Answer Set Programming (ASP) and the mixed discrete-continuos variant of the Planning Domain Definition Language (PDDL+) are well-known knowledge representation and reasoning methodologies to solve configuration, scheduling and planning problems arising in real-life applications. In this paper, we focus on the recent problems modeled and solved with ASP and PDDL+ in the context of urban trafic distribution and control, including Trafic Signal Optimization and the optimization of pre-computed routes of Connected Autonomous Vehicles in urban networks.
Answer Set Programming (ASP) 1[] and the mixed discrete-continuous variant of the Planning Domain
Definition Language (PDDL+) [
In this paper, we overview the recent problems modelled and solved with ASP and PDDL+ in the context of urban trafic distribution and control. We first present the applications in which ASP is employed, i.e., Trafic Signal Optimization, which aims of minimizing the average trafic delay for a region of interest by determining the optimal green length, and the optimization of pre-computed routes of Connected Autonomous Vehicles in urban areas. Then, we move to PDDL+, and review several models, from flexible to deployable, related to the Trafic Signal Optimization problem. The paper ends by drawing some conclusions and by discussing a solution which integrates ASP and PDDL+ for eficiently solving the Trafic Signal Optimization problem.
CEUR Workshop
ISSN1613-0073
In this section, we describe ASP applications for tackling problems appearing in Urban Trafic Distribution.
In [30] we introduced an ASP solution to tackle thTerafic Signal Optimization Problem , which aims
to minimize the average trafic delay for a region of interest by determining the optimal green length
for each trafic signal in a given set. We focus on a major corridor in the Kirklees council area
within West Yorkshire (UK) containin6g junctions and34 road links. We frame the problem using a
mesoscopic trafic model, where the approximate number of vehicles in road links is considered (instead
of individual vehicles) and the trafic signals in a junction are abstractedsatasges, each representing
a set of simultaneous trafic movements. A cycle is the ordered sequence of these stages, separated
by intergreen times. Constraints apply to the minimum and maximum durations of both stages and
full cycles, while the order of stages remains fixed. The SCOOT3[
In the simulation, we discretise the time in seconds and consider a horizon of90u0pseconds (i.e.,15 minutes), which is the maximum duration for which the SCOOT data remains representative. The ASP solver decisions only concern which configuration is selected for each junction. Sin()cethe duration of each cycle is the same (regardless of the configuration), an(d) once a configuration is selected, it cannot be changed unless the cycle ends, the decision points can be precomputed, drastically reducing their number. Despite this observation, calculating the occupancy and counter for34threoad links at each second produces a number of ground rules that cannot be handled, even for short horizons (around100 seconds). To handle this issue, we applied the systemclingcon 3 [32], which extends the well-known ASP solverclingo with theory atoms and propagators for linear constraints. By redefining through theory atoms the link road occupancies and counters, we managed to model and efectively apply Constraint ASP (CASP) to tackle the trafic signal optimization problem with meaningful horizons. In particular, thanks to ASP expressivity, we can maximize the counters for a set of given links, or even define more articulated optimization criteria.
In [33], we proposed a new framework to explicitly incorporate sustainability aspects in the context of
AI-Enabled Connected and Autonomous Vehicles (CAVs). We assumed a Vehicle-to-Infrastructure (V2I)
communication system that allows trafic controllers to interact with individual vehicles, hence actively
managing trafic flow by directing vehicles along optimised routes to their destinations. The framework
is an extension of 3[
∑ ( ∈{1… } 1, 2, , , ) ⋅ ln() where( 1, 2, , , ) is the emission value obtained from a Urban Mobility Simulator (e.g., using sumo) at time step for a specific pair of streets, emission class, and congestion condition. To ensure that highly congested streets are likely to be avoided, a logarithmic weighting falcnt(o)r is introduced, emphasizing later stages of the simulation, and using time also as an indicator of potentially highly congested streets. Given a road (i.e., a sequence of connected streets), the overall emission risk is computed by summing all the local risks for each pair of streets. More formally, the pollution risk associated with assigning vehicleto route is defined as: (, ) =
∑ ( 1, 2, , ) ( 1, 2)∈
After the Preprocessing a Search phase is introduced: its objective is to define a set of candidate solutions, which will be then provided to the optimization phase. The Optimisation module was our second contribution with respect to the initial framework. Given the set of candidate routes produced by the search step, the Optimisation step is responsible for selecting the most suitable route to assign to each vehicle. In our approach, for each new vehicle entering the network, a new route is provided, taking into account all routes already assigned to previously scheduled vehicles. A3s4i]n,w[ e adopted ASP [35] in the underlying approach for the optimisation phase. The pollution risk associated with a segment, previously defined as the function ( 1, 2, , ) , is represented as:
risk(S1,S2,EC,CL,RV) where S1 and S2 are consecutive streetsE,C identifies the emission class of the vehicle, CL reflects the local trafic condition, and RV quantifies the pollution risk. To define the objective function for minimising emission risk associated with a specific candidate route (given by the Search module), the following optimation in term of ASP weak constraint has been introduced: :∼ vehicle(V), solutionRoute(V,R), emissionClass(V,EC), indexStreetOnRoute(S1,R,I),indexStreetOnRoute(S2,R,I+1), enter(V,S1,T), congestion(S1,T,CL), risk(S1,S2,EC,CL,RV). [RV@1]
This rule penalises the selection of routes with a high pollution risk, whReVreis a numeric value that measures the pollution risk. The optimiser will favour solutions that minimise this objective function, which are solutions with a lower pollution risk.
To evaluate the efectiveness of our proposed approach, we employed thsuemo trafic simulator and examined two distinct urban environments: Bologna and Milton Keynes. In both scenarios, our approach performed well. The results were particularly strong in the Milton Keynes scenario, likely due to the wider range of possible routes from a single starting point, confirming the ability of the designed optimisation approaches in reducing emissions.
This section reviews PDDL+-based approaches to urban trafic distribution. Our research has focused on designing planning models that capture diverse requirements, which shape the resulting formulations. Moreover, it has also focused on developing domain-specific heuristics to eficiently solve these models within the “planning as heuristic search” approac3h6][. The models described here can be roughly characterized aflesxible ordeployable, according to the needs that guided their design.
Flexible models maximize controller freedom by admitting a large set of valid signal plans. This class of models is useful for exploring the upper-bound performance in an idealized setting. In particular, they highlight the potential gains that can be achieved by upgrading the underlying control infrastructure and removing specific technological constraints. However, prioritizing flexibility can undermine the deployability of the approach in relation to the target UTC infrastructure. Indeed, these models abstract away controller capabilities, e.g., limits on how frequently configurations may change, and let the solver dynamically generate the most suitable configurations. Therefore, signal plans can prove inefective in practice or fail trafic authority validation. From a computational perspective, these expressive models are typically more challenging to solve and require the use of domain-dependent heuristics to obtain eficient performance in plan signal generation.
In contrast, deployable models are designed with a target UTC infrastructure in mind, for example, SCOOT [31], widely adopted in the UK. They restrict control to a limited set of pre-validated signal configurations, typically uploaded at least one day in advance, and require all cycles to have identical durations to preserve network synchronization and green wave coordination. Moreover, they limit how often configurations may change over time. Consequently, signal plans derived from such models are more likely to be executable as-is on existing UTC systems, requiring little or no additional validation efort. Prioritizing deployability over flexibility, however, comes at the cost of reduced control and potential performance losses due to suboptimal use of green time. Deployable models, by significantly restricting the space of valid signal plans, are typically more manageable to solve and allow the use of domain-independent approaches.
All the models we studied share a common representation of the environment in terms of PDDL+ processes and events, which capture how the urban network evolves over time under the applied control. In particular, each link of the network is associated with a numeric variable, the occupancy, representing the number of vehicles present. Processes model vehicle flows across network links. Specifically, for each permitted movement defined by a stage of a cycle, there is a process that linearly decreases the occupancy of the corresponding incoming link and increases the occupancy of the associated outgoing link whenever the green phase for that stage is active. Events model the triggering of stages within the cycles, capturing the discrete transitions between diferent trafic light phases. Finally, actions influence signal configurations, indirectly afecting the occurrence of events and the evolution of processes.
In the following, while keeping the environmental representation fixed, we describe how the diferent models implement trafic signal control.
In the first proposed model [22, 37], trafic control relies on the switchPhase action, which represents the planner’s decision to change the state of an intersection. Each intersection is associated with a token that identifies which stage is currently running; executinsgwitchPhase increases the token value and transfers the green phase to the next stage. The action is subject to temporal constraints: it can only be applied once the minimum green time has elapsed; reaching the maximum green time triggers an exogenous event that enforces the phase change. This model was solved using the PDDL+ planner UPMurphi [38], extended with a domain-specific heuristic to guide the exploration.
Whereas the first model allowed control of minimum and maximum stage durations, it did not enforce any constraint on the overall cycle length. To address this, the subsequEexnttend and Reduce model (ExRe) we propose [39] introduced two actionse,xtendStage andreduceStage, which allow increasing or decreasing the duration of the current stage with a fixed granularity. By acting on global variables associated with the cycle under consideration, these actions ensure that the resulting configurations remain within specified minimum and maximum bounds for the overall cycle length.
To solve this model, we developed a domain-specific heuristic implemented on top of ENHSP8[] and combined it with a Greedy Best First Search (GBFS). In simulation, the resulting signal plans showed competitive performance on real-world scenarios, specifically on a major trafic corridor in the city of Huddersfield, when compared with the reactive plans generated by the existing infrastructure.
The signal plans generated by thEexRe model, although competitive, are characterized by variable cycle lengths and frequent configuration changes, which make them undeployable. This limitation motivated the design of the following deployable models.
In [23], we addressed the gap between theoretical trafic signal optimisation and real-world deployment by providing three new PDDL+ models that enable to generate deployable signal plans compatible with the UTC infrastructure. The three models, denotedCayscle by Cycle (CbC), Fixed Repetition (FiRe), and Variable Repetition (VaRe), difer in their flexibility regarding when and how often predefined cycle configurations can be changed.
Specifically, CbC model allows the configuration of a junction to be selected at every cycle transition (during the intergreen phase of the cycle’s final stage), therefore significantly increasing the number of decision points compared to other models. This behaviour is achieved bycthhaengeConfiguration action, which switches a junction from one predefined cycle configuration to another when triggered at the end of a cycle.
FiRe model requires a selected configuration to be retained for a minimum number of cycles before allowing a change, reducing the number of decision points compared to CthbeC model. A cycle counter tracks the number of completed cycles for the current configuration, and tchheangeConfiguration action is used to switch configurations only after the minimum cycle limit is reached.
VaRe model allows decisions on how many times a configuration repeats at each junction. The minimum number of repetitions can vary within a specified range and is set using thcehangeLimit action, which executes after changeConfiguration to assign the repetition count for that junction. Once this repetition limit is established, the handling of stage durations and cycle counts followsFitRhe model unchanged.
As a preliminary step, we compared the three models under diferent search strategies and heuristic configurations to identify the most suitable one. This analysis revealed thFaitRe was the most promising candidate, particularly when solved using GBFS combined with the heurisℎtmicax. We then evaluated this configuration of FiRe on the same benchmark used to assess theExRe [39]. Under this setting, FiRe produced signal plans that are better than those historically generated by SCOOT and proved competitive with the plans obtained from theExRe model equipped with a domain-specific heuristic.
Although deployable models, and in particulaFriRe, ofer clear benefits, such as generating signal plans that are almost ready for execution, they are strongly limited in expressiveness, since they rely on a ifxed set of precompiled configurations. For this reason, in a recent work40[], we investigated a model that lies in between in terms of flexibility and deployability, namelyTrade. Our goal was to increase the expressiveness of deployable models while preserving a certain degree of deployability. Intuitively, Trade works by keeping the cycle length fixed while allowing the redistribution of green time among its stages. In practice, this is achieved through thtreadeTime action, which, during intergreen periods, transfers a small amount of time from one future stage to another. The model is parameterised by two factors: the granularity of time that can be traded between stages and the maximum number of trades allowed per cycle. From an experimental perspective, we observed that, when solved with the same heuristic originally designed foErxRe, the Trade model achieves performance comparable tEoxRe while maintaining a higher degree of deployability.
So far, we have characterised the presented models in qualitative terms. However, the distinction between deployable and flexible models can also be formally understood by relating the sets of valid signal plans that they generate for the same trafic scenario. Consider a trafic scenario encoded with the various models, and leptlans() denote the set of valid signal plans under mod el. Roughly, the spaces of valid plans can be expressed aplsans(FiRe) ⊆ plans(VaRe) ⊆ plans(CbC) ⊆ plans(Trade) ⊆ plans(ExRe).
In this paper, we have overviewed the recent problems in the context of urban trafic distribution and control modelled and solved with ASP and PDDL+, separately. However, ASP and PDDL+ can be also combined to exploit the benefits provided by both approaches to solve the Trafic Signal Optimization problem. For instance, in3[0], we showed how theclingcon encoding can be used to improve the quality of solutions returned by the PDDL+ approach. We implemented an automated pipeline to exploit the synergies of the two approaches as follows: from the solution found by the PDDL+ planner, the values of counter at a given horizon of each target link can be extracted. This information can be encoded as ASP atoms that appear in constraints forcinclgingcon to return a solution that is strictly better than the PDDL+ one. In this way, thanks to PDDL+, we have the guarantee to have a solution quickly, while the subsequent application of ASP allows for trying to improve it (which was observed for almost half of the instances in the benchmark).
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