
+
+        result = Matrix.sparse(BOOL, num_vert_graph, num_vert_graph)
+
+        # create a mask of source vertices vector
+        m_src_v = Vector.from_lists(src_verts, [True for _ in range(len(src_verts))], size=num_vert_graph)
+
+        # initialize matrices for multiple source bfs
+        ident = self.create_masks_matrix()
+        vect = ident.dup()
+        found = ident.dup()
+
+        # fill start states
+        for reg_start_state in regex_start_states:
+            for gr_start_state in src_verts:
+                found[reg_start_state, num_vert_regex + gr_start_state] = True
+
+        # matrix which contains newly found nodes on each iteration
+        found_on_iter = found.dup()
+
+        # Algo's body
+        not_empty = True
+        level = 0
+        while not_empty and level < num_verts_inter:
+            # for each symbol we are going to store if any new nodes were found during traversal.
+            # if none are found, then 'not_empty' flag turns False, which means that no matrices change anymore
+            # and we can stop the traversal
+            not_empty_for_at_least_one_symbol = False
+
+            vect.assign_matrix(found_on_iter, mask=vect, desc=descriptor.RC)
+            vect.assign_scalar(True, mask=ident)
+
+            # stores found nodes for each symbol
+            found_on_iter.assign_matrix(ident)
+
+            for symbol in regex:
+                if symbol in graph:
+                    with BOOL.ANY_PAIR:
+                        found = vect.mxm(diag_matrices[symbol])
+
+                    with Accum(binaryop.MAX_BOOL):
+                        # extract left (grammar) part of the masks matrix and rearrange rows
+                        i_x, i_y, _ = found.extract_matrix(col_index=slice(0, num_vert_regex - 1)).to_lists()
+                        for i in range(len(i_y)):
+                            found_on_iter.assign_row(i_y[i], found.extract_row(i_x[i]))
+
+                    # check if new nodes were found. if positive, switch the flag
+                    if not found_on_iter.iseq(vect):
+                        not_empty_for_at_least_one_symbol = True
+
+            # extract right (graph) part of the masks matrix and get a row of reachable nodes in a graph
+            reachable = found_on_iter.extract_matrix(
+                col_index=slice(num_vert_regex, num_verts_diag - 1)
+            ).T.reduce_vector(BOOL.ANY_MONOID) # reduce by columns
+
+            # update graph boolean matrix for every source vertex
+            # result matrix contains reachability for every symbol combined
+            with Accum(binaryop.MAX_BOOL):
+                for st_v in src_verts:
+                    result.assign_row(st_v, reachable, mask=m_src_v, desc=descriptor.C)
+
+            not_empty = not_empty_for_at_least_one_symbol
+            level += 1
+
+        return result
\ No newline at end of file
diff --git a/src/problems/MultipleSource/algo/matrix_bfs/matrix_bfs.py b/src/problems/MultipleSource/algo/matrix_bfs/matrix_bfs.py
new file mode 100644
index 0000000..13f626e
--- /dev/null
+++ b/src/problems/MultipleSource/algo/matrix_bfs/matrix_bfs.py
@@ -0,0 +1,31 @@
+from pyformlang.cfg import CFG
+
+from src.grammar.rsa import RecursiveAutomaton
+
+from src.graph.graph import Graph
+from src.graph.label_graph import LabelGraph
+
+from src.problems.AllPaths.AllPaths import AllPathsProblem
+from src.problems.utils import ResultAlgo
+
+
+class ProblemAlgo(AllPathsProblem):
+    """
+    For now we have regular grammar only in this algo.
+    Hence this is to be implemented with CFG.
+    """
+
+    def prepare(self, graph: Graph, grammar: CFG):
+        pass
+
+    def prepare_for_solve(self):
+        pass
+
+    def solve(self):
+        pass
+
+    def prepare_for_exctract_paths(self):
+        pass
+
+    def getPaths(self, v_start: int, v_finish: int, nonterminal: str, max_len: int):
+        pass
diff --git a/src/problems/MultipleSource/algo/matrix_bfs/reg_automaton.py b/src/problems/MultipleSource/algo/matrix_bfs/reg_automaton.py
new file mode 100644
index 0000000..2883a2d
--- /dev/null
+++ b/src/problems/MultipleSource/algo/matrix_bfs/reg_automaton.py
@@ -0,0 +1,49 @@
+from __future__ import annotations
+
+from pyformlang.regular_expression.regex import Regex
+
+from pygraphblas.matrix import Matrix
+from pygraphblas.types import BOOL
+
+
+class RegAutomaton:
+    """
+    Automata representation of regular grammar
+    """
+
+    def __init__(self, regex: Regex):
+        self.enfa = regex.to_epsilon_nfa().minimize()
+
+        self.states = self.enfa.states
+        self.num_states = len(self.states)
+
+        self.enum_states = dict(zip(self.states, range(self.num_states)))
+        self.start_states = [
+            self.enum_states[state] for state in self.enfa.start_states
+        ]
+        self.final_states = [
+            self.enum_states[state] for state in self.enfa.final_states
+        ]
+
+        self.matrices = dict()
+        self.load_bool_matrices()
+
+    def from_regex_txt(path) -> RegAutomaton:
+        with open(path, "r") as file:
+            regex = Regex(file.readline())
+
+            return RegAutomaton(regex)
+
+    def load_bool_matrices(self) -> None:
+        """
+        Creates boolean matrices for self automata
+        """
+        for src_node, transition in self.enfa.to_dict().items():
+            for symbol, tgt_node in transition.items():
+                if symbol not in self.matrices:
+                    self.matrices[symbol] = Matrix.sparse(
+                        BOOL, self.num_states, self.num_states
+                    )
+
+                matr = self.matrices[symbol]
+                matr[self.enum_states[src_node], self.enum_states[tgt_node]] = True
diff --git a/test/MultipleSource/test_bfs.py b/test/MultipleSource/test_bfs.py
new file mode 100644
index 0000000..eb76752
--- /dev/null
+++ b/test/MultipleSource/test_bfs.py
@@ -0,0 +1,93 @@
+import pytest
+
+from src.graph.graph import Graph
+
+from src.problems.MultipleSource.algo.matrix_bfs.intersection import Intersection
+from src.problems.MultipleSource.algo.matrix_bfs.reg_automaton import RegAutomaton
+
+from src.utils.useful_paths import LOCAL_CFPQ_DATA
+
+
+@pytest.mark.CI
+def test_case_regular_cycle():
+    test_data_path = LOCAL_CFPQ_DATA.joinpath("regular/cycle")
+
+    graph = Graph.from_txt(test_data_path.joinpath("Graphs/graph_1.txt"))
+    grammar = RegAutomaton.from_regex_txt(
+        test_data_path.joinpath("Grammars/regex_1.txt")
+    )
+
+    intersection = Intersection(graph, grammar)
+    
+    source_verts = [0]
+    result = intersection.intersect_bfs(source_verts)
+
+    assert result.nvals == 2 * len(source_verts)
+
+
+@pytest.mark.CI
+def test_case_regular_disconnected():
+    test_data_path = LOCAL_CFPQ_DATA.joinpath("regular/disconnected")
+
+    graph = Graph.from_txt(test_data_path.joinpath("Graphs/graph_1.txt"))
+    grammar = RegAutomaton.from_regex_txt(
+        test_data_path.joinpath("Grammars/regex_1.txt")
+    )
+
+    intersection = Intersection(graph, grammar)
+    
+    source_verts = [0, 3]
+    result = intersection.intersect_bfs(source_verts)
+
+    assert result.nvals == 2 * len(source_verts)
+
+
+@pytest.mark.CI
+def test_case_regular_loop():
+    test_data_path = LOCAL_CFPQ_DATA.joinpath("regular/loop")
+
+    graph = Graph.from_txt(test_data_path.joinpath("Graphs/graph_1.txt"))
+    grammar = RegAutomaton.from_regex_txt(
+        test_data_path.joinpath("Grammars/regex_1.txt")
+    )
+
+    intersection = Intersection(graph, grammar)
+    
+    source_verts = [0, 2]
+    result = intersection.intersect_bfs(source_verts)
+
+    assert result.nvals == 0 * len(source_verts)
+
+
+@pytest.mark.CI
+def test_case_regular_midsymbol():
+    test_data_path = LOCAL_CFPQ_DATA.joinpath("regular/midsymbol")
+
+    graph = Graph.from_txt(test_data_path.joinpath("Graphs/graph_1.txt"))
+    grammar = RegAutomaton.from_regex_txt(
+        test_data_path.joinpath("Grammars/regex_1.txt")
+    )
+
+    intersection = Intersection(graph, grammar)
+    
+    source_verts = [0]
+    result = intersection.intersect_bfs(source_verts)
+
+    assert result.nvals == 1 * len(source_verts)
+
+
+@pytest.mark.CI
+def test_case_regular_two_cycles():
+    test_data_path = LOCAL_CFPQ_DATA.joinpath("regular/two_cycles")
+
+    graph = Graph.from_txt(test_data_path.joinpath("Graphs/graph_1.txt"))
+    grammar = RegAutomaton.from_regex_txt(
+        test_data_path.joinpath("Grammars/regex_1.txt")
+    )
+
+    intersection = Intersection(graph, grammar)
+    
+    source_verts = [0, 3]
+    result = intersection.intersect_bfs(source_verts)
+
+    assert result.nvals == 2 * len(source_verts)
diff --git a/test/data/regular/cycle/Grammars/regex_1.txt b/test/data/regular/cycle/Grammars/regex_1.txt
new file mode 100644
index 0000000..08c2643
--- /dev/null
+++ b/test/data/regular/cycle/Grammars/regex_1.txt
@@ -0,0 +1 @@
+a a*
\ No newline at end of file
diff --git a/test/data/regular/cycle/Graphs/graph_1.txt b/test/data/regular/cycle/Graphs/graph_1.txt
new file mode 100644
index 0000000..a675802
--- /dev/null
+++ b/test/data/regular/cycle/Graphs/graph_1.txt
@@ -0,0 +1,3 @@
+0 a 1
+1 a 2
+2 a 0
\ No newline at end of file
diff --git a/test/data/regular/disconnected/Grammars/regex_1.txt b/test/data/regular/disconnected/Grammars/regex_1.txt
new file mode 100644
index 0000000..35456ac
--- /dev/null
+++ b/test/data/regular/disconnected/Grammars/regex_1.txt
@@ -0,0 +1 @@
+a* b (a|b)*
\ No newline at end of file
diff --git a/test/data/regular/disconnected/Graphs/graph_1.txt b/test/data/regular/disconnected/Graphs/graph_1.txt
new file mode 100644
index 0000000..7296430
--- /dev/null
+++ b/test/data/regular/disconnected/Graphs/graph_1.txt
@@ -0,0 +1,8 @@
+0 a 1
+0 b 0
+1 a 1
+1 b 2
+2 a 2
+2 b 2
+3 a 3
+3 b 3
\ No newline at end of file
diff --git a/test/data/regular/loop/Grammars/regex_1.txt b/test/data/regular/loop/Grammars/regex_1.txt
new file mode 100644
index 0000000..08c2643
--- /dev/null
+++ b/test/data/regular/loop/Grammars/regex_1.txt
@@ -0,0 +1 @@
+a a*
\ No newline at end of file
diff --git a/test/data/regular/loop/Graphs/graph_1.txt b/test/data/regular/loop/Graphs/graph_1.txt
new file mode 100644
index 0000000..17294dd
--- /dev/null
+++ b/test/data/regular/loop/Graphs/graph_1.txt
@@ -0,0 +1,3 @@
+0 a 0
+0 b 1
+1 c 2
\ No newline at end of file
diff --git a/test/data/regular/midsymbol/Grammars/regex_1.txt b/test/data/regular/midsymbol/Grammars/regex_1.txt
new file mode 100644
index 0000000..76ece65
--- /dev/null
+++ b/test/data/regular/midsymbol/Grammars/regex_1.txt
@@ -0,0 +1 @@
+a* b c*
\ No newline at end of file
diff --git a/test/data/regular/midsymbol/Graphs/graph_1.txt b/test/data/regular/midsymbol/Graphs/graph_1.txt
new file mode 100644
index 0000000..3fe9500
--- /dev/null
+++ b/test/data/regular/midsymbol/Graphs/graph_1.txt
@@ -0,0 +1,3 @@
+0 a 0
+0 b 1
+1 c 1
\ No newline at end of file
diff --git a/test/data/regular/two_cycles/Grammars/regex_1.txt b/test/data/regular/two_cycles/Grammars/regex_1.txt
new file mode 100644
index 0000000..08c2643
--- /dev/null
+++ b/test/data/regular/two_cycles/Grammars/regex_1.txt
@@ -0,0 +1 @@
+a a*
\ No newline at end of file
diff --git a/test/data/regular/two_cycles/Graphs/graph_1.txt b/test/data/regular/two_cycles/Graphs/graph_1.txt
new file mode 100644
index 0000000..17d1fba
--- /dev/null
+++ b/test/data/regular/two_cycles/Graphs/graph_1.txt
@@ -0,0 +1,6 @@
+0 a 1
+1 a 2
+2 b 1
+2 b 3
+3 a 2
+3 c 4
\ No newline at end of file