Optimal Symbolic Controllers Determinization for BDD storage

Abstract

Controller synthesis techniques based on symbolic abstractions appeal by producing correct-by-design controllers, under intricate behavioural constraints. Yet, being relations between abstract states and inputs, such controllers are immense in size, which makes them futile for embedded platforms. Control-synthesis tools such as PESSOA, SCOTS, and CoSyMA tackle the problem by storing controllers as binary decision diagrams (BDDs). However, due to redundantly keeping multiple inputs per-state, the resulting controllers are still too large. In this work, we first show that choosing an optimal controller determinization is an NPcomplete problem. Further, we consider the previously known controller determinization technique and discuss its weaknesses. We suggest several new approaches to the problem, based on greedy algorithms, symbolic regression, and (muli-terminal) BDDs. Finally, we empirically compare the techniques and show that some of the new algorithms can produce up to ≈ 85% smaller controllers than those obtained with the previous technique. Preliminaries Minimum set coverThe minimum set cover problem (MSC) is formulated as:Problem 2.1 (MSC). Given a set X and a cover {S j } j∈I , i.e. X ⊆ j∈I S j , where |X|, |I| < ∞, find the smallest subcover I * ⊆ I : X ⊆ j∈I * S j .Both, the decision and selection versions of (MSC, are known to be NPcomplete. The first approximate poly-nomial-time solution for MSC was given by [12]. Later, [5] suggested an approximate poly-nomial-time solution for the generalized "minimum set weight cover problem" (MWSC); which extends MSC by that each set S k is assigned a weight s k ≥ 0 and the question is to find the smallest sub-cover with the minimum total weight. According to [6], the Chvátal's algorithm time complexity is: O (|I| · |X| · min (|I|, |X|)).1 Instead of storing the control law as an explicit map, we search for a symbolic function that for a given state computes the input value.2 Up to the found optimal BDD variable reordering.

SOS: Safe, Optimal and Small Strategies for Hybrid Markov Decision Processes

“…As illustrated in Remark 1, this could lead to further reduction in size and improved understandability. Additionally, isomorphic/similar subtrees could be merged as in decision diagrams and further optimizations for algebraic decision diagrams [57] could be employed. Finally, we plan to visualize the DT representation of the strategies directly in Uppaal Stratego + for convenience of the users.…”

“…Moreover, these decision trees can only predict a single action for a state configuration whereas in this work, we allow the trees to predict more than one action for a single configuration. In control theory, [57] proves that the problem of computing size-optimal determinisiation of controllers is NP-complete and hence discuss various heuristic-based determinisation algorithms. None of these works consider the optimization aspect, which being a soft constraint enables the trade-offs.…”

Formal synthesis of closed-form sampled-data controllers for nonlinear continuous-time systems under STL specifications

“…For each region within this partition, a finite set of admissible inputs are stored, resulting in a nondeterministic controller. This controller is stored as a binary decision diagram (BDD) with the size of 2.87 MB and can be further reduced to 0,15 MB by removing nondeterminism (see [9]). Additionally, to parse the BDD format, special libraries are required.…”

“…As it originally dealt with finite systems, (bi-)simulation approaches have been proposed to abstract infinite systems to finite systems [3,4]. However, as a downside, these approaches (e.g., [5][6][7][8]) typically suffer from the curse of dimensionality and return controllers in the form of enormous lookup tables [9].…”

Formal Controller Synthesis via Genetic Programming

“…Most of these methods fit into one of three main paradigms: synthesis by means of 1) finite (bi-)simulation abstractions [2,3], 2) control Lyapunov and/or barrier functions [4,5] and 3) online optimization-based methods [6]. The first paradigm relies on discretization of the state space and therefore suffers from the curse of dimensionality, resulting in controllers taking the form of enormous look-up tables, which complicates their implementation [7]. Control approaches using this paradigm include [8][9][10][11][12], whereas tools implementing this paradigm include PESSOA [13], SCOTS [14], CoSyMa [15] and ROCS [16].…”