# Automata-based Optimal Planning with Relaxed Specifications

In this paper, we introduce an automata-based framework for planning with relaxed specifications. User relaxation preferences are represented as weighted finite state edit systems that capture permissible operations on the specification, substitution and deletion of tasks, with complex constraints on ordering and grouping. We propose a three-way product automaton construction method that allows us to compute minimal relaxation policies for the robots using standard shortest path algorithms. The three-way automaton captures the robot's motion, specification satisfaction, and available relaxations at the same time. Additionally, we consider a bi-objective problem that balances temporal relaxation of deadlines within specifications with changing and deleting tasks. Finally, we present the runtime performance and a case study that highlights different modalities of our framework.

## Authors

• 1 publication
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03/04/2021

### MT* : Multi-Robot Path Planning for Temporal Logic Specifications

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### Robustness-by-Construction Synthesis: Adapting to the Environment at Runtime

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### An SDP dual relaxation for the Robust Shortest Path Problem with ellipsoidal uncertainty: Pierra's decomposition method and a new primal Frank-Wolfe-type heuristics for duality

This work addresses the Robust counterpart of the Shortest Path Problem ...
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### Continuous Motion Planning with Temporal Logic Specifications using Deep Neural Networks

In this paper, we propose a model-free reinforcement learning method to ...
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### Reactive Control Meets Runtime Verification: A Case Study of Navigation

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### On the Expressive Power of the Normal Form for Branching-Time Temporal Logics

With the emerging applications that involve complex distributed systems ...
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## I Introduction

Robots are increasingly required to perform complex tasks with rich temporal and logical structure. In recent years, automata-based approaches have been widely used for solving robotic path planning problems wherein an automaton is constructed from mission specifications posed as temporal logic formulae (e.g. LTL, scLTL, STL, TWTL) [23, 6, 22, 1]. Using shortest path algorithms on the product models between abstract robot motion models and specification automata, optimal satisfying trajectories are synthesized.

This traditional approach, albeit useful, does not consider modifying mission specifications in case satisfaction is infeasible. This implies that even the sub-parts of the specification that might be feasible will not be executed. In many real-world scenarios, it is often preferable that the robot performs at least some part of the assigned task even if it cannot be satisfied in its entirety. Consider the following mission specification for data collection: “Collect data from region and then region and then upload it at . Collect data from and upload it at . Always avoid obstacles.” In case an obstacle makes it impossible to reach , it is still preferred to receive the data from and . Thus, we need to consider relaxed satisfaction semantics to handle infeasible mission specifications.

In literature, the problem of specification relaxation has been formulated in various ways. Minimum violation is considered in [21, 25, 20] for self-driving cars, where policies are computed with minimal rules of the road violation based on priorities. Their approach is based on the removal of violating symbols from the input of the specification automata to produce satisfying runs. A related problem is considered in [8, 9], where the focus is on removing fewest geometric constraints for object manipulation. In [15], minimum revision of tasks for office robots is explored. Their approach allows changing of tasks based on user-provided substitution costs. A similar problem is tackled in [12], but for infinite horizon case with Büchi automata. Both works modify the input stream of the specification automaton to induce feasibility. Partial satisfaction [13, 14] approaches aim to compute policies that minimize distance to satisfaction given by paths to accepting states in specification automata. In a different direction, [25, 1, 17] consider temporal relaxation

of deadlines to complete missions. Their approach introduces annotated automata that capture all deadline relaxations from specifications, to compute policies with minimal delays. Some of these works combine relaxation of specifications with maximizing satisfaction probability

[13, 15, 7]. All these works use automata-based techniques. However, all have specialized approaches that can not be readily combined. Moreover, they operate on a symbol-by-symbol basis rather than words translations that capture rich relaxation preferences on groups of tasks.

In this paper, we introduce an automata-based framework that captures the notion of relaxation from several existing approaches and generalizes them to operate on groups of tasks (words). We decompose the problem of robot motion planning into a high-level planning and a low-level control problem. As in [18, 20, 25, 24, 15], our focus is on symbolic path planning. The robot motion is abstracted as a weighted transition system (TS) with the regions of interest as states. Mission specification are given as deterministic finite state automata obtained from finite horizon temporal logic formulae such as syntactially co-safe LTL (scLTL) and TWTL formulae. Users also specify relaxation preferences in the form of regular expressions (RE) which we translate to weighted finite state edit systems (WFSE) [11] that capture differences between pairs of input words. The WFSEs determine the sets of permissible edit operations (substitution or skipping), on single or groups of tasks, along with their costs when the mission specification is infeasible. The user-specified relaxation rules enable the framework to be used in complex situations without much computational expense on semantically understanding the environment and deriving the rules. We introduce a three-way product automaton construction method that captures the motion of the robot, the specification satisfaction, and possible relaxations at the same time. We compute minimal relaxation robot trajectories using shortest path algorithms on the proposed product model. Additionally, we leverage the framework in [24] for temporal relaxation, and consider bi-objective optimal synthesis problem that balances relaxation of deadlines with task relaxations.

This work proposes a framework that brings together the core notions of several automata-based methods for planning with relaxation and allows for handling complex specifications and relaxation preferences. The main contributions of the paper are: (1) the formulation of the minimum relaxation problem that unifies several problems from the literature, and generalizes them to relaxation rules with memory; (2) an automata-based formalism to capture user relaxation preferences via WFSEs; (3) an automata-based planning framework that uses a novel three-way product automaton construction between the motion, specification, and relaxation preference models; and (4) case studies that demonstrate different instances of specification relaxation and the runtime performance. To the best of our knowledge, this is the first time relaxation rules are considered that account for complex ordering and grouping of sub-tasks when revising mission specifications.

## Ii Preliminaries

In this section, we introduce notation used throughout the paper, and briefly review the main concepts from formal languages, automata theory, and formal verification. For a detailed exposition of these topics, we refer the reader to [2, 10] and the references therein.

We denote the range of integer numbers as , and .

Let be a finite set. We denote the cardinality and the power set of by and , respectively. A word over is a finite or infinite sequence of elements from . In this context, is also called an alphabet. The length of a word is denoted by . Let , be non-negative integers. The -th element of is denoted by , and the sub-word is denoted by . Let . The sub-sequence is denoted by . A set of words over an alphabet is called a language over . The language of all finite words over is denoted by .

###### Definition II.1 (Deterministic Finite State Automaton)

A deterministic finite state automaton (DFA) is a tuple , where: is a finite set of states; is the initial state; is the input alphabet; is the transition function; is the set of accepting states.

A trajectory of the DFA is generated by a finite sequence of symbols if is the initial state of and for all . A finite input word over is said to be accepted by a finite state automaton if the trajectory of generated by ends in a state belonging to the set of accepting states, i.e., . The (accepted) language of a DFA is the set of accepted input words denoted by .

###### Definition II.2 (Transition System)

A weighted transition system (TS) is a tuple , where: is a finite set of states; is the initial state; is a set of transitions; is a set of properties (atomic propositions); is a labeling function; is a weight function.

A trajectory (or run) of the system is an infinite sequence of states such that for all , and . The set of all trajectories of is . A state trajectory generates an output trajectory , where for all . We also denote an output trajectory by . The (generated) language corresponding to a TS is the set of all generated output words, which we denote by . We define the weight of a trajectory as .

## Iii Background on Planning With Relaxed Specifications

In this section, we review temporal logic-based planning problems that consider specification relaxation in case of infeasibility. In the subsequent sections, we unify and generalize all these problems, and propose an automata-based framework amenable to off-the-shelf synthesis methods instead of customized solutions. For cohesiviness and clarity, we present the core features of the relaxed TL planning problems, in some cases, adapted to finite-time.

Throughout the paper, we assume that the motion of a robot is captured by a finite weighted transition system .

We consider finite-time specifications expressed using temporal logics (TL), e.g. scLTL [16, 5], TWTL [24], BLTL [19], and Finite LTL [4], and regular expressions (RE) [3, 10]. We do not provide details on their syntax and semantics, and instead point the reader to relevant references. All of these representations can be translated to DFAs using off-the-shelf tools. Thus, we consider specifications given as a DFA .

### Iii-a Canonical Problem (CP)

###### Problem III.1 (Canonical)

Find a trajectory for such that the output trajectory is accepted by .
Optimality: Minimize the weight of the trajectory.

In the canonical problem, no relaxations are permitted.

### Iii-B Minimum Violation Problem (MVP)

Let be a word over , and per symbol violation cost. The violation cost of with respect to is s.t. . The violation cost of a TS trajectory is induced by the output word .

###### Problem III.2 (Minimum violation)

Find a trajectory for such that a sub-sequence of the output trajectory is accepted by .
Optimality: Minimize the violation cost of the trajectory.

### Iii-C Minimum Revision Problem (MRP)

Let be a word over , and the symbol substitution cost. The revision cost of with respect to is s.t. , where is the revised word.

The symbol substitution cost function is defined such that there is no penalty for no substitution, i.e., for all . In most cases, is a non-negative symmetric function, for all .

###### Problem III.3 (Minimum revision)

Let be the symbol substitution cost. Find a trajectory for such that a revision of the output trajectory is accepted by .
Optimality: Minimize the revision cost of the trajectory.

### Iii-D Hard-Soft Constraints Problem (HSC)

###### Problem III.4

Let and be two specification DFAs. Find a trajectory for such that the output trajectory is accepted by , and, if possible, by .
Optimality: Minimize the cost of the trajectory.

We adapt the HSC problem from [6] for finite-time specifications, where we replace Büchi automata with DFAs.

### Iii-E Partial Satisfaction (PS)

Let . The continuation cost of with respect to is s.t. , where is a continuation of .

###### Problem III.5

Find a trajectory for such that a continuation of the output trajectory is accepted by .
Optimality: Minimize the cost of the continuation.

The problem minimizes the amount of work still needed to satisfy the specification from partial trajectories.

### Iii-F Temporal Relaxation (TR)

In this section, we review Time window temporal logic (TWTL) [23], a rich specification language for robotics applications with explicit time bounds. As opposed to previous relaxation semantics, temporal relaxation for TWTL is defined based on the formulae structure (i.e., relaxation of deadlines) rather than symbol operations on satisfying words. For brevity, we omit most details and refer to [24].

The syntax of TWTL formulae over a set of atomic propositions is:

 ϕ::=Hds∣Hd¬s∣ϕ1∧ϕ2∣ϕ1∨ϕ2∣¬ϕ1∣ϕ1⋅ϕ2∣[ϕ1][a,b],

where is either the “true” constant or an atomic proposition in ; , , and are the conjunction, disjunction, and negation Boolean operators, respectively; is the concatenation operator; with is the hold operator; and is the within operator, and . See [24] for the full description of semantics and examples.

Let be a TWTL formula and , where is the number of within operators contained in . The -relaxation of is a TWTL formula , where each subformula of the form is replaced by .

Bottleneck and linear truncated temporal relaxation were introduced in [24, 17], respectively. For brevity, we consider the linear temporal relaxation (LTR). The LTR of is , where is a -relaxation of .

###### Problem III.6

Find a trajectory for such that the output trajectory satisfies the relaxed formula for some relaxation of the deadlines in .
Optimality: Minimize the linear temporal relaxation.

### Iii-G Planning

All the aforementioned problems are solved by constructing a standard product automaton between the motion model and the specification DFA . Planning with relaxed semantics is achieved via custom pre-processing procedures of , and custom shortest path algorithms. In the following, we show that all these problems can be captured via an additional automata-based model for user task relaxation, and solved using standard shortest path methods applied on a novel 3-way product. Moreover, MVP and MRP are restricted to relaxations of a single symbol at a time. Our framework can handle rich relaxation rules that involve changing groups of symbols (words).

## Iv Problem Formulation

In this section, we introduce an optimal planning problem for finite system abstractions with temporal logic goals. The specifications are expressed as DFAs which can be obtained from multiple temporal logics, e.g., scLTL [16, 5], BLTL [19], Finite LTL [4], TWTL [24], and regular expressions [3, 10] using off-the-shelf tools, e.g., spot [5], scheck [16], pytwtl [24]. We define a cost function based on user preferences on task removal and substitution in case satisfying the given specification is infeasible. Using the user task preference we define an optimal planning problem over the finite motion model, where the specification language is enlarged to ensure feasibility with appropriate penalties.

###### Definition IV.1 (User Task Preference)

Let be a language over the alphabet . A user task preference is a pair , where is a relation that captures how words in can be transformed to words from , and represents the cost of the word transformations.

The relation can also be understood as a multi-valued function .

###### Assumption 1

The representation of relation requires bounded memory.

Asm. 1 is a reasonable requirement in practice, and allows similar expressivity as finite-time TLs and DFAs. With general relations , we run into decidability issues [10].

Robot motion is captured by TSs whose weights represent either duration, distance, energy, or control effort. For simplicity, in this paper, weights are transition durations.

###### Problem IV.1 (Minimum Relaxation)

Given a transition system , a specification DFA , and a task relaxation preference , find a trajectory that minimizes the task cost. Formally, we have

 minx∈Runs(T) wR(o,o′) (1) s.t. o∈L(A) o′=h(x)∈L(T)∩R(o)

where and .

Task preferences can be used to substitute and delete tasks which are associated with words. These generalize edit-space operations on single symbols to words, and the optimization problem Problem IV.1 generalizes the Levenstein distance between languages of finite words.

User task preferences can be represented in many ways. We consider the user preferences for relaxation provided as regular expressions (RE) and regular grammars that can be readily translated to automata using standard methods [10]. Consider the following example.

###### Example IV.1

Suppose the task is to visit region P1 for 1 time unit followed by P2 for 2 time units. Should visiting either or both be not possible, the substitution rules are: Substitute the visit to P1 by visiting Q1 for 2 time units with a penalty of d1, and the visit to P2 by visiting S1 for 2 time units followed by S2 for 1 time unit with a penalty of d2. Formally, , where represents any symbol in , denotes a empty symbol, denotes that is substituted by , and d1, d2 denote the penalties for the corresponding substitutions. Note that the transformation can be performed multiple times due to the outer Kleene star operator. Alternatively, the transformation rules are and . and the possible alternatives are: a) , b) , and c) . From Fig. 3, it is evident that the existing approaches that allow symbol-symbol translations (Fig. (b)b) cannot capture these relaxations as opposed to the WFSE for word-word translations (Fig. (a)a).

Penalties in user task preferences have multiple interpretations; they can be additive, multiplicative, or percentage rate with respect to the TS weights depending on the nature of the tasks and preferred relaxations. We use weight computation functions that combine TS and WFSE weights to capture these multiple semantics.

Next, we show that Prob. IV.1 captures all problems from Sec. III.

###### Proposition IV.1

Problems III.1III.2III.3III.4III.5, and III.6 are instances of Prob. IV.1

We provide a constructive proof for each case. In all cases, we consider additive penalties.

##### Cp

, and , where is the trajectory generating , i.e., . Equivalently, .

##### Mvp

and . Equivalently, .

##### Mrp

and . Equivalently, .

##### Hsc

, , is the trajectory of generating , and .

##### Ps

, and . Equivalently, where, and .

##### Tr

, and , where is the temporal relaxation of associated with , and , are the DFAs for and , see [24] for details.

## V Unified Automata-Based Framework

In this section, we introduce a unified automata-based framework to capture user preference specifications, and to synthesize minimal relaxation policies.

### V-a Relaxation Specification

We consider two classes of problems related to task changes and deadline relaxations.

In this problem class, we allow parts of the specification to be substituted and/or removed. Preferences can be given in many formats, e.g., regular expressions and grammars, see Ex. IV.1 We introduce weighted finite state edit systems to represent user task relaxation preferences with bounded memory (Asm. 1), where weights capture translation penalties.

###### Definition V.1 (Weighted Finite State Edit System)

A weighted finite state edit system (WFSE) is a weighted DFA , where , denotes a missing or deleted symbol, and is the transition weight function.

The alphabet captures word edit operations (addition, substitution, or deletion of symbols). A transition has input, output symbols and . Given a word , , we call and obtained by removing only the symbol , the input and output words, where , , and . Moreover, we say that transforms into .

Note that WFSE is a special type of finite state transducer where the input and output alphabets are the same, and the empty symbol cannot be mapped to itself. Moreover, the weights capture translation penalties and can depend on the states and symbol translation pairs.

We can use standard methods [10] to translate REs, expressing relaxation rules, into WFSEs.

#### V-A2 Temporal Relaxation

Temporal relaxation allows delays with respect to deadlines in the satisfaction of specifications. In the following, we consider annotated automata computed from TWTL formulae [24] that capture all possible deadline relaxations. Formally, given TWTL formula , an annotated DFA is a DFA such that , where is satisfied by a word if and only if s.t. . When the transition weights of the TS represent (integer) durations, we construct an extended TS from such that all transitions have unit weight (duration). This additional step ensures that transitions of and are synchronized. See [24] for details.

### V-B Product Automaton Construction

The optimal control policy that takes into account the user preferences is computed based on a product automaton between three models: (a) the motion model (TS) of the robot ; (b) the user preferences WFSE ; and (c) the specification DFA .

###### Definition V.2 (Three-way Product Automaton)

Given a TS , a WFSE system , and a specification DFA , the product automaton is a tuple also denoted by , where:

• is the state space;

• is the initial state;

• is the set of transitions;

• is the set of accepting states;

• is the transition weight function.

A transition if or , , , and . Note that we introduce a virtual TS state connected to to avoid the definition of a set of initial states and associated start weights. State is only used to simplify notation and implementation, and does not correspond to an actual state of the robot. The weight function is , where is an arbitrary function, , , and by convention. A trajectory of is said to be accepting only if it ends in a state that belongs to the set of final states . The projection of the trajectory onto the TS is , where is the initial state of , and , for all . Similar to [23, 24], we construct such that only states that are reachable from the initial state, and reach a final state.

### V-C Optimal Planning

The general planning procedure is outlined in Alg. 1. Given a TS , TL specification , and user task preference , Alg. 1 first translates to DFA and to WFSE (lines 1-2). Next, it computes the three-way product automaton (line 3). Similar to the standard procedure, the trajectory is obtained by projecting onto the shortest path from the initial state of the PA to an accepting state in (lines 2-3). The weights of used for computing the shortest path depend on whether we wish to minimize task or deadline relaxation as shown next.

Let be a trajectory of . The task cost of is

 CE(q) =n−1∑k=0cE(qk,qk+1), (2) cE(qk,qk+1) =fw(wT(xk,xk+1),wE(zk,zk+1)), (3)

where is the transition weight, for all . The task cost takes into account the penalties associated with substitution and deletion of tasks represented as sub-words of the TS’s output words. The optimal trajectory is computed as using Alg. 1.

#### V-C2 Temporal Relaxation Cost

In this case, the cost is captured by LTR introduced in Sec. III-F that aggregates all delays captured by the annotated specification DFA . The PA is denoted by and the optimal trajectory is computed as , where is the extended TS, and is a trivial WFSE with a single node and a pass-through self-loop (leaves symbols unchanged and has weight 0). The temporal relaxation cost of is

 CTR(q) =|q|, (4) cTR(q,q′) =1,∀(q,q′)∈δE0P, (5)

where is a trajectory of . Minimizing the length of trajectory is equivalent to minimizing . This follows from the results in [24].

#### V-C3 Bi-objective Cost

We consider cases where a robot can trade-off between changing tasks and delaying their satisfaction. The solution combines an annotated specification automaton with a (non-trivial) relaxation preference WFSE to compute policies in using . The blended cost of a trajectory is

 Cbi(q∣λ) (6) cbi(q,q′∣λ) =λcE(q,q′)+(1−λ)cTR(q,q′), (7)

where the is a parameter that trades-off between the two objectives, and is the bi-objective transition cost.

We compute the Pareto-optimal trajectories and the Pareto-front using a parametric Dijkstra’s algorithm [26]. The Pareto-front for our bi-objective problem is composed of a finite number of isolated points in the cost space. This follows from the finite size of , and the fact that any Pareto-optimal policy must be a simple path in . This implies that as a function of is a continuous piecewise-affine function, where each piece corresponds to a point on the Pareto-front and an interval of values.

Note that in this setup both types of relaxations are allowed to happen at the same time.

###### Remark V.1

The proposed framework reduces planning to standard shortest path problems on rather than various custom methods. Thus, automata-based methods for the canonical case can be immediately used to solve the minimum relaxation problem Pb. IV.1.

###### Remark V.2

As the original MRP problem operates on symbol-symbol basis, the length of the original word and relaxed word needs to be the same (See proof of Prop. 4.1(c)). This is not a requirement for our framework as it operates on groups of symbols (words). Thus, in the following sections, we call the substitution problem as Minimum Word Revision Problem (MWRP).

### V-D Complexity Analysis

The construction of the three-way PA , line 3 in Alg. 1, takes . Computing the shortest path (line 4) is done with Dijkstra’s algorithm which takes . Lastly, projection onto (line 5) is linear in the size of .

Crucially, our framework has the same asymptotic complexity as custom planning methods for the problems in Sec. III. MVP, MRP, and PS operate one symbol at a time, see the proof of Prop. IV.1. Their associated WFSEs have a single state with a self-loop, i.e., (see Fig.(b)b). Thus, the PA construction complexity degenerates to . For TR, the complexity also reduces since the WFSE has a single state; the deadline relaxation is captured by  [24]. Lastly, for HSC, we can choose as specification DFA, and the WFSE can have same structure as with penalty if the soft constraint is not satisfied. For brevity, we omit the formal details. Thus, the complexity of PA construction becomes , the same as for custom methods (due to the quadratic complexity of language intersection [10]).

## Vi Case Studies

In this section, we present a case study highlighting different instances of specification relaxations. We consider a self-driving car in an urban environment as shown in Fig. (a)a tasked with visiting specified task regions (green rectangles) while avoiding obstacles. The possible routes that a vehicle can follow are shown using blue lines, the permissible directions indicated using arrows, whereas the waypoints are shown using black circles. The obstacle shown as a red cross as well as the local obstacles and shown using orange crosses (Fig.(a)a) may or may not be present. The ‘No entry’ symbol and the obstacle (if present) together make it impossible to reach and in turn, make the red dotted path infeasible. Similarly, even if the local obstacle is not present, the smaller region next to cannot be stayed at due to the no parking zone.

The motion of the robot is modeled as a weighted TS (Fig.(b)b) with states and transitions representing the waypoints (white states) and task locations (green states), and roads between them, respectively. The weights associated with transitions represent their duration. The transitions to the green task location states have weight one, and may, e.g., capture parking. Note that for all states in Fig. (b)b, self-transitions exist, but have not been included in the figure for simplicity. Self-transitions allow the robot to be stationary at all location, except for the purple states 14 and 15. The purple states correspond to local obstacles and whereas the obstacle is shown using a red state. The initial state of the robot is state . The transitions shown using yellow arrows and obstacle are present only for problems 6-8, node 15 only for problem 8.

In the following, we present multiple scenarios in the self-driving setting that showcase the CP, MVP, MWRP, HSC, TR, and bi-objective problems. The specifications, user preferences and the costs for each problem are provided in Tab. I.

All specifications are translated to DFAs using off-the-shelf tools [5, 24]. For MWRP, HSC-MWRP, bi-objective problems, the relaxation preferences allow the substitution of with , , , and with costs 5, 8, 11, and 14, respectively. For MVP and HSC-MVP, the cost of not visiting , deletion cost, is 10. For HSC, the cost of not taking the , violation of the soft constraint, is 10. The preferences and costs are captured by a WFSE with .

We first consider a canonical scenario. Subsequently, we present extensions within the same environment wherein satisfying the given specification becomes infeasible without relaxation. Feasibility depends on the presence of obstacles indicated in Tab. I for each scenario.

For MVP, MWRP, and HSC problems, we consider scenarios with present (e.g., road construction, temporary closure), thereby making visits to infeasible.

#### Vi-1 Canonical Problem (CP)

The task specification is “Visit T1”, i.e., , where is the eventually operator. As obstacle is absent and thus, no relaxations are required, this case corresponds to a pass-through operation (no substitutions or deletions) in . The optimal trajectory is which corresponds to the shortest path to in Fig. (b)b with the total cost .

#### Vi-2 Minimum Violation Problem (MVP)

We consider the specification “Visit and while avoiding obstacles” that translates to the scLTL formula . The optimal trajectory of is as is not reachable in the presence of . The optimal cost is 15, including the cost of 10 for skipping .

#### Vi-3 Minimum Word Revision Problem (MWRP)

In this case, the task specification is “Visit while avoiding obstacles”. If the task is not feasible, revise the task according to the preferences given above. Here . The optimal trajectory accepted by is with an optimal cost = 11, where is substituted by with cost 5.

#### Vi-4 Hard-Soft Constraints (HSC)

This problem is implemented both in the presence and absence of obstacle . The task specification for both scenarios is “Visit while avoiding obstacles, and, if possible, take the ”. The specification is , where and are the hard and soft constraints, respectively. The cost of not satisfying is 10 and is added to the WFSE.
HSC-CP: Without obstacle , the case is analogous to CP and, thus, the optimal trajectory is with optimal cost .
HSC-MWRP: In this case, is substituted by which has the lowest substitution cost, again due to . Thus, the optimal trajectory is with optimal cost that includes the substitution cost and the violation cost for not going over the .
HSC-MVP: , . With obstacle , only can be visited. In the MVP case, the optimal trajectory is with an optimal cost of that includes the costs of 10 for not visiting and of 10 for not taking the .

The penalty for not taking the bridge is added to the WFSE when taking transitions and , since these indicate when the robot has diverted from satisfying .

#### Vi-5 Temporal Relaxation (TR)

In this example, the specification “Visit and stay in for 2 time units within 6 time units.” translates to a TWTL formula , where is the hold operator. Note that, the minimum travel time to from state is 7 time units, see Fig (b)b. Thus, the specification is relaxed to with obtained by the optimal trajectory . The optimal cost is corresponding to .

#### Vi-6 Multiple word-word translations

Now consider that the local obstacle is present. If the task specification is: “Visit T4 for 2 consecutive instances and next, visit regions T4 and then T2 and next, visit regions T4 and T1. Avoid obstacles all the time.” The corresponding scLTL specification is: . If the task is infeasible, the substitution rules are: Substitute the first two instances of T4 (i.e. X T4) by T5 with a total penalty of 6. Substitute the next occurrence of T4 by one T5 and two T3s with a total penalty of 4. Finally, delete the last occurrence of T4 with a penalty of 10 and substitute T1 by T2 with a penalty of 7. Fig. 7 shows a wfse constructed from these relaxation rules. Note that all self loops correspond to the translation ({},{},1). z0-z6 denote the states and the edges represent the allowed edit operations. The trajectory obtained after relaxation is: (0,6,5,5,5,6,8,12,3,3,12,13,13,2,2) with a total cost .

The above example demonstrates that our framework allows for multiple rules to be taken into account for different instances of the same symbol/word. Also, it highlights how the ordering is considered and retained during relaxation.

#### Vi-7 Bi-objective Problem

In the absence of the : “Visit for 3 time units between 0 to 5 time units”. If not feasible, use the substitution preferences. The DFA is obtained from the TWTL formula . We obtain a set of Pareto-optimal trajectories and the corresponding intervals for parameter values. The intervals indicate the range of possible trade-offs between the two objectives and that correspond to the same Pareto-optimal trajectory. The set of solutions are: (1) , , corresponds to the minimum temporal relaxation with and ; (2) , , strikes a balance between task and temporal relaxations with , , and ; (3) , , achieves minimum task cost , and . Having established the core idea, we now consider a word-word translation preference rule. Consider the specification “Visit T5 for 1s within first 3s from the start and immediately next, proceed to visit T4 for 2s within first 7s followed by T1 for 1s within first 5s of the mission. The local obstacle O2 should be avoided for the first 4s whereas the global obstacle O should be avoided for all 20s duration of the task.” The corresponding TWTL formula is: “”. The substitution rules are as follows: , , , . The trajectories obtained after relaxation are: 1) ,  2) ,  3) .

#### Vi-8 Difference between symbol-symbol and word-word translations

Given that obstacle is present and is absent, the task is to visit T1 for 2 time units. As the route through node 15 is a no parking zone, there are no self-transitions on T1 at node 15. Given same substitution preferences as for MRP and if modelled as a wfse with a single state (see e.g., Fig. (b)b) which is analogous to relaxations performed by the existing solutions, the shortest path obtained is (0,6,8,12,13,15,2) which violates the specification as it can pass through T1 (node 15) but not stay there. However, our framework with a wfse model similar to Fig.(a)a allows for this situation to be taken into account and the resultant trajectory is (0,6,8,12,13,2,2).