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Paper Info

Title: Incorporating Human Path Preferences in Robot Navigation with Minimal Interventions
Publish Date: 16 Mar 2023
Author List: Oriana Peltzer, Dylan Asmar, Mac Schwager, Mykel Kochenderfer

Figure

Hyperplane arrangement of a twodimensional space containing two obstacles (colored in gray).The robot is located inside the pink polytope, surrounded by three adjacent obstacle-free polytopes.Each hyperplane on the boundary of the robot's polytope corresponds to one of the nonredundant constraints in eq.(4).(b)Graph derived from the hyperplane arrangement.The nodes on the graph designate polytopes, and edges designate transitions to adjacent polytopes.To estimate the human's preference, the robot updates a posterior over the goal and over which of the graph transitions φ 1 , φ 2 and φ 3 is preferred by the human.(c)Example preference defined over the graph.The location of the goal is indicated in yellow in the lower right polytope.For each node, the outgoing pink arrow designates the edge on the graph corresponding to the preferred transition between polytopes.
Simple, 10 × 10, 8 polytopes.(b) Map 2: Office, 10 × 10, 56 polytopes.(c) Map 3: Classroom, 20 × 20, 73 polytopes.(d) Sampled observations and robot's executed trajectories.
Fig.5: Maps used for simulating the robot navigation problem with path preferences.In (d), the heading angles observed are indicated with arrows.The goal is indicated with a pink circle, and the orange robot corresponds to the starting location.The blue robot follows a policy that accounts for path preference, while the green robot does not.The opacity of the robots increases with time.
Map 1 problem setup and example realizations for goal-only (green) and path preference (blue) solution methods.The robot starts at the lower left corner of the environment, and the goal of the task (pink circle) is in the upper left area.The robot does not know which goal, among 10 options (shown in light blue squares), is the correct goal.The human provides noisy observations, indicated by arrows, at each iteration.The green robot selects actions according to the goal-only baseline, and the blue robot uses our proposed method to infer path preferences.The polytopes composing G are drawn in blue.Probability of correct goal.WLPHVWHS +J (c) Entropy of goal distribution g.
Fig. 6: Probability of the correct goal, fig.6b, and entropy of the goal belief distribution P (g), fig.6c, for the same problem setup, fig.6a.In this problem instance, the human's preference is to go to the goal by passing on the right side of the obstacle.Results are averaged over 50 runs and the area filled represents one standard deviation above and below the mean value.The goal-only baseline shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference.
Success rates in the simple environment (Map 1).The results are averaged over 6 randomly sampled problem instances (start location, goal location, and goal possibilities), and over 50 runs per problem instance.∆T is the number of time steps separating two consecutive human inputs.The robot's mission time is Tmax = 30 time steps.We selected γ h = 1.5, corresponding to relatively noisy human inputs and making the problem more difficult to solve for the robot.
Computation times for Goal Only and Path Preference methods on Map 1 (fig.5a),Map 2 (fig.5b), and Map 3 (fig.5c),averaged over 100 runs with randomly sampled problem instances.The 95 % confidence interval is provided with the mean.We evaluate computation time at the first iteration of each run (where the search depth takes on its highest value Tmax).

abstract

Robots that can effectively understand human intentions from actions are crucial for successful human-robot collaboration. In this work, we address the challenge of a robot navigating towards an unknown goal while also accounting for a human's preference for a particular path in the presence of obstacles.
This problem is particularly challenging when both the goal and path preference are unknown a priori. To overcome this challenge, we propose a method for encoding and inferring path preference online using a partitioning of the space into polytopes. Our approach enables joint inference over the goal and path preference using a stochastic observation model for the human.
We evaluate our method on an unknown-goal navigation problem with sparse human interventions, and find that it outperforms baseline approaches as the human's inputs become increasingly sparse. We find that the time required to update the robot's belief does not increase with the complexity of the environment, which makes our method suitable for online applications.

INTRODUCTION

Collaboration between humans and robots has become increasingly important and one key aspect of this collaboration is the ability for robots to adapt to human decisions. In many scenarios, such as a robot navigating through a busy room to deliver an item, it is important for the robot to take into account human preferences.
For instance, humans may prefer a specific path that would allow their colleagues to notice the item being delivered, but this preference may change dynamically based on various factors such as changes in the environment or unforeseen circumstances. While some preferences can be incorporated into the path-planning process, accommodating dynamic user preferences in real-time remains challenging.
In this paper, we propose a way to enable robots to adapt to human preferences dynamically by leveraging real-time feedback to inform decision-making. In this work, we tackle the problem of robot navigation in which the robot cannot observe the goal or the preferred path to the goal, but must make navigation decisions that are influenced by humans through recommended actions.
Prior work has explored how to adapt to a human's preference through feedback, but such approaches often require a high level of intervention, which can be time-consuming and impractical in real-world scenarios. To optimize the use of human input and quickly infer the human's preference, Fig. : An autonomous robot navigates in a simulated classroom towards a goal location (pink circle).
At the start of its mission, it receives direction indications (arrows) from a human that indicate which path it should take to get to the goal. In this scenario, the human wants the robot to go around the desks on the right side of the classroom. A robot that does not reason over path preferences (green) will take the shortest path to the goal regardless of the human's input.
Our method (blue) infers the human's path preference from these indications and adapts to their recommendations. we propose an approach that leverages probabilistic representations of human preference and incorporates real-time feedback. Previous research by Bajcsy et al. considered an online adaptation problem in a manipulation task, where the person can apply forces to the robot to indicate their preferences.
By allowing the robot to continue its task while taking into account a probabilistic representation of human preference, their approach does not require frequent inputs. Building on this idea, we adopt a similar approach to adapt to a human's preference in the context of a robot autonomously navigating through a known environment, such as a cluttered office space.
Specifically, we focus on allowing the human to influence the robot's trajectory with respect to obstacles, by providing guidance on preferred routes or paths, while the robot continues to execute its task. Paths can be represented using homotopy classes . However, homotopies can pose computational challenges when used to encode and infer human preferences.
When the robot maintains a belief over homotopy classes, the inference problem can become exponentially complex with the number of obstacles in the space. Additionally, when the goal is unknown, the number of variables increases with the number of candidate destinations. This complexity can render the decision-making problem intractable.
Our solution is to encode path preference based on a partitioning of the environment into polytopes . This representation allows path preferences to be expressed as sets of preferred transitions between adjacent polytopes. Paths belonging to different homotopy classes correspond to different sequences of transitions.
By leveraging conditional independence assumptions, we can make the Bayesian inference problem tractable. These assumptions exploit the fact that human actions provide information about the path in a piece-wise manner. For example, indicating a preference for navigating around a particular obstacle only provides information about the local area and not the entire path.
Finally, after updating its belief representation over the human's preference, the robot can adapt to indications by replanning online. Our contributions are as follows. • We formulate the human-robot collaboration problem as a Partially Observable Markov Decision Process (POMDP) where both the goal of the task and the human's path preference are unknown random variables.
• We propose an encoding of a human's path preference using a partitioning of the environment into polytopes, along with conditional independence assumptions that make the Bayesian inference problem tractable to infer the task goal and path preference online. • Through simulations in two environments of different sizes and complexity, we show that our method is effective for solving problems where the robot must reach a goal that is unknown a-priori while simultaneously adapting to a human's indications.
Our method shows higher success rates compared to baseline approaches when the human inputs are sparse. Our approach enables a robot to make effective navigation decisions in collaboration with a human, even when the goal and path preference are not known in advance, and with minimal human input. In recent years, there has been a growing interest in shared autonomy and interactive systems, where humans and robots work together to accomplish tasks.
Several approaches have been proposed to address the challenge of enabling effective collaboration between human and robot agents while still achieving high task performance. Losey et al. and Jeon, Losey, and Sadigh propose a framework where a human operator is given control of a task-relevant latent action space while an autonomous system handles the rest.
Dragan and Srinivasa present a formalism for arbitrating between a user's input and a robot's policy when both human and robot share control of the same action space. Cognetti et al. [7] provide a method for real-time modifications of a path, . . . Fig. : We model the intent inference problem with the above diagram.
At each step in time, the robot receives an observation ot from the human conditioned on its current location st, the intended goal g, and the human's path preference θ. The robot updates its belief over g and θ and transitions to a next location st+1. while Hagenow et al. present a method that allows an outside agent to modify key robot state variables and blends the changes with the original control.
However, a common challenge of these approaches is the high level of intervention required from humans. Best and Fitch propose a method for predicting an agent's intended trajectory from observations. Rather than maintaining a belief over the agent's future path, they infer the agent's intended goal among a set of candidate locations at the boundary of the space.
This approach provides information on where the agent is heading and generates a distribution of candidate future trajectories for the agent. Inferring the goal of the task among a discrete set of candidates is also relevant to the area of shared autonomy. Javdani, Srinivasa, and Bagnell propose a formalism for shared control of a robotic arm, where the robot must assist the human in picking up an object but needs to infer which object the human has chosen from joystick inputs.
Planning with homotopy class constraints is useful in problems where the robot's requirements are given with respect to obstacles, and Yi, Goodrich, and Seppi consider topological constraints provided by human operators. Bhattacharya propose an efficient algorithm for solving pathplanning problems under homotopic constraints.
However, the number of homotopy classes for a given problem can be infinite, and as the robot changes location and updates its representation of the world, carrying out inference over homotopy classes in a dynamic environment requires recomputing the set of homotopies at every iteration, making the belief update challenging.
Prior work has addressed the challenge of shared autonomy by considering how robots can infer a human's intended goal, or how they can infer the preferred path to a goal. However, we argue that inferring the goal and the path as separate problems can lead to over-confidence in incorrect beliefs about the user's preferences.
To illustrate this point, consider the following scenario: a robot and a human are collaborating to move an object from one end of a room to Fig. : Using the hyperplanes composing the H-representation of each obstacle, we construct a hyperplane arrangement of the obstacle-free space (a). We define the human's preference for the robot's one step action choices as the posterior distribution (given all human input up to that point) over transitions from the current to the neighboring polytopes, i.e. edges on the graph.
Each time the robot transitions to a new polytope, the set of neighbor polytopes and the distribution over human preferences are updated. another, but there is an obstacle in the way. The human would like the robot to take a path around the obstacle on the left, even though the goal is on the right. If the robot only infers the goal from the human's inputs, it may incorrectly assume that the goal is on the right, and become over-confident in this belief.
On the other hand, if the robot only infers the preferred path, it may mistakenly assume that the goal is on the left, leading to a failure in completing the task. To overcome these challenges, our work proposes a joint inference approach that considers both the human's intended goal and their preferred path to that goal.
Specifically, we model the human's preference over different homotopy classes and leverage a conditional independence assumption to provide a tractable solution. In our approach, we assume that the human's inputs are noisily rational conditioned on both the goal and the preference. By jointly inferring the goal and path preference, we can avoid over-confidence in incorrect beliefs about the user's preferences, leading to improved system performance.
We consider the problem of robot navigation in a known environment to an unknown destination, where a human can intervene and provide a heading direction to the robot using a joystick or force cues. The human also has a preference on which path the robot should take with respect to obstacles, and our objective is for the robot to understand the human's intentions and execute the task with minimal interventions.
Let g be a discrete random variable denoting the goal of the task, belonging to a set of candidates Ω g , and let θ be a discrete-valued random variable representing the human's path preference, belonging to a set of possible preferences Θ. The physical location of the robot at time index t is denoted by s t ∈ R 2 , and the robot's action at time index t, belonging to some action space A, is denoted by a t .
The transition model T (s t+1 | s t , a t ) is deterministic, meaning the robot has full control over its future location. At any time step, the human may provide an observation to the robot. When the human intervenes, the robot receives a direction (heading angle) that can be mapped to a future location in space.
More specifically, we map the direction to an intended location, which is the resulting robot location after advancing in the indicated direction for one time step. For simplicity, we consider that the robot directly makes an observation o t of the location indicated by the human. We assume that the robot has a stochastic observation model for the human P (o t | s t , g, θ) that is conditioned on both the goal of the task g and the human's preferred path θ.
We further assume that having chosen a goal and path preference, the human takes actions to noisily minimize a cost function C g,θ that measures the cost of moving from the robot's current location to the goal along the preferred path. For example, C g,θ (s t , o t ) can be the length of the shortest path from location s t to the goal g after taking a first step to o t , and constrained by path preference θ.
We use C g,θ to induce a probability distribution over observations, given by: where γ h is a hyperparameter that designates the rationality coefficient. This model assumes the human will pick the lowest cost action with the highest probability and the likelihood of an action decreases exponentially with the increase in cost .
Our inclusion of the path preference θ sets our approach apart from . The model is shown in fig. represented as a Bayesian Network.

Inference

At each time step where the human provides an observation, the posterior P (g, θ) is given through the Bayesian update We note that the number of Bayesian updates required at each iteration to update the belief is equal to the cardinality of Ω g × Θ. In addition, each Bayesian update involves computing C g,θ ( .
, . ) in eq. ( ), which involves solving an optimization problem (such as a shortest path problem). In section IV, we propose a specific encoding of preference θ for resolving eq. ( ), while ensuring the number of computations of the cost C g,θ (., .) per update does not grow exponentially with the number of obstacles.

Decision Making

We consider a navigation problem where the robot receives reward according to the model R(s t , g, θ, a t ). We wish to find the optimal policy π that maximizes the expected discounted sum of future rewards, with discount factor γ. The above problem is a Partially Observable Markov Decision Process (POMDP) .
In this section, we propose an encoding of human's path preference θ for computing the posterior in eq. ( ). Devifrom the concept of homotopy classes, we define the preference according to a partitioning of the environment into polytopes, as shown in fig. , creating a hyperplane arrangement of the space.
Hyperplane arrangements have been used by Vincent and Schwager in the context of Neural Network verification. In our setting, we leverage this representation to define path preferences as preferred transitions between adjacent regions of the space.

Hyperplane Arrangement

We assume a two-dimensional environment composed of m polytopic obstacles, each defined by their half-space representation (H-representation) where A i ∈ R di×2 and b i ∈ R di , and where d i is the number of edges (hyperplanes) composing polytope i. Let n = i d i be the total number of hyperplanes. We leverage each obstacle's H-representation to construct a hyperplane arrangement of the environment as shown in fig.
.e. a partitioning of the space into polytopes. More specifically, each location in space belongs to a polytope j for which we can write an H-representation of the form where α j i ∈ {−1, 1} di is a vector specific to polytope j and obstacle i corresponding to the relative position of any point in the set with respect to each hyperplane in O i .
Fig. : Intent inference model in a hyperplane arrangement of the obstacle free space. We spatially decompose the preference θ into a set of preferred neighboring polytopes per region of the space. Within each polytope j, the human preference pj is a discrete distribution over the preferred neighbor in N (j).
We assume that for a location st belonging to polytope j, and given goal g and preference pj, the observation ot and any other preference p i,i =j are conditionally independent. Concatenating elements from each obstacle's Hrepresentation, we can write polytope j's H-representation as where Some of the constraints in eq. ( ) (corresponding to rows of A, b and α j ) are redundant, i.e. the set P j does not change upon their removal.
We can further reduce the Hrepresentation of a polytope to include only non-redundant constraints. By removing the rows corresponding to redundant constraints, we obtain new matrices A j e , b j e and α j e such that we can write the polytope's reduced H-representation as The non-redundant constraints correspond to edges of the polytope.
In other words, as the robot continually moves in space, the first hyperplane that it will cross upon exiting the polytope will correspond to one of the polytope's nonredundant constraints. Vincent and Schwager outline an iterative method for removing redundant constraints by solving n linear programs.
We use this method in practice for computing α j e for each polytope. We can now characterize each polytope by a vector α j e ∈ {−1, 1} n j e , where n j e ≤ n is the number of essential constraints of the polytope. The polytopes P j partition the environment into a hyperplane arrangement.

Path Preference

In this section, we provide a definition of preference θ according to a graphical representation of the environment based on the hyperplane arrangement. Under this representation, a path preference corresponds to a set of preferred transitions. In other words, for each polytope in the space, the human will have a preference to which neighboring polytope they wish to transition.
Let G := (V, E) be an undirected graph, where vertices are obstacle-free polytopes, and edges connect two adjacent polytopes. Each polytope is described by a unique vector α j as defined in eq. ( ). Two polytopes are adjacent if they share non-redundant constraints (rows in eq. ( )) corresponding to the same hyperplane (i.e. they are on opposite sides of the hyperplane).
Let N (v) be the set of neighbors of a vertex v. For each vertex, we denote p v the discrete-valued random variable describing which edge in N (v) the human intends to transition to. Using this formalism, we define a path preference as the set of preferred transitions over all nodes in the graph, Let m θ = v∈V |N (v)| be the cardinality of Θ, and m g = |Ω g | the number of possible goals.
A priori, the number of Bayesian updates required to update the belief at every iteration should be m θ × m g . Now, let us assume the conditional independence relationships described by the new problem diagram in fig. . More specifically, we introduce the assumption that conditioned on a robot location s t , the goal g, and the preference for the corresponding vertex p v in the graph, the observation o t and the preference for any other vertex are conditionally independent.
In other words, the observations the human provides can be defined conditioned only on the robot location, the goal, and the human's preference for its current vertex p v . By introducing this assumption, each update step only requires updating the joint (p v , g), reducing the number of cost computations to |N (v)| × m g .
We can notice that by introducing this assumption, we removed the direct relationship between the number of polytopes in the environment and the complexity of the Bayesian update in eq. ( ). In practice, components of θ are not mutually independent. For example, if the human preference at a vertex v 1 is
, it is unlikely that the human will also prefer p v2 = (v 2 , v 1 ) (turning back). We can improve our model by assuming a dependent relationship between preferences for adjacent edges, which does not significantly increase the complexity of the inference problem. An interesting property of our encoding is that any two paths that belong to different homotopy classes will cross different sequences of polytopes, i.e. they correspond to a different sequence of edges on G.
This can be proved by contradiction. Let us suppose that two continuous trajectories ξ 1 and ξ 2 , with the same start and end points and that do not intersect any obstacle, traverse the same regions in G in the same order. From the construction of the hyperplane arrangement, each polytope that the paths traverse through is obstacle-free.
Therefore, within each polytope, there is no obstacle in the area located in between the portions of ξ 1 and ξ 2 that belong to the region. A smooth transformation of ξ 1 into ξ 2 can be obtained by transforming each portion of ξ 1 belonging to the polytopes it intersects into the corresponding portion of ξ 2 for the same polytopes, where the extremities of the trajectory portions are connected to one another along the polytope's edges (where the same edge is crossed by both paths).
Along this transformation, the paths do not intersect any obstacle, and therefore ξ 1 and ξ 2 belong to the same homotopy class.

EXPERIMENTS

We evaluate our model on a simulated navigation task where the robot must reach a goal that is unknown a priori while respecting the path preferences indicated by a human. The robot navigates in a grid world containing obstacles. The transition model is deterministic: the robot selects an adjacent location on the grid to reach at the next time step.
The robot is also allowed to take diagonal actions. Each location s t in the map can be mapped to a vertex v t ∈ G. Therefore, the actions leading to locations mapped to different vertices correspond to edges on the graph. We note f (s t , a t ) the edge crossed by taking action a t from location s t .
The robot is given a mission time limit T max for reaching the goal. In this problem, we assume that the human selects actions to noisily minimize a cost function C g,θ , where θ is defined as per eq. ( ), corresponding to the length of the shortest path to the goal constrained by the preference (where the robot is only allowed to make transitions on G along preferred edges).
More specifically, where δ(s t , g | o t , p vt ) designates the length of the shortest path from s t to g passing by o t and constrained by preference p vt . This is a slight variant of the cost function proposed by Best and Fitch , where we add in a conditioning on the path preference. We compute costs by running the A path planning algorithm on the environment maps (grid worlds with diagonal actions) and impose preference constraints by pruning invalid transitions from the search tree.
Reward model. At each step in time, the robot receives a reward which is a sum of three components: a goal-specific reward a preference-specific reward or penalty We compute solutions to the POMDP defined in section III-B with the online solver POMCP , and with the particularity that within the rollouts, the robot does not expect to collect human inputs.
Each time a solution is computed, the robot takes an action and may receive an observation. If it does, it updates its belief distribution over the unknown problem variables and resolves the POMDP over a receding horizon.

Baselines

• Goal only. The robot solves the POMDP while ignoring the effects of path preference. Similarly to , we assume the human is taking action to minimize a goaldependent cost C g (s t , o t ) = δ(s t , g | o t ), where the conditioning on the preference is removed. We also omit the path preference's contribution to the reward R pref .
• Compliant. The robot complies with the human input, but does not take an initiative. If the user stops providing information, the robot continues in the last direction indicated for 5 time steps (conserving its momentum), then stops. • Blended. We designed an arbitration function to decide between our proposed policy (accounting for path preferences) and the user's recommendation when the robot receives inputs.
Our metric to evaluate confidence in the robot's prediction for the purpose of arbitration is the entropy of the intention distribution H(g, p i ), where p i denotes the preferred neighbor for the current region. Because our representation of the world is discrete, the arbitration is given by a step function.
Denoted by U , the action corresponding to the human's input, and P , the robot's prediction for the optimal action, we write the policy where we chose h = 1.6 as the confidence threshold.

Results

When evaluating the algorithm, we consider that a run is successful if the robot reached the goal within its allocated mission time T max and only made transitions between graph vertices corresponding to the human's preferences. We vary the time delay between human inputs, from constant guidance (∆ T = 1) to only a single observation (∆ T ≥ T max ).
Success rates. Table I reports the success rates for experiments conducted over six randomly sampled problem instances and 50 runs per instance in Map 1 (fig. ). When the human provides inputs at every iteration, the compliant policy shows the highest success rates. However, as ∆ T increases, the compliant robot is not able to accomplish the task within the allotted time as it does not receive sufficient inputs to do so, and performance decreases compared to the autonomous baselines.
We find that in these runs, accounting for path preference consistently improves performance compared with the goal-only baseline. Results also show that blending the user's input with the robot's policy (Path Preference + Blend) when the human provides information leads to improved performance. Belief entropy.
Figure shows a challenging problem instance where the directions the human provides do not align directly with the shortest path to the goal. By ignoring the effects of preferences in the problem model (goal only), the robot quickly infers from observations that the upper left goal is less likely than others (P (g) drops).
The strong decrease in entropy shows that the robot becomes overconfident in this prediction. Overconfidence in an incorrect goal will prevent the agent from finding the correct goal once the human's indications directly align with it, as it needs to correct for the wrong predictions, as shown in the path realization (fig.
). In this realization, the goal-only method (green robot) fails to search the upper left area within the allotted time. By accounting for path preferences in its model, the blue robot's entropy over the goal distribution decreases more steadily, allowing for it to leverage the human's latest observations and reach the goal successfully.
shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference. Computation time. In table II we provide the time required to solve the POMDP, and the time required to update the robot's belief as it receives new observations.
We compute solutions on three maps: a simple 10 × 10 grid world with 8 polytopes (fig. ), a 10 × 10 grid world with 56 polytopes (fig. ), and a 20×20 grid world with 73 polytopes (fig. ). The latter environment being larger, we increase the mission time and the depth of the search tree in POMCP from T max = 30 (Map 1 and Map 2) to T max = 60 (Map 3).
We do not notice an increase in the time required to update the robot's belief with an increase in problem complexity, which is consistent with our observation that the complexity of the Bayesian update should not increase with the number of obstacles or polytopes. On the contrary, the belief update time on Map 2 and Map 3, containing more obstacles, is reduced compared to the first map.
More obstacles result in fewer iterations when solving the constrained shortest path problem with A . Adding constraints due to the obstacles and polytopes reduces the size of the A search tree. C. Limitations Simulation environments. In our simulations, we hardcoded the preference policy over the maps (e.g. in Map 1, go around the table counter-clockwise).
We randomly sampled problem instances (start and goal locations, and goal options) to reduce the bias introduced by these preference choices. To best evaluate and compare the different approaches, it would be best to sample preferences among a distribution of preferences chosen by a human (for example, from benchmarks resulting from a collection of data).
Creating such a benchmark is an interesting direction for future work. Hyperplane arrangement construction. The main limitation of our approach is that the size and geometry of each polytope depends strongly on the geometry of the obstacles, as seen in fig. . Because of this, the robot can make predictions over preferences that are too refined compared with the topology of the environment.
A direct consequence is that when the size of the polytopes is small, the information provided by the human can be incorrectly interpreted as a preference on the robot's immediate action. Our method can be improved by changing the structure of the hyperplane arrangement so that it relies on the topology of the environment, but does not vary strongly with the geometry of the features in the environment.
For this purpose, topometric maps and region construction algorithms are promising directions. We presented an approach for encoding and inferring a human's path preference in an environment with obstacles. By leveraging a partitioning of the space into polytopes and a stochastic observation model, our method allows for joint inference over the goal and path preference even when both are unknown a-priori.
Our experiments on an unknown-goal navigation problem with sparse human interventions demonstrate the effectiveness of our approach and its suitability for online applications. The time required to update the robot's belief does not increase with the complexity of the environment, which further highlights the practicality of our method.

Answer: The time required to update the belief does not increase with the complexity of the environment.


Input: What did Justice Kennedy argue about Quill in Direct Marketing Ass'n v. Brohl?

Context:
South Dakota v. Wayfair, Inc. - Harvard Law Review
Fourth Circuit Invalidates Maryland Statute Regulating Price Gouging in the Sale of Generic Drugs.
South Dakota Supreme Court Holds Unconstitutional State Law Requiring Internet Retailers Without In-State Physical Presence to Remit Sales Tax.
Judicial junk, the Court has long thought, is easier to scrap when the erroneous precedent cannot be fixed by Congress, as in constitutional cases.1× 1. See Burnet v. Coronado Oil & Gas Co., 285 U.S. 393, 405–10 (1932) (Brandeis, J., dissenting); Lee Epstein, William M. Landes & Adam Liptak, The Decision to Depart (or Not) from Constitutional Precedent: An Empirical Study of the Roberts Court, 90 N.Y.U. L. Rev. 1115, 1116 (2015) (“[Justice Brandeis’s] dissenting opinion . . . now has the status of black letter law.”). On the flip side, whenever a bad precedent can be corrected by Congress, stare decisis applies with “special force.”2× 2. See Patterson v. McLean Credit Union, 491 U.S. 164, 172–73 (1989). The Court, following Justice Brandeis, usually articulates the rule as distinguishing between “constitutional” and “statutory” precedents. See, e.g., id. But the distinction is occasionally said to be between “constitutional” and “nonconstitutional cases.” See, e.g., Glidden Co. v. Zdanok, 370 U.S. 530, 543 (1962) (plurality opinion). Nomenclature aside, the Court has — until now — adhered to Justice Brandeis’s key insight that the important factor is whether or not the mistake may be legislatively corrected. Last Term, in South Dakota v. Wayfair, Inc.,3× 3. 138 S. Ct. 2080 (2018). the Court tinkered with this thinking in overruling an outdated dormant commerce clause precedent. Dormant commerce clause decisions technically produce constitutional holdings, but Congress may override them at will.4× 4. See Prudential Ins. Co. v. Benjamin, 328 U.S. 408, 421–27 (1946). Under the usual logic of stare decisis, it should take special force to dislodge such precedents. But Wayfair applied the weakened stare decisis of constitutional cases, asserting that the Court must “address a false constitutional premise . . . . whether or not Congress can or will act.”5× 5. Wayfair, 138 S. Ct. at 2096–97.
Emerging from Wayfair is an odd and ominous development in stare decisis doctrine. Odd, because it turns on a formal classification instead of on Congress’s practical ability to fix the problem. Ominous, because the Court’s logic leads far past the dormant commerce clause. Wayfair grants only feeble stare decisis to precedents that set a “constitutional default rule,”6× 6. Id. at 2096 (“While . . . Congress has the authority to change the physical presence rule, Congress cannot change the constitutional default rule.”). meaning constitutional decisions that allow for legislative adjustment or override. This new stare decisis analysis makes other precedents setting constitutional default rules more vulnerable — including, perhaps, mainstays of criminal procedure like Miranda v. Arizona7× 7. 384 U.S. 436 (1966). and Mapp v. Ohio.8× 8. 367 U.S. 643 (1961).
Since its 1967 decision in National Bellas Hess, Inc. v. Department of Revenue,9× 9. 386 U.S. 753 (1967). the Court has held that, under the “dormant” or “negative” implication of the Commerce Clause,10× 10. The dormant or negative commerce clause is a judicial derivation from the Commerce Clause “prohibiting States from discriminating against or imposing excessive burdens on interstate commerce without congressional approval,” which “strikes at one of the chief evils that led to the adoption of the Constitution, namely, state tariffs and other laws that burdened interstate commerce.” Comptroller of the Treasury of Md. v. Wynne, 135 S. Ct. 1787, 1794 (2015). states may not compel remote sellers with no physical presence in the state to collect and remit sales taxes.11× 11. See Bellas Hess, 386 U.S. at 759–60. In Quill Corp. v. North Dakota,12× 12. 504 U.S. 298 (1992). the Court refused to overrule the “bright-line, physical-presence requirement” of Bellas Hess, leaning heavily on stare decisis.13× 13. Id. at 317–18. Three Justices joined a concurrence explaining that their decision rested solely “on the basis of stare decisis.” Id. at 320 (Scalia, J., concurring in part and concurring in the judgment). So the physical presence test remained the law of the land while the internet conquered the earth. Justice Kennedy had joined the Quill majority and Justice Scalia’s concurring opinion emphasizing stare decisis, but by 2015 he had second thoughts. Writing separately in Direct Marketing Ass’n v. Brohl,14× 14. 135 S. Ct. 1124 (2015). Justice Kennedy acknowledged that “[t]he Internet has caused far-reaching systemic and structural changes in the economy” and therefore “Quill now harms States to a degree far greater than could have been anticipated earlier.”15× 15. Id. at 1135 (Kennedy, J., concurring). He concluded with the wish that “[t]he legal system should find an appropriate case for this Court to reexamine Quill and Bellas Hess.”16× 16. Id.
Seldom has a concurring opinion signed by a lone Justice prompted a state to officially declare an emergency. Yet in 2016, in response to Justice Kennedy’s overture, the South Dakota legislature passed a law, S.B. 106, “to provide for the collection of sales taxes from certain remote sellers . . . and to declare an emergency.”17× 17. 2016 S.D. Sess. Laws ch. 70 pmbl. 217 (codified at S.D. Codified Laws § 10-64 (2017)). It required every remote seller to collect and remit sales tax if the seller’s business in South Dakota comprised either a “gross revenue” greater than $100,000 or at least 200 “separate transactions” within one calendar year.18× 18. Id. § 1. Significantly, the law did not apply retroactively.19× 19. Id. § 5. The “emergency” declaration was necessary to give the law immediate effect, for the purpose of “permitting the most expeditious possible review of the constitutionality of this law” by the U.S. Supreme Court.20× 20. Id. § 8(8). As Justice Alito put it, the “South Dakota law [was] obviously a test case.”21× 21. Transcript of Oral Argument at 27, Wayfair, 138 S. Ct. 2080 (No. 17-494), https://www.supremecourt.gov/oral_arguments/argument_transcripts/2017/17-494_7lho.pdf [https://perma.cc/8HYH-VU8N].
Expeditiously, a group of remote sellers challenged the law. After being sued by South Dakota for refusing to register for the newly required sales tax license, Wayfair, Inc., Overstock.com, Inc., and Newegg, Inc. moved for summary judgment in South Dakota circuit court on the grounds that S.B. 106 was unconstitutional under Quill and Bellas Hess — a point South Dakota conceded, indicating that it was seeking review by the U.S. Supreme Court to overturn Quill.22× 22. State v. Wayfair Inc., 2017 SD 56, ¶¶ 9–11, 901 N.W.2d 754, 759–60. Accordingly, the South Dakota circuit court granted the motion for summary judgment and South Dakota appealed to the state’s highest court.23× 23. Id. ¶ 12, 901 N.W.2d at 760. The South Dakota Supreme Court unanimously affirmed, recognizing that South Dakota’s “arguments on the merits” may be “persuasive” but “Quill remains the controlling precedent.”24× 24. Id. ¶ 18, 901 N.W.2d at 761. See generally Recent Case, State v. Wayfair Inc., 2017 SD 56, 901 N.W.2d 754 (S.D. 2017), 131 Harv. L. Rev. 2089 (2018).
The U.S. Supreme Court vacated and remanded.25× 25. Wayfair, 138 S. Ct. at 2100. Writing for the Court one last time, Justice Kennedy26× 26. Justices Thomas, Ginsburg, Alito, and Gorsuch joined Justice Kennedy’s opinion. pilloried Quill’s physical presence rule as “arbitrary, formalistic,” “anachronistic,” and “unfair and unjust” to both states and brick-and-mortar retailers.27× 27. Wayfair, 138 S. Ct. at 2092, 2095. After all, the rationale of Quill was that remote sellers lacked a sufficiently “substantial nexus” with the state to justify imposing a duty of tax collection.28× 28. Quill Corp. v. North Dakota, 504 U.S. 298, 311 (1992) (quoting Complete Auto Transit, Inc. v. Brady, 430 U.S. 274, 279 (1977)). This was wrong even in the mail-order catalog days of 1967 and 1992, but “the Internet revolution has made [Quill’s] earlier error all the more egregious and harmful.”29× 29. Wayfair, 138 S. Ct. at 2097; see also id. at 2092. The rule deprived the states of billions of dollars, since they could not force remote sellers to collect the tax and consumers hardly ever paid it on their own.30× 30. Id. at 2088 (“[C]onsumer compliance rates are notoriously low.”). Quill “serve[d] as a judicially created tax shelter” for remote retailers who do a great deal of business online.31× 31. Id. at 2094.
Satisfied that Bellas Hess and Quill were wrongly decided, the Court then jumped the hurdle of stare decisis. The Quill Court had feared upsetting reliance interests.32× 32. Quill, 504 U.S. at 317 (“Bellas Hess . . . has engendered substantial reliance and has become part of the basic framework of a sizable industry.”). Wayfair shrugged off this concern, noting that “stare decisis accommodates only ‘legitimate reliance interest[s]’”; by contrast, reliance on the physical presence rule was largely due to consumers evading their use-tax obligations.33× 33. Wayfair, 138 S. Ct. at 2098 (alteration in original) (quoting United States v. Ross, 456 U.S. 798, 824 (1982)). Quill had also appealed to Congress’s ultimate authority over interstate commerce as a reason to abide by a precedent, even if wrongly decided.34× 34. See Quill, 504 U.S. at 318–19; id. at 320 (Scalia, J., concurring in part and concurring in the judgment) (“Congress . . . can change the rule of Bellas Hess by simply saying so.”). But Wayfair denied that Congress’s ability to change the law was a proper consideration:
While it can be conceded that Congress has the authority to change the physical presence rule, Congress cannot change the constitutional default rule. It is inconsistent with the Court’s proper role to ask Congress to address a false constitutional premise of this Court’s own creation. Courts have acted as the front line of review in this limited sphere; and hence it is important that their principles be accurate and logical, whether or not Congress can or will act in response.35× 35. Wayfair, 138 S. Ct. at 2096–97.
Having dispensed with the physical presence rule, the Court remanded the case to the South Dakota courts to determine in the first instance “whether some other principle in the Court’s Commerce Clause doctrine might invalidate the Act.”36× 36. Id. at 2099. But the Court listed “several features [of South Dakota law] that appear[ed] designed to prevent discrimination against or undue burdens upon interstate commerce.” Id.
Justices Thomas and Gorsuch each filed concurring opinions. Justice Thomas wistfully likened himself to Justice White — who voted for Bellas Hess but against Quill a quarter-century later — and confessed that he “should have joined [Justice White’s dissenting] opinion.”37× 37. Id. at 2100 (Thomas, J., concurring). Justice Thomas added that the “Court’s entire negative Commerce Clause jurisprudence” is wrong and should be abandoned.38× 38. Id. Justice Gorsuch also wrote separately to express skepticism of the Court’s dormant commerce clause jurisprudence, raising “questions for another day” of whether the doctrine “can be squared with the text of the Commerce Clause, justified by stare decisis, or defended as misbranded products of federalism or antidiscrimination imperatives flowing from Article IV’s Privileges and Immunities Clause.”39× 39. Id. at 2100–01 (Gorsuch, J., concurring).
Chief Justice Roberts dissented.40× 40. Justices Breyer, Sotomayor, and Kagan joined the Chief Justice’s dissent. Surprisingly, the dissenting Justices “agree[d] that Bellas Hess was wrongly decided, for many of the reasons given by the Court.”41× 41. Wayfair, 138 S. Ct. at 2101 (Roberts, C.J., dissenting). The dispute between the majority and the dissent turned entirely on the principles and application of stare decisis. Chief Justice Roberts argued that whether or how to reverse Quill should be left to Congress, which “has the flexibility to address these questions in a wide variety of ways” and “can focus directly on current policy concerns rather than past legal mistakes.”42× 42. Id. at 2104. He also pointed to the “baffling” burdens of compliance with the idiosyncratic tax codes of “[o]ver 10,000 jurisdictions,” particularly for small businesses, and doubted that new “software” — the majority’s proposed solution to this mess43× 43. Id. at 2098 (majority opinion) (“Eventually, software that is available at a reasonable cost may make it easier for small businesses to cope with these problems.”). — would soon solve the problem.44× 44. Id. at 2103–04 (Roberts, C.J., dissenting). In Bellas Hess, the Court reasoned that the dormant commerce clause protects interstate business from being “entangle[d] . . . in a virtual welter of complicated obligations to local jurisdictions.” Nat’l Bellas Hess, Inc. v. Dep’t of Revenue, 386 U.S. 753, 759–60 (1967). The dissent replied that the Court “vastly underestimate[d] the skill of contemporary man and his machines.” Id. at 766 (Fortas, J., dissenting). The dispute in Wayfair over whether software is up to the task effectively reprised the old debate from Bellas Hess, only this time couched as part of the stare decisis inquiry’s concern for reliance interests rather than as a matter of dormant commerce clause doctrine. While Wayfair acknowledged that “[c]omplex state tax systems could have the effect of discriminating against interstate commerce,” 138 S. Ct. at 2099, the Court remarked that “[t]he physical presence rule is a poor proxy” for an inquiry into any actual burdens imposed on interstate commerce, id. at 2093.
Chief Justice Roberts emphasized that a “heightened form of stare decisis”45× 45. Wayfair, 138 S. Ct. at 2102 (Roberts, C.J., dissenting). applies when “Congress . . . can, if it wishes, override this Court’s decisions with contrary legislation.”46× 46. Id. at 2101 (first citing Michigan v. Bay Mills Indian Cmty., 134 S. Ct. 2024, 2036 (2014) (tribal sovereign immunity); then citing Kimble v. Marvel Entm’t, LLC, 135 S. Ct. 2401, 2409 (2015) (statutory interpretation); and then citing Halliburton Co. v. Erica P. John Fund, Inc., 134 S. Ct. 2398, 2411 (2014) (judicially created doctrine implementing a judicially created cause of action)). In Quill, the Chief Justice noted, the Court had taken to heart that “Congress may be better qualified” and “has the ultimate power to resolve” the question47× 47. Id. at 2102 (quoting Quill Corp. v. North Dakota, 504 U.S. 279, 318 (1992)). while Justice Scalia had “recogniz[ed] that stare decisis has ‘special force’ in the dormant Commerce Clause context due to Congress’s ‘final say over regulation of interstate commerce.’”48× 48. Id. (quoting Quill, 504 U.S. at 320 (Scalia, J., concurring in part and concurring in the judgment)). Moreover, “[i]f stare decisis applied with special force in Quill, it should be an even greater impediment” afterward since Quill effectively “tossed [the ball] into Congress’s court.”49× 49. Id. (alteration in original) (quoting Kimble, 135 S. Ct. at 2409); cf. Bay Mills, 134 S. Ct. at 2039 n.12 (“When we inform Congress that it has primary responsibility over a sphere of law, and invite Congress to consider a specific issue within that sphere, we cannot deem irrelevant how Congress responds.”). Because the Court invited Congress to act and then “suddenly chang[ed] the ground rules, the Court may have waylaid Congress’s consideration of the issue.”50× 50. Wayfair, 138 S. Ct. at 2102–03 (Roberts, C.J., dissenting).
In Wayfair, the Court applied the flimsier form of stare decisis to a precedent that could have been overruled by Congress. It did so in the context of a dormant commerce clause case, but Wayfair’s logic extends to all constitutional default rules — that is, constitutional decisions that Congress remains free to change. Not only does Wayfair deviate from the Court’s decades-old stare decisis analysis, it also imperils other precedents that set constitutional default rules.
The Court’s reasoning in Wayfair departs from its prior stare decisis analysis. In 1932, Justice Brandeis posited that stare decisis must bend “in cases involving the Federal Constitution, where correction through legislative action is practically impossible.”51× 51. Burnet v. Coronado Oil & Gas Co., 285 U.S. 393, 406–07 (1932) (Brandeis, J., dissenting). The Court has long since adopted his argument,52× 52. See, e.g., Smith v. Allwright, 321 U.S. 649, 665 (1944). as well as its corollary — that stare decisis commands “special force in the area of statutory interpretation” where “Congress remains free to alter what [the Court has] done.”53× 53. Patterson v. McLean Credit Union, 491 U.S. 164, 172–73 (1989). For normative evaluations of heightened stare decisis for statutory precedents, see generally Einer Elhauge, Statutory Default Rules: How to Interpret Unclear Legislation 211–23 (2008); and William N. Eskridge, Jr., Overruling Statutory Precedents, 76 Geo. L.J. 1361, 1364–1409 (1988). Justice Brandeis’s logic demands that dormant commerce clause cases, where Congress is free to act, be granted the weightier stare decisis.54× 54. Scholars have noted the curious fact that Justice Brandeis included many dormant commerce clause cases as examples of overruled constitutional precedents. See, e.g., Earl M. Maltz, Commentary, Some Thoughts on the Death of Stare Decisis in Constitutional Law, 1980 Wis. L. Rev. 467, 468–469, 469 n.11. One explanation for this is that Justice Brandeis sought the authority of Chief Justice Taney’s dictum that the Court’s “opinion upon the construction of the Constitution is always open to discussion” — which referred to the dormant commerce clause. See Burnet, 285 U.S. at 408 n.3 (Brandeis, J., dissenting) (quoting The Passenger Cases, 48 U.S. (7 How.) 283, 470 (1849) (Taney, C.J., dissenting)). In Chief Justice Taney’s time, it was thought that Congress could not override the Court’s dormant commerce clause decisions, see Cooley v. Bd. of Wardens, 53 U.S. (12 How.) 299, 321 (1852), so the context of Chief Justice Taney’s dictum does not conflict with Justice Brandeis’s theory of stare decisis. The Court applied this reasoning in Quill, as Chief Justice Roberts underscored.55× 55. Wayfair, 138 S. Ct. at 2102 (Roberts, C.J., dissenting).
Yet the Wayfair majority refused to consider Congress’s authority to legislate as a relevant factor for stare decisis.56× 56. Even Justice Kennedy’s earlier opinion in Direct Marketing contemplated judicially overruling Quill, conspicuously neglecting a possible legislative solution. See supra p. 278. The Court even insisted that to do so “is inconsistent with the Court’s proper role,” since Quill embodied “a false constitutional premise of th[e] Court’s own creation.”57× 57. Wayfair, 138 S. Ct. at 2096 (emphasis added). This refusal breaks from the practical Brandeisian wisdom that has guided the Court’s treatment of precedent for the better part of a century. The point is not that stare decisis should have ultimately propped up Bellas Hess yet again, as Wayfair’s dissenting Justices maintained. After all, a realistic approach that is alert to each branch’s institutional capacities might have led to the conclusion that Congress was actually ill-equipped to overrule Quill. In this vein, the Court could have sensibly pointed out that Congress is unlikely to stick its neck out with a tax hike (or a look-alike) from which only the states would benefit.58× 58. For two practical arguments to this effect, see Brian Galle, Essay, Kill Quill, Keep the Dormant Commerce Clause: History’s Lessons on Congressional Control of State Taxation, 70 Stan. L. Rev. Online 158, 160–62 (2018), https://review.law.stanford.edu/wp-content/uploads/sites/3/2018/03/70-Stan.-L.-Rev.-Online-158-Galle.pdf [https://perma.cc/22YP-P4V5]; Edward A. Zelinsky, The Political Process Argument for Overruling Quill, 82 Brook. L. Rev. 1177, 1191–92 (2017). Indeed, South Dakota advanced such practical arguments in its brief.59× 59. See Petitioner’s Brief at 54, Wayfair, 138 S. Ct. 2080 (No. 17-494) (“Congress has little incentive to act here because it would be (or appear to be) authorizing new or greater tax collections from its constituents, while receiving none of the revenue in return.”). More generally, the Court might have discussed the limits of the states’ influence in the federal system as a reason not to wait for congressional intervention, a topic it has debated on other occasions.60× 60. See Richard H. Pildes, Institutional Formalism and Realism in Constitutional and Public Law, 2013 Sup. Ct. Rev. 1, 30–32; see also Galle, supra note 58, at 159 (“Congress is not a trustworthy guardian of state fiscal power, making continuing judicial involvement a more appealing prospect.”). Or it could have argued that new facts on the ground — namely, the blast of e-commerce that hit like a comet after Quill — overpowered stare decisis of any force, special or plain.61× 61. Two recent studies of stare decisis highlighted the physical presence rule as exemplifying a precedent that may reasonably be overruled due to changed facts. See Bryan A. Garner et al., The Law of Judicial Precedent 364–65 (2016); Randy J. Kozel, Settled Versus Right: A Theory of Precedent 112–13 (2017). It should be noted that the authors of The Law of Judicial Precedent classify the physical presence rule as a constitutional precedent for stare decisis purposes, thus anticipating the Court’s misstep in Wayfair. Garner et al., supra, at 354–65. Because even statutory precedents may sometimes be overruled,62× 62. See Patterson v. McLean Credit Union, 491 U.S. 164, 173–74 (1989) (discussing justifications for overruling statutory precedents). Contra Lawrence C. Marshall, “Let Congress Do It”: The Case for an Absolute Rule of Statutory Stare Decisis, 88 Mich. L. Rev. 177 (1989). the Court could have killed Quill without first planting its constitutional kiss of death.63× 63. Cf. Thomas R. Lee, Stare Decisis in Historical Perspective: From the Founding Era to the Rehnquist Court, 52 Vand. L. Rev. 647, 704 (1999) (“Justice Brandeis’ . . . memorable prose has since become a mandatory part of the burial rite for any constitutional precedent.”).
The Court resisted such arguments. Instead, Wayfair reasoned that Congress’s total ability to correct an erroneous decision counts for nothing when the Court gets the Constitution wrong. That such a theory sprouts from a case like Wayfair, which repudiated a “formalistic distinction,”64× 64. Wayfair, 138 S. Ct. at 2092. is ironic. Wayfair’s stare decisis analysis resorts to the formalism of making constitutional a “magic” word65× 65. See Transcript of Oral Argument, supra note 21, at 12. rather than asking whether Congress can step in.
Moreover, the Court’s new thinking on stare decisis threatens other constitutional default rules. Wayfair now stands for the proposition that a “constitutional default rule” — a term the Court apparently lifted from South Dakota’s reply brief on the merits66× 66. Reply Brief at 22, Wayfair, 138 S. Ct. 2080 (No. 17-494) (“Congress is polarized, which makes it critical . . . to get the constitutional default rule right.”). — gets only weakened stare decisis. To appreciate why this holding matters, it is worth exploring the concept and scope of constitutional default rules. Contract theory describes default rules as legal rules that the parties may “contract around.”67× 67. See, e.g., Ian Ayres & Robert Gertner, Filling Gaps in Incomplete Contracts: An Economic Theory of Default Rules, 99 Yale L.J. 87, 87 (1989). Although “constitutional default rule” could be read broadly to include a variety of actors and contracting mechanisms,68× 68. See John Ferejohn & Barry Friedman, Toward a Political Theory of Constitutional Default Rules, 33 Fla. St. U. L. Rev. 825, 826 (2006) (“When we speak of default rules in constitutional law, we typically are talking about specifications of ways the government can act (or modify its behavior) to get around a constitutional prohibition.”). the Court’s use of the term for purposes of stare decisis may be narrowly defined as judicial precedents of constitutional law that “are ultimately subject to congressional control.”69× 69. Gillian E. Metzger, Congress, Article IV, and Interstate Relations, 120 Harv. L. Rev. 1468, 1525 (2007) (describing judicially enforceable “constitutional default rules imposing obligations on the states in the name of union [that] are ultimately subject to congressional control”). The dormant commerce clause is a paradigmatic constitutional default rule because what the Court does today Congress may undo tomorrow. Justice Scalia declared this fact “[t]he clearest sign that the negative Commerce Clause is a judicial fraud,” for “[h]ow could congressional consent lift a constitutional prohibition?”70× 70. Comptroller of the Treasury of Md. v. Wynne, 135 S. Ct. 1787, 1808 (2015) (Scalia, J., dissenting). But that’s what a constitutional default rule is. The Court has allowed Congress to overturn its dormant commerce clause cases since 1891.71× 71. See In re Rahrer, 140 U.S. 545, 560–62 (1891).
Dormant commerce clause cases are not the only constitutional default rules. Professor Laurence Tribe’s treatise identifies two others.72× 72. 1 Laurence H. Tribe, American Constitutional Law § 6-35 (3d ed. 2000). And in a groundbreaking article, Professor Henry Monaghan revealed “a substructure of substantive, procedural, and remedial rules” forming “a constitutional common law subject to amendment, modification, or even reversal by Congress.”73× 73. Henry P. Monaghan, The Supreme Court, 1974 Term — Foreword: Constitutional Common Law, 89 Harv. L. Rev. 1, 2–3 (1975); see also Susan R. Klein, Identifying and (Re)Formulating Prophylactic Rules, Safe Harbors, and Incidental Rights in Constitutional Criminal Procedure, 99 Mich. L. Rev. 1030 (2001) (further developing Monaghan’s theory in criminal procedure context). What follows is a list of six lines of cases beyond the dormant commerce clause that may be fairly described as constitutional default rules. The first two are drawn from Tribe’s treatise while the next four are found in Monaghan’s article:
(1) State Taxation of Federal Instrumentalities: States may not tax instrumentalities of the federal government74× 74. McCulloch v. Maryland, 17 U.S. (4 Wheat.) 316, 436 (1819). — unless Congress consents.75× 75. See, e.g., Helvering v. Gerhardt, 304 U.S. 405, 411 n.1 (1938) (“Congress may curtail an immunity which might otherwise be implied or enlarge it beyond the point where, Congress being silent, the Court would set its limits.” (citations omitted)). One court has described such judicial decisions as setting a “constitutional default rule.” United States v. Delaware, 958 F.2d 555, 560 n.9 (3d Cir. 1992) (“[W]e must decide the constitutional default rule for this type of tax, fully aware that Congress could decide at any time to reverse our decision statutorily.”). (2) Article I, Section 10 Cases: Article I, Section 10 provides that certain prohibitions on the states may be waived by Congress.76× 76. See U.S. Const. art. I, § 10, cls. 2–3. The Court has taken note of this when considering whether to overrule, for instance, an Import-Export Clause precedent.77× 77. See Hooven & Allison Co. v. Evatt, 324 U.S. 652, 668 (1945) (“In view of the fact that the Constitution gives Congress authority to consent to state taxation of imports and hence to lay down its own test for determining when the immunity ends, we see no convincing practical reason for abandoning the test which has been applied for more than a century . . . .”), overruled on other grounds by Limbach v. Hooven & Allison Co., 466 U.S. 353 (1984). In Michelin Tire Corp. v. Wages, 423 U.S. 276 (1976), the Court left open the question whether “Congress may authorize, under the Import-Export Clause, an exaction that it could not directly impose under the Tax Clause.” Id. at 301 n.13. Metzger, however, argues that the Import-Export Clause is free of other clauses’ limits on congressional power. See Metzger, supra note 69, at 1500 & n.120. (3) Bivens Cases: In Bivens v. Six Unknown Named Agents of Federal Bureau of Narcotics,78× 78. 403 U.S. 388 (1971). the Court held that a violation of the Fourth Amendment gives rise to a right to sue for damages.79× 79. Id. at 397. But the Court has also held that “[s]uch a cause of action may be defeated . . . when . . . Congress has provided an alternative remedy which it explicitly declared to be a substitute for recovery directly under the Constitution and viewed as equally effective.”80× 80. Carlson v. Green, 446 U.S. 14, 18–19 (1980). (4) Miranda Cases: The Miranda Court famously “encourage[d]” Congress and the states to explore alternative “procedures which are at least as effective in apprising accused persons of their right of silence and in assuring a continuous opportunity to exercise it.”81× 81. Miranda v. Arizona, 384 U.S. 436, 467 (1966). In Dickerson v. United States, 530 U.S. 428 (2000), the Court struck down a congressional attempt to effectively abolish Miranda, holding that “Miranda announced a constitutional rule that Congress may not supersede legislatively.” Id. at 444. But Dickerson also stood by Miranda’s “invitation for legislative action” to replace Miranda with an adequate substitute. Id. at 440; see also Michael C. Dorf & Barry Friedman, Shared Constitutional Interpretation, 2000 Sup. Ct. Rev. 61 (discussing legislative alternatives to Miranda). (5) The Police Lineup Case: In United States v. Wade,82× 82. 388 U.S. 218 (1967). the Court created an exclusionary rule for evidence obtained from a police lineup in violation of the Sixth Amendment right to counsel but acknowledged that it could be replaced by “[l]egislative or other regulations . . . which eliminate the risks of abuse.”83× 83. Id. at 239. (6) The Exclusionary Rule Cases: Mapp v. Ohio made the Fourth Amendment “exclusionary rule” binding on the states,84× 84. 367 U.S. 643, 655 (1961). yet Congress is thought to have the power to replace it.85× 85. See Bivens v. Six Unknown Named Agents of Fed. Bureau of Narcotics, 403 U.S. 388, 422–24 (1971) (Burger, C.J., dissenting) (inviting Congress to replace the Fourth Amendment exclusionary rule); Harold J. Krent, How to Move Beyond the Exclusionary Rule: Structuring Judicial Response to Legislative Reform Efforts, 26 Pepp. L. Rev. 855, 864–71 (1999).
All of the above are arguably constitutional default rules set by the Court that remain, to one degree or another, open to congressional revision. The list could be longer or shorter, depending on which default rules the Court will view as constitutional86× 86. A shorter list could be produced by whittling away at the constitutional status of the cases identified by Monaghan. While the Court has held that Miranda is a constitutional decision, Dickerson, 530 U.S. at 444, some of the other cases may be viewed as nonconstitutional. See, e.g., Collins v. Virginia, 138 S. Ct. 1663, 1675–80 (2018) (Thomas, J., concurring) (arguing that Mapp is “nonconstitutional,” id. at 1678 n.5); Richard H. Fallon, Jr. et al., Hart and Wechsler’s The Federal Courts and the Federal System 775–77 (7th ed. 2015) (discussing whether Bivens is constitutionally required). Conversely, a longer list might include any constitutional right that can be waived by a party. See, e.g., Daniel A. Farber, Another View of the Quagmire: Unconstitutional Conditions and Contract Theory, 33 Fla. St. U. L. Rev. 913, 918 (2006) (describing the Eleventh Amendment as “just a contractual default rule that the states are free to barter away”). Such a list might also include various constitutionally inspired judicial presumptions. See, e.g., Jack Goldsmith & John F. Manning, The President’s Completion Power, 115 Yale L.J. 2280, 2299 (2006) (describing the Chevron presumption of delegated interpretive power to administrative agencies as “a constitutionally inspired default rule”); Nicholas Quinn Rosenkranz, Federal Rules of Statutory Interpretation, 115 Harv. L. Rev. 2085, 2097–98 (2002) (describing clear statement rules as “constitutional default rules” reversible by Congress). Many other decisions could likely be characterized as constitutional default rules; the list above is only an initial stab. and on how it will answer open questions about congressional authority over certain constitutional provisions.87× 87. See, e.g., Thomas v. Wash. Gas Light Co., 448 U.S. 261, 272 n.18 (1980) (plurality opinion) (leaving unresolved whether Congress may limit constitutional full faith and credit obligations); White v. Mass. Council of Constr. Emp’rs, Inc., 460 U.S. 204, 215 n.1 (1983) (Blackmun, J., concurring in part and dissenting in part) (leaving unresolved “whether Congress may authorize . . . what otherwise would be a violation” of the Privileges and Immunities Clause); 1 Tribe, supra note 72, § 6-35, at 1243–44 (arguing that Congress cannot override judicial constructions of the Privileges and Immunities Clause); Metzger, supra note 69, at 1486–89 (arguing the opposite). But the takeaway is clear: weaker stare decisis for constitutional default rules. Pre-Wayfair, one would have thought that stare decisis applies with special force to such precedents, given congressional power to set them straight. Not anymore. Why? Because it is improper to “ask Congress to address a false constitutional premise of th[e] Court’s own creation.”88× 88. Wayfair, 138 S. Ct. at 2096. The Latin for Wayfair’s doctrine is not stare decisis, which should reflect a realistic, working relationship between the legislative and judicial branches. It is mea culpa.
In its zeal to update the Constitution for “the Cyber Age,”89× 89. Id. at 2097. the Court deleted Congress from stare decisis doctrine in constitutional cases. The Court had better options. It could have left Quill on Congress’s doorstep, as the dissent argued. Or it could have justified overruling Quill notwithstanding the special force of stare decisis. Instead, the Court reasoned that it doesn’t matter whether Congress is willing and able to do the job: a constitutional mess calls for a judicial clean-up crew. For constitutional default rules — a category of decisions embracing the dormant commerce clause and sweeping far beyond — Wayfair’s new theory of stare decisis makes the Court’s precedents less sticky and Congress less relevant.

Answer: Quill harmed states more than anticipated due to the Internet.


Input: What is the main focus of the research paper?

Context:
Paper Info

Title: Nuclear Liquid-Gas Transition in the Strong Coupling Regime of Lattice QCD
Publish Date: 28 Mar 2023
Author List: J Kim (from Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich), P Pattanaik (from Fakultät für Physik, Bielefeld University), W Unger (from Fakultät für Physik, Bielefeld University)

Figure

FIG. 1.Typical 2-dimension configuration at β = 1.0, at non-zero quark mass, temperature, chemical potential.The black dots are monomers, the blue lines are dimers, the red arrows are baryon loop segments (or triplets g b + f b = ±3 if adjacent to a non-trivial plaquette), and the green squares are plaquette occupations ±1.The actual configurations are 3+1-dimensional.
FIG.2.Chiral susceptibility on a 2 4 volume for various quark masses, as a function of the bare anisotropy γ (with aT = γ 2 /2), analytic results from enumeration compared to numerical data from simulations via the worm algorithm.
FIG.3.Various observables in the µB-T plane on a 2 4 volume at amq = 0.1.The back-bending of the first order transition at temperatures below aT = 0.5 in all observables is an artifact of the small volume, and vanishes in the thermodynamic limit.The temperature aT = 1/2 corresponds to the isotropic lattice here.
FIG. 4. The chiral condensate (left) and the baryon density (right) for quark mass m = 1.5 as a function of the chemical potential and for various temperatures.
FIG. 7. ∆f at amq = 0.2 as a function of chemical potential and β the on a 6 3 × 4 lattice
FIG. 8. Baryon mass from ∆E as a function of the quark mass amq, and contributions from different dual variables: monomers, dimers and baryon segments.
FIG. 9. Baryon density for volume 4 3 × 8 in the full µB − mq plane, illustrating the strong quark mass dependence of the onset to nuclear matter.
FIG. 10.Baryonic observables on various volumes in the first order region amq = 1.5.Vertical bands indicate the mean and error of the nuclear transition.
FIG. 12. Left: Extrapolation of the pseudo-critical values of µB for the various volumes into the thermodynamic limit.Right: Critical baryon chemical potential for different quark masses.The first order transition region is shown in blue, the crossover region is shown in red and the range for critical end point is marked in black.
FIG. 17. Nuclear interaction scaled with baryon mass.As the quark mass increases, it tends to zero.
FIG. 18. Critical baryon chemical potential and baryon mass from different approaches.
Parameters for the Monte Carlo runs to determine the nuclear transition at strong coupling, with statistics after thermalization.

abstract

The nuclear liquid-gas transition from a gas of hadrons to a nuclear phase cannot be determined numerically from conventional lattice QCD due to the severe sign problem at large values of the baryon chemical potential. In the strong coupling regime of lattice QCD with staggered quarks, the dual formulation is suitable to address the nuclear liquid gas transition.
We determine this first order transition at low temperatures and as a function of the quark mass and the inverse gauge coupling β. We also determine the baryon mass and discuss the nuclear interactions as a function of the quark mass, and compare to mean field results. It is known from experiments that at low temperatures, there is a phase transition between dilute hadron gas and dense nuclear matter as the baryon chemical potential increases.
This transition is of first order and terminates at about T c = 16 MeV in a critical end point. The value of the chemical potential µ 1st B at zero temperature is given roughly by the baryon mass m B , where the difference of µ 1st B −m B is due to nuclear interactions. For a review on nuclear interactions see .
As the nuclear force between baryons to form nuclear matter is due to the residual strong interactions between quarks and gluons, it should be accurately described by QCD. We choose to study the nuclear transition and nuclear interaction via lattice QCD , with its Lagrangian being a function of the quark mass and the inverse gauge coupling.
In order to understand the nature of the transition, it is helpful to study its dependence on these parameters. However, at finite baryon density, lattice QCD has the infamous sign problem which does not allow us to perform direct Monte Carlo simulations on the lattice. Various methods have been proposed to overcome the numerical sign problem, but they are either limited to µ B /T 3 or can not yet address full QCD in 3+1 dimensions in the whole µ B − T plane , in particular the nuclear transition is out of reach.
An alternative method is to study lattice QCD via the strong coupling expansion. There are two established effective theories for lattice QCD based on this: (1) the 3-dim. effective theory for Wilson fermions in terms of Polyakov loops, arising from a joint strong coupling and hopping parameter expansion , the dual representation for staggered fermions in 3+1 dimensions, with dual degrees of freedom describing mesons and baryons.
Both effective theories have their limitations: is limited to rather heavy quarks (but is valid for large values of β) whereas ( ) is limited to the strong coupling regime β 1 (but is valid for any quark mass). We study lattice QCD in the dual formulation, both at infinite bare gauge coupling, β = 0, and at leading order of the strong coupling expansion in the regime β < 1, which is far from the continuum limit.
But since strong coupling lattice QCD shares important features with QCD, such as confinement, and chiral symmetry breaking and its restoration at the chiral transition temperature, and a nuclear liquid gas transition, we may get insights into the mechanisms, in particular as the dual variables give more information in terms of its world lines, as compared to the usual fermion determinant that depends on the gauge variables.
To establish a region of overlap of both effective theories, we have chosen to perform the Monte Carlo simulations in the dual formulation extending to rather large quark masses. This paper is organized as follows: in the first part we explain the dual formulation in the strong coupling regime, in the second part we provide analytic results based on exact enumeration and mean field theory, in the third part we explain the setup of our Monte Carlo simulations and present result on the m q -and β-dependence of the nuclear transition.
Since the strong coupling regime does not have a well defined lattice spacing, we also determine the baryon mass am B to set the parameters of the grand-canonical partition function, aT and aµ B , in units of am B . We conclude by discussing the resulting nuclear interactions, and compare our findings with other results.

Staggered action of strong coupling QCD and its dual representation

In the strong coupling regime, the gauge integration is performed first, followed by the Grassmann integration to obtain a dual formulation. This was pioneered for the strong coupling limit in and has been extended by one of us to include gauge corrections . The sign problem is mild in the strong coupling limit and still under control for β < 1, where we can apply sign reweighting.
The dual degrees of freedom are color-singlet mesons and baryons, which are point-like in the strong coupling limit, and become extended about a lattice spacing by incorporating leading order gauge corrections. The partition function of lattice QCD is given by where DU is the Haar measure, U ∈ SU(3) are the gauge fields on the lattice links (x, μ) and { χx , χ x } are the unrooted staggered fermions at the lattice sites x.
The gauge action S G [U] is given by the Wilson plaquette action and the staggered fermion action S F [ χ, χ, U] is: where the gauge action depends on the inverse gauge coupling β = 2Nc g 2 and the fermion action depends on the quark chemical potential aµ q which favors quarks in the positive temporal direction, and the bare quark mass am q .
First we consider the strong coupling limit where the inverse gauge coupling β=0 and hence the gauge action S G [U] drops out from the partition function in this limit. The gauge integration is over terms depending only on the individual links (x, μ) so the partition function factorizes into a product of one-link integrals and we can write it as:
with z(x, μ) the one-link gauge integral that can be eval-uated from invariant integration, as discussed in , where we write the one-link integral in terms of new hadronic variables: Only terms of the form (M (x)M (y)) k x, μ (with k x,μ called dimers which count the number of meson hoppings) and B(y)B(x) and B(x)B(y) (called baryon links) are present in the solution of the one-link integral.
The sites x and y = x + μ are adjacent lattice sites. It remains to perform the Grassmann integral of the fermion fields χ, χ. This requires to expand the exponential containing the quark mass in Eq. (4) (left), which results in the terms (2am q M (x)) nx (with n x called monomers). To obtain non-vanishing results, at every site, the 2N c Grassman variables χ x,i and χx,i have to appear exactly once, resulting in the Grassmann constraint (GC):
where n x is the number of monomers, k x,μ is the number of dimers and the baryons form self-avoiding loops x,μ , which due to the constraint cannot coexist with monomers or dimers. With this, we obtain an exact rewriting of the partition function Eq. ( ) for N c = 3, in terms of integer-valued dual degrees of freedom {n, k, }:
where the sum over valid configurations has to respect the constraint (GC). The first term in the partition function is the contribution from dimers and the second term is the contribution from monomers. The weight factor w( ) for each baryon loop depends on the baryon chemical potential µ B = 3µ q and induces a sign factor σ( ) which depends on the geometry of :
Here, ω is the winding number of the loop . The total sign factor σ( ) ∈ {±1} is explicitly calculated for every configuration. We apply sign reweighting as the dual formulation has a mild sign problem: baryons are non-relativistic and usually have loop geometries that have a positive signs. The dual partition function of the strong coupling limit is simulated with the worm algorithm (see Section III A) and the sign problem is essentially solved in this limit.

Extension to finite β

The leading order gauge corrections O(β) to the strong coupling limit are obtained by expanding the Wilson gauge action Eq. ( ) before integrating out the gauge links. A formal expression is obtained by changing the order of integration (first gauge links, then Grassmann-valued fermions) within the QCD partition function:
With this the O (β) partition function is The challenge in computing Z (1) is to address the SU(N c ) integrals that receive contributions from the elementary plaquette U P . Link integration no longer factorizes, however the tr[U P ] can be decomposed before integration: Integrals of the type J ij with two open color indices -as compared to link integration at strong coupling -have been derived from generating functions
for either J = 0 or for G = U(N c ) . The SU(3) result was discussed in , in terms of the dual variables, neglecting rotation and reflection symmetries, there are 19 distinct diagrams to be considered. The resulting partition function, valid to O(β), is with q P ∈ {0, ±1}, and the site weights w x → ŵx , bond weights w b → ŵb and baryon loop weights w → ŵ receive modifications compared to the strong coupling limit Eq. ( ) for sites and bonds adjacent to an excited plaquette q P = 1.
The weights are given in , and are rederived for any gauge group in . The configurations {n, k, , q p } must satisfy at each site x the constraint inherited from Grassmann integration: which is the modified version of Eq. ( ) with q x = 1 if located at the corner of an excited plaquette q p = 0, otherwise q x = 0.
A more general expression that we obtained via group theory and is valid to higher orders of the strong coupling expansion is discussed in terms of tensor networks . A typical 2-dimensional configuration that arises at β = 1 in the Monte Carlo simulations is given in Fig. . Note that if a baryon loop enters a non-trivial plaquette, one quark is separated from the two other quarks, resulting in the baryon being extended object, rather being point-like in the strong coupling limit.
The O(β) partition function has been used in the chiral limit to study the full µ B − T plane via reweighting from the strong coupling ensemble. Whereas the second order chiral transition for small values of the aµ B decreased up to the tri-critical point, the first order nuclear transition was invariant: aµ 1st B 1.78(1) at zero temperature has no β-dependence.
For the ratio T (µ B = 0)/µ 1st B (T 0) we found the values 0.787 for β = 0 and 0.529 β = 1, which should be compared to T c / 0.165 for full QCD . However, since reweighting cannot be fully trusted across a first order boundary, direct simulations at nonzero β are necessary. The Monte Carlo technique to update plaquette variables is discussed in Section III A.
In this section, we provide analytic results from exact enumeration for small volumes, and mean field results based on the 1/d expansion, valid in the thermodynamic limit. The main purpose is to compare our Monte Carlo results to these analytic predictions.

Exact enumeration

To establish that our Monte Carlo simulations indeed sample the partition functions Eq. ( ) and Eq. ( ), we have obtained analytic results on a 2 4 volume at strong coupling, and at finite beta in two dimensions on a 4 × 4 volume, comparing O (β) and O β 2 truncations. Our strategy to obtain an exact enumeration of the partition function Z is to enumerate plaquette configurations first, then fixing the fermion fluxes which together with the gauge fluxes that are induced by the plaquettes form a singlet, a triplet or anti-triplet, i.e. on a given bond b, g b + f b ∈ {−3, 0, 3}, and last we perform the monomerdimer enumeration on the available sites not saturated by fermions yet by a depth-first algorithm .
At strong coupling, with no plaquettes, g b = 0 and f b are baryonic fluxes. All observables that can be written in terms of derivatives of log(z), such as the baryon density, the chiral condensate, the energy density, and also the average sign, are shown in Fig.

Expectations from mean field theory

Another analytical method to study strong coupling lattice QCD is the mean field approach, where the partition function is expanded in 1 d (d is the spatial dimension) and then a Hubbard-Stratonovich transformation performed . After this procedure, the free energy is a function of temperature T , the chiral condensate σ and chemical potential µ B :
here E[m] is one-dimensional quark excitation energy which is a function of the quark mass m = am q . For N c = 3 and d = 3 we determined the minimum of the free energy with respect to the chiral condensate. This gives us the equilibrium chiral condensate as a function of (T, m, µ B ). The chiral condensate and the baryon density as a function of the baryon chemical potential in lattice units aµ B and for various temperatures at quark mass m = 1.5 is shown in Fig. . We have determined the critical temperature to be aT c = 0.23 , which is characterized by an infinite slope of the chiral condensate.
For lower temperatures, there is a clear discontinuity of the chiral con-densate, separating the low density phase from the high density phase. For temperatures above and in the vicinity of aT c the chiral condensate and baryon density has no discontinuity but rapidly changes, corresponding to a crossover transition.
With this method, the phase diagram is plotted for different quark masses in Fig. . The second order phase transition in the chiral limit is plotted in solid blue line, the dotted lines show the first order phase transition for different quark masses and the solid red line indicates the critical end point for the different quark masses.
Mean field theory also gives an expression for the pion mass am π and the baryon mass am B : The mean field baryon mass for N c = 3, d = 3 is also plotted in red in Fig. . Whereas the baryon mass is around N c in the chiral limit (am B 3.12 for N c = 3), it approximately doubles at m = 3.5 (am B 6.28) which corresponds to the pion mass am π = 4.45, i.e. m π /m B = 0.708.
Hence, at around bare mass m = 3.5, the valence quark mass of the baryon corresponds roughly to 1/3 of the chiral limit value of the baryon mass. The first Monte Carlo simulations that could extend in the µ B − T plane was the MDP algorithm , but it required the introduction of the worm algorithm to make substantial progress.
First studies of the worm algorithm applied to the strong coupling limit QCD (with gauge group U(3)) are , and for gauge group SU . Monte Carlo simulations to extend the worm to incorporate leading order corrections were first proposed in . We will shortly review the setup of or Monte Carlo strategy for the nuclear transition, with an emphasis on the challenges to address large quark masses.

Strong Coupling

Without any further resummation, there is a mild sign problem in the dual formulation of lattice QCD in the strong coupling limit. When the average sign σ is not too small (close to zero), it implies that most of the configurations have a positive weight thus allowing us to perform sign reweighting strategies.
In Fig. , ∆f is plotted as a function of the baryon chemical potential and the quark masses. It is seen that ∆f is close to zero for most cases except near the critical chemical potential and for small quark masses, but never exceeds 5 × 10 −4 . Hence sign reweighting can be performed in the full parameter space.
The result that the sign problem becomes even milder when increasing the mass is related to the fact that larger critical chemical potentials result in a larger fraction of static baryons (spatial baryon hoppings become rare). FIG. . ∆F at strong coupling as a function of chemical potential and quark mass on a 6 3 × 8.
The sign problem becomes milder as the quark mass increases.

Finite β

All runs at finite β have been obtained for N τ = 4, which corresponds to a moderately low temperature aT = 0.25 compared to the value of the chiral transition aT 1.54. Those simulations were too expensive to attempt N τ = 8 runs, in particular as a higher statistics was required. The spatial volumes are 4 3 , 6 3 and 8 3 .
For β values are from 0.0 to 1.0 with step size 0.1, and for am q values from 0.00 to 1.00 with step size 0.01. The values of aµ were chosen close to the nuclear transition, the scanning range is shifted to large values as am q increases. At small quark masses the scanning range is from aµ = 0.4 to 1.0 and for the large quark masses, it is from 0.6 to 1.2 with step size 0.01.
The statistics used for are 15 × 10 4 measurements and between measurement, 40 × N 3 s worm updates.

Residual sign problem

Although it is possible to resum the sign problem at strong coupling with a resummation of baryon and pion world lines, this is not possible when including gauge corrections. In order to compare both sign problems, we kept the original dual formulation to monitor the severity of the sign problem. This is done via the relation
between the average sign σ and the difference of the free energy density ∆f between the full ensemble f and of the sign-quenched ensemble f || .

Nuclear interactions

We have found that aµ 1st B is very different from the baryon mass. This must be due to strong attractive interactions of nucleons. In contrast to continuum physics, in the strong coupling limit there is no pion exchange due to the Grassmann constraint. Instead, nucleons are point like and hard core repulsive.
However, the pion bath, which is modified by the presence of static baryons, results in an attractive interaction. In , this has been analyzed in the chiral limit using the snake algorithm, and it has been found that the attractive force is of entropic origin. Here, we do not quantify the nuclear interaction via the nuclear potential, but via the difference between critical baryon chemical potential and baryon mass, in units baryon mass, as shown in Fig. , given the am B as measured in Section III C.
This compares better to the 3dim. effective theory. The nuclear interaction is maximal and more than 40% in the chiral limit, which is related to pions being massless: the modification of the pion bath is maximal. We clearly find that the nuclear interaction decreases drastically and almost linearly until it almost approaches zero at about am q = 2.0, corresponding to a pion mass am π = 3.36, see Section II B. The large error bars for larger quark masses, that are due to the subtraction of almost same magnitudes, makes it difficult to extract a non-zero nuclear interaction at the largest quark masses.
In this work, we have determined the baryon mass and the nuclear transition via Monte Carlo: the worm algorithm based on the dual formulation, at finite β equipped with additional updates. All those numerical results and various analytic expressions are summarized in Fig. . We find that as the quark mass becomes large, spatial mesons hoppings (i.e.
spatial dimers) become rare, which makes this 3+1-dimensional system closer to 1dim. QCD . Also, both the baryon mass and the baryon chemical potential obtained in our dual representation, i.e. for staggered fermions, approaches the baryon mass of the 3-dim. effective theory which is based on Wilson fermions.
Another comparison that summarizes the validity of the mean field approach discussed in Section II B is shown in Fig. . It is evident that mean field theory has strong deviations for small quark masses, but this discrepancy becomes smaller for larger quark masses. The extension of the study of the nuclear transition to finite inverse gauge coupling β is summarized in Fig. , which shows the β-dependence of aµ c B for various quark masses.
For all quark masses ranging from am q = 0 to am q = 1.0, there is only a very weak β-dependence, confirming the expectation from mean field theory . This works was restricted to isotropic lattices ξ = a/a t = 1, i.e. we performed simulations at fixed temperature. Non-isotropic lattices are necessary to vary the temperature at fixed values of β.
This requires to include two bare anisotropies, γ for the fermionic action and γ G for the gauge action. Finite β has only been studied by us in the chiral limit . Clearly, it is interesting to study the location of the nuclear critical point also including higher order gauge corrections and at finite quark mass.
Simulations including O(β 2 ) are under preparation.

Answer: Nuclear liquid-gas transition in lattice QCD.