def testSimplePredicateOptimization(self): nr_samples=100 ltnw.variable("?data_A",numpy.random.uniform([0.,0.],[.1,1.],(nr_samples,2)).astype("float32")) ltnw.variable("?data_not_A",numpy.random.uniform([2.,0],[3.,1.],(nr_samples,2)).astype("float32")) ltnw.predicate("A",2) ltnw.axiom("forall ?data_A: A(?data_A)") ltnw.axiom("forall ?data_not_A: ~A(?data_not_A)") ltnw.initialize_knowledgebase(initial_sat_level_threshold=.1) sat_level=ltnw.train(track_sat_levels=10000,sat_level_epsilon=.99) self.assertGreater(sat_level,.8) ltnw.constant("a",[0.5,0.5]) ltnw.constant("b",[2.5,0.5]) self.assertGreater(ltnw.ask("A(a)")[0],.8) self.assertGreater(ltnw.ask("~A(b)")[0],.8) result=ltnw.ask_m(["A(a)","~A(b)"]) for r in result: self.assertGreater(r[0],.8) self.assertGreater(r[0],.8)
W = tf.Variable(np.random.randn(), name="weight") b = tf.Variable(np.random.randn(), name="bias") ltnw.function("f", 1, 1, fun_definition=lambda X: tf.add(tf.multiply(X, W), b)) # defining an equal predicate based on the euclidian distance of two vectors ltnw.predicate("eq", 2, ltnl.equal_euclidian) # defining the theory for f in ["eq(f(x_%s),y_%s)" % (i, i) for i in range(len(train_X))]: ltnw.axiom(f) print("\n".join(sorted(ltnw.AXIOMS.keys()))) # initializing knowledgebase and optimizing ltnw.initialize_knowledgebase(optimizer=tf.train.GradientDescentOptimizer( learning_rate=learning_rate)) ltnw.train(max_epochs=epochs) # visualize results on training data ltnw.variable("?x", 1) prediction = ltnw.ask("f(?x)", feed_dict={"?x": train_X.reshape(len(train_X), 1)}) plt.figure(figsize=(12, 5)) plt.subplot(1, 2, 1) plt.plot(train_X, train_Y, 'bo', label='Training data', color="black") plt.plot(train_X, ltnw.SESSION.run(W) * train_X + ltnw.SESSION.run(b), label='Fitted line') plt.plot(train_X, prediction, 'bo', label='prediction', color="red") plt.legend() # generate test data and visualize regressor results
for i in range(nr_of_clusters): ltnw.axiom("exists ?x: C_%s(?x)" % i) ltnw.axiom( "forall ?close_x_y: C_%s(first(?close_x_y)) %% C_%s(second(?close_x_y))" % (i, i)) ltnw.axiom( "forall ?distant_x_y: C_%s(first(?distant_x_y)) %% C_%s(second(?distant_x_y))" % (i, i)) for j in range(i + 1, nr_of_clusters): ltnw.axiom("forall ?x: ~(C_%s(?x) & C_%s(?x))" % (i, j)) print("%s" % "\n".join(ltnw.AXIOMS.keys())) # initialize the knowledgebase and train ltnw.initialize_knowledgebase(optimizer=torch.optim.RMSprop, initial_sat_level_threshold=.5) ltnw.train(max_epochs=200) # retrieve the truth values for all samples and all clusters, i.e. check membership prC = [ltnw.ask("C_%s(?x)" % i) for i in range(nr_of_clusters)] n = 2 m = (nr_of_clusters + 1) // n + 1 fig = plt.figure(figsize=(3 * 3, m * 3)) fig.add_subplot(m, n, 1) plt.title("groundtruth") for c in clusters: plt.scatter(c[:, 0], c[:, 1]) data = np.concatenate(clusters) x0 = data[:, 0] x1 = data[:, 1]
]).to(device), verbose=False) if ep + b == 0: # Initialise LTN at very beginning of training print('******* Initialising LTN ******') sat_level = ltnw.initialize_knowledgebase( initial_sat_level_threshold=.5, device=device, learn_rate=learning_rate, perception_mode=perception_mode) print("Initial Satisfiability %f" % (sat_level)) print("Initial p-Value %f" % (p_factor * (sat_level.item()**2))) ltnw.set_p_value(p_factor * (sat_level.item()**2)) sat_level = ltnw.train( max_epochs=1, sat_level_epsilon=.01, track_values=False, device=device, show_progress=False) #, early_stop_level=0.00001) dictw.writerow({ key: value.detach().cpu().numpy()[0] for (key, value) in ltnw.AXIOMS.items() }) if sat_level > 0.997: break pbar.set_description("Current Satisfiability %f" % (sat_level)) pbar.update(1) print("Final p-Value %f" % (p_factor * (sat_level.item()**2))) #################### ### Save the LTN ### ####################
ltnw.axiom("exists ?x: C_%s(?x)" % i) ltnw.axiom("forall ?x,?y: (C_%s(?x) & close(?x,?y)) -> C_%s(?y)" % (i, i)) ltnw.axiom( "forall ?x,?y: (C_%s(?x) & ~close(?x,?y)) -> (%s)" % (i, "|".join(["C_%s(?y)" % j for j in range(nr_of_clusters) if i != j]))) for j in range(i + 1, nr_of_clusters): ltnw.axiom("forall ?x: ~(C_%s(?x) & C_%s(?x))" % (i, j)) print("\n".join(sorted(ltnw.AXIOMS.keys()))) ## initialize and optimize ltnw.initialize_knowledgebase(optimizer=tf.train.RMSPropOptimizer( learning_rate=0.1, decay=.9), initial_sat_level_threshold=.0) ltnw.train(max_epochs=1000) ## visualize results nr_of_clusters = len(clusters) prC = [ltnw.ask("C_%s(?x)" % i) for i in range(nr_of_clusters)] n = 2 m = (nr_of_clusters + 1) // n + 1 fig = plt.figure(figsize=(3 * 3, m * 3)) fig.add_subplot(m, n, 1) plt.title("groundtruth") for c in clusters: plt.scatter(c[:, 0], c[:, 1]) data = np.concatenate(clusters) x0 = data[:, 0]
data=np.random.uniform([0,0],[1.,1.],(nr_samples,2)).astype(np.float32) data_A=data[np.where(np.sum(np.square(data-[.5,.5]),axis=1)<.09)] data_not_A=data[np.where(np.sum(np.square(data-[.5,.5]),axis=1)>=.09)] ltnw.variable("?data_A",data_A) ltnw.variable("?data_not_A",data_not_A) ltnw.variable("?data",data) ltnw.predicate("A",2) ltnw.axiom("forall ?data_A: A(?data_A)") ltnw.axiom("forall ?data_not_A: ~A(?data_not_A)") ltnw.initialize_knowledgebase(initial_sat_level_threshold=.1) sat_level=ltnw.train(track_sat_levels=1000,sat_level_epsilon=.99) plt.figure(figsize=(12,8)) result=ltnw.ask("A(?data)") plt.subplot(2,2,1) plt.scatter(data[:,0],data[:,1],c=result.squeeze()) plt.colorbar() plt.title("A(x) - training data") result=ltnw.ask("~A(?data)") plt.subplot(2,2,2) plt.scatter(data[:,0],data[:,1],c=result.squeeze()) plt.colorbar() plt.title("~A(x) - training data") data_test=np.random.uniform([0,0],[1.,1.],(500,2)).astype(np.float32)
"?not_above_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.is_not_above), "?below_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.is_below), "?not_below_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.is_not_below), "?contains_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.contains), "?not_contains_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.not_contains), "?contained_in_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.is_in), "?not_contained_in_xy" : spatial_relations_data.generate_data(nr_examples,spatial_relations_data.is_not_in), "?x" : spatial_relations_data.generate_rectangles(nr_examples), "?y" : spatial_relations_data.generate_rectangles(nr_examples), "?z" : spatial_relations_data.generate_rectangles(nr_examples)} # 4) train the model ltnw.initialize_knowledgebase(feed_dict=feed_dict, optimizer=tf.train.AdamOptimizer(0.05), formula_aggregator=lambda *x: tf.reduce_min(tf.concat(x,axis=0))) ltnw.train(feed_dict=feed_dict,max_epochs=10000) # 5) evaluate the truth of formulas not given directly to the model for f in ["forall ?x,?y,?z: Contained_in(?x,?y) -> (Left(?y,?z)->Left(?x,?z))", "forall ?x,?y,?z: Contained_in(?x,?y) -> (Right(?y,?z)->Right(?x,?z))", "forall ?x,?y,?z: Contained_in(?x,?y) -> (Above(?y,?z)->Above(?x,?z))", "forall ?x,?y,?z: Contained_in(?x,?y) -> (Below(?y,?z)->Below(?x,?z))", "forall ?x,?y,?z: Contained_in(?x,?y) -> (Contains(?y,?z)->Contains(?x,?z))", "forall ?x,?y,?z: Contained_in(?x,?y) -> (Contained_in(?y,?z)->Contained_in(?x,?z))"]: print("%s: %s" % (f,ltnw.ask(f,feed_dict=feed_dict))) # 6) plot some examples truth values of P(ct,t) where ct is a central rectangle, and # t is a set of randomly generated rectangles ltnw.constant("ct",[.5,.5,.3,.3]) test_data=spatial_relations_data.generate_rectangles(nr_test_examples) ltnw.variable("?t",test_data)
##################### ### Train the LTN ### ##################### time_diff = time.time() - start_time print('Time to complete : ', time_diff) start_time = time.time() print('******* Initialising LTN ******') ltnw.initialize_knowledgebase(initial_sat_level_threshold=.5, learn_rate=learning_rate) time_diff = time.time() - start_time print('Time to complete : ', time_diff) start_time = time.time() print('******* Training LTN ******') sat_level = ltnw.train(max_epochs=max_epochs, sat_level_epsilon=.005, track_values=True) #, early_stop_level=0.00001) #################### ### Test the LTN ### #################### # ask queries about objects in image_val_00000.png # print('\nIs object0 (large brown cylinder) in front of object3 (large purple sphere)? ', ltnw.ask('Front(object3,object0)')) # print('Is object3 (large purple sphere) not to the left of object2 (small green cylinder)? ', ltnw.ask('~Left(object2,object3)')) # print('Is object2 (small green cylinder) to the left of object1 (large gray cube)? ', ltnw.ask('Left(object1,object2)')) # print('Is object4 (small gray cube) to the right of object0 (large brown cylinder)? ', ltnw.ask('Right(object0, object4)')) # print('Is object2 (small green cylinder) small? ', ltnw.ask('Small(object2)')) # print('Is object1 (large gray cube) a sphere? ', ltnw.ask('Sphere(object1)')) #print('Is there an object to the right of object1 (large gray cube)?', ltnw.ask('exists ?obj: Right(object1,?obj)'))
ltnw.predicate("A",2) ltnw.predicate("B",2) ltnw.axiom("forall ?data_A: A(?data_A)") ltnw.axiom("forall ?data_B: ~A(?data_B)") ltnw.axiom("forall ?data_B: B(?data_B)") ltnw.axiom("forall ?data_A: ~B(?data_A)") ltnw.axiom("forall ?data: A(?data) -> ~B(?data)") ltnw.axiom("forall ?data: B(?data) -> ~A(?data)") ltnw.initialize_knowledgebase(optimizer=torch.optim.RMSprop, initial_sat_level_threshold=.99) # The number of iterations were dramatically reduced. sat_level=ltnw.train(track_sat_levels=1000,sat_level_epsilon=.01,max_epochs=2000) result=ltnw.ask("A(?data)") plt.figure(figsize=(10,8)) plt.subplot(2,2,1) plt.title("A(x) - training") plt.scatter(data[:,0],data[:,1],c=result.squeeze()) plt.colorbar() plt.subplot(2,2,2) result=ltnw.ask("B(?data)") plt.title("B(x) - training") plt.scatter(data[:,0],data[:,1],c=result.squeeze()) plt.colorbar() data_test=np.random.uniform([0,0],[1.,1.],(nr_samples,2)).astype(np.float32)
ltnw.predicate("R_A_A",4) ltnw.predicate("R_B_B",4) ltnw.predicate("R_A_B",4) ltnw.axiom("forall ?data, ?data_2: (A(?data) & A(?data_2)) -> R_A_A(?data,?data_2)") ltnw.axiom("forall ?data, ?data_2: R_A_A(?data,?data_2) -> (A(?data) & A(?data_2))") ltnw.axiom("forall ?data, ?data_2: (B(?data) & B(?data_2)) -> R_B_B(?data,?data_2)") ltnw.axiom("forall ?data, ?data_2: R_B_B(?data,?data_2) -> (B(?data) & B(?data_2))") ltnw.axiom("forall ?data, ?data_2: (A(?data) & B(?data_2)) -> R_A_B(?data,?data_2)") ltnw.axiom("forall ?data, ?data_2: R_A_B(?data,?data_2) -> (A(?data) & B(?data_2))") ltnw.initialize_knowledgebase(initial_sat_level_threshold=.1) sat_level=ltnw.train(track_sat_levels=1000,sat_level_epsilon=.99,max_epochs=epochs) plt.figure(figsize=(12,8)) plt.subplot(2,2,1) plt.title("data A/B") plt.scatter(data_A[:,0],data_A[:,1],c="red",alpha=1.,label="A") plt.scatter(data_B[:,0],data_B[:,1],c="blue",alpha=1.,label="B") plt.legend() idx=2 for pred in ["R_A_A","R_A_B","R_B_B"]: result_A_A=ltnw.ask("%s(?data_A,?data_A_2)" % pred) result_A_B=ltnw.ask("%s(?data_A,?data_B)" % pred) result_B_B=ltnw.ask("%s(?data_B,?data_B_2)" % pred) plt.subplot(2,2,idx) idx+=1
ltnw.predicate("A", 2) ltnw.predicate("B", 2) ltnw.variable("?data_A", data_A) ltnw.variable("?data_B", data_B) ltnw.variable("?data", data) ltnw.axiom("forall ?data_A: A(?data_A)") ltnw.axiom("forall ?data_B: B(?data_B)") ltnw.axiom("forall ?data: A(?data) -> ~B(?data)") ltnw.axiom("forall ?data: ~B(?data) -> A(?data)") ltnw.initialize_knowledgebase(initial_sat_level_threshold=.1) sat_level = ltnw.train(max_epochs=max_epochs, track_sat_levels=track_sat_levels) plt.figure(figsize=(10, 8)) result = ltnw.ask("A(?data)") plt.subplot(2, 2, 1) plt.title("A(x) - training") plt.scatter(data[:, 0], data[:, 1], c=result.squeeze()) plt.colorbar() result = ltnw.ask("B(?data)") plt.subplot(2, 2, 2) plt.title("B(x) - training") plt.scatter(data[:, 0], data[:, 1], c=result.squeeze()) plt.colorbar() data_test = np.random.uniform([0, 0], [1., 1.],