# Completely and partially executable sequences of actions in deontic context

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## Abstract

The paper offers a logical characterisation of multi-step actions in the context of deontic notions of obligation, permission and prohibition. Deontic notions for sequentially composed actions (procedures or instructions) are founded on deontic notions for one-step actions. The present work includes a formal study of situations where execution of a multi-step action has been unsuccessful and provides normative analysis of such actions.

### Keywords

Deontic action logic Sequential composition of actions Successful and unsuccessful actions Logic of procedures## 1 Introduction

When complex planned actions are executed, it sometimes happens that they can be started but cannot be continued until their successful end. In other words, a planned action, a procedure or an instruction, may be only partially executable if its realisation meets unexpected obstacles. The subject is especially important since nowadays procedures govern more and more activities in business, administration and other fields of social life. People are often assessed on the basis of the way how they follow procedures rather than the results they achieve.

What should one do when a procedure or an instruction cannot be continued at some point of its execution? How to make sure that a given procedure does not have such a dead end trap? According to our best knowledge those issues have attracted little attention in action logic and action theory in general so far. They differ from the problems of standard planning research, where one is primarily focused on organising actions to achieve “as best as possible some prestated objectives” (Ghallab et al. 2004) and, on the other hand, from analysing preconditions which make attempts successful, as in Lorini and Herzig (2008).

In the paper we set up formal tools by means of which those questions can be expressed, hoping that further research will provide the answers. We found a deontic action logic adequate for the analysis of the execution of actions according to procedures or instructions.

In the early eighties of the twentieth century, through the works of Segerberg (1982) and Meyer (1987), deontic logic was brought back to von Wright’s initial idea of studying deontic notions in close relation to actions. The results of research, described in the works (Trypuz and Kulicki 2009, 2010, 2011; Trypuz 2014) contributed to a better understanding of deontic action logic systems, especially those based on finite Boolean algebra of actions. In the systems in question Boolean algebra is treated as a simple theory of actions providing a precise meaning for action constructors such as: action complement, indeterministic choice between actions and parallel execution. We want to apply the results of that research to setting up the logic of possibly unsuccessful (partially successful) complex actions.

We want to focus particularly on a sequential composition of actions. We build a framework where we can analyse a way in which sequences of actions interplay with Boolean and deontic operators. We are able to express what we find most interesting and challenging – the deontic characterisation of partially executable complex actions, for instance, what it means to impose the regulation “you ought to do \(\alpha \) and then \(\beta \)” or “you are allowed to do \(\alpha \) and then \(\beta \)”, taking into account the fact that action \(\alpha \) can be carried out in several ways, some of which may exclude the possibility of executing \(\beta \) afterwards. Thus, we have to start with modelling sequentially composed actions themselves to obtain their representation adequate for answering the questions.

We derive our theory, directly or indirectly, from the modelling of sequences of actions, widely known in computer science in the form of such formalisms as Kleene algebra, propositional dynamic logic (PDL) (in which Boolean and Kleene algebras are directly or indirectly essential components) or Hoare logic (Kleene 1956; Fischer and Ladner 1979; Hoare 1969; Hoare et al. 2011). Those systems show their usefulness in the analysis of algorithms, programs and more general phenomena of agents’ behaviour. They were also connected with deontic notions in e.g. Meyer (1987), Dignum et al. (1996), Meyden (1996), Castro and Maibaum (2009) and recently Dong and Li (2103); Prisacariu and Schneider (2012). Especially the article (Prisacariu and Schneider 2012) presents an approach similar to the one adopted in this paper. Its authors, Prisacariu and Schneider, built a deontic action logic on the foundation of an algebraic structure which they call synchronous Kleene algebra. The structure combines parallel and sequential execution of actions and a free choice operator on actions. The deontic characterisation of complex actions, including multi-step actions, is defined on the basis of deontic values of simple state-to-state transitions. They introduce an algebra of actions independent of their model-theoretic structure and then define the model for an algebraically defined normal form. Moreover, they combine their deontic action logic with PDL.

In the present paper we propose an alternative system based on results presented in Trypuz and Kulicki (2011). We focus on the algebraic description of complex actions (especially sequentially composed ones) with a special interest in action identity defined on the basis of a model-theoretic structure for actions.

In most of the recent papers concerning deontic action logic PDL operators are employed. It is essential for PDL to consider what is true after a given action has been performed. We have decided not to express our results in that framework for two reasons. The first reason is that in our considerations we disregard the results of actions. We are only interested in finding out which complex action (consisting of many consecutive steps) should start at a certain point and be continued in consecutive situations. The second reason is simplicity: to solve our scientific problem we decided to use as simple formal tools as possible. We have found Segerberg-style deontic action logic from the 1980s less complex than PDL and at the same time adequate for our purposes.

The first attempt to design the proposed system was published in Kulicki and Trypuz (2012). Now we present a slightly different version avoiding the drawbacks of the previous one. We have also significantly changed the way the system is shown.

The paper is structured as follows. In Sect. 2 we formulate basic intuitions concerning the theory of successful and unsuccessful actions, build its model and provide an action algebra, sound and complete with respect to the model. We present a new interpretation of actions which takes into account their successful and unsuccessful manifestations. The key issues of action identity are also discussed here.

In Sect. 3 we put forward a semantic characterisation of deontic operators for sequentially composed actions. We build it on the deontic characterisation of one-step actions, i.e., we assume that for each situation we know which one-step actions are permitted, forbidden and obligatory (Sect. 3.2). While defining the deontic values of actions we take into account the distinction between successful and unsuccessful realisations of sequentially composed actions—we point out that it is of particular importance in the indeterministic environment. Satisfaction conditions for the basic formulas of deontic logic of multi-step actions are defined in Sect. 3.3.

In Sect. 4 we show how deontic properties of multi-step actions emerge from the properties of one-step actions. In particular, in Sect. 4.1, we introduce an example of one-step deontic logic being sound and complete with respect to a model of the kind. Then, in two subsequent sections, we derive a model for sequentially composed actions and the respective logic.

## 2 Action model and its formalisation

### 2.1 Basic intuitions

The basic element which we use to build a formal model for a deontic action logic is a representation of two types of entities: actions and states (situations). When we write about actions we understand them primarily as action types, i.e., kinds of action, not action tokens, i.e., individual events.

- (1)
Is it possible that a transition between the same two states is a result of carrying out two or more different actions?

- (2)
Should we distinguish between an action which an agent chooses to carry out in a given situation and its possible realisations understood as the different possible ways the action might unfold in time

^{1}(depending on the conditions)?

- (1)
the first step of an action is impossible in a given situation or

- (2)
an action can be started but, after completing a part of its performance, it cannot be continued in the intended direction.

The conception of actions as (successful or unsuccessful) attempts was discussed by Lorini and Herzig in (2008). According to the authors an agent performs an action in an unsuccessful way in a given situation if and only if it cannot try to perform the action there or its execution precondition does not hold. Ignoring other assumptions accepted by the authors (like, for example, the determinism of actions and the fact that all actions are one step transitions) their understanding of unsuccessfulness of actions conceptually corresponds to our first meaning of unsuccessfulness above.

Finally, let us note that in order to combine sequentially composed actions using parallel composition we adopt (like in Prisacariu and Schneider 2012, Lorini and Herzig 2008) a synchronicity assumption that all basic actions take the same amount of time. Thus we can think of our model as a system similar to an indeterministic Turing machine.

### 2.2 Formal model

Every element of \({\mathcal {S}tep}\) is a triple \(\langle w_1, w_2, e \rangle \), where \(w_1, w_2 \in {\mathcal {W}}\) are initial and final states respectively and \(e \in {\mathcal {E}}\) is a label of an action causing a transition from \(w_1\) to \(w_2\). Intuitively elements of \({\mathcal {S}tep}\) represent different ways of performing actions in certain situations. We shall call them *action steps*. Subsets of \({\mathcal {S}tep}\) represent *actions*, so we can model a parallel execution of two actions by the intersection of respective sets of action steps and a free choice between two actions by a sum of the respective sets.

As mentioned in the previous subsection we do not impose any restrictions on a model. Thus, it may happen that we have an indeterministic execution of an action, e.g., \(\langle w_1, w_2, e\rangle \in {\mathcal {S}tep}\) and \(\langle w_1, w_3, e\rangle \in {\mathcal {S}tep}\) and \(w_2 \ne w_3\) and, on the other hand, that a transition between the same two states is a result of execution of two different actions, e.g., \(\langle w_1, w_2, e_1\rangle \in {\mathcal {S}tep}\) and \(\langle w_1, w_2, e_2\rangle \in {\mathcal {S}tep}\) and \(e_1 \ne e_2\).

A *transition* is a triple \(\langle w_1, w_2, s \rangle \), where \(w_1, w_2 \in S\) and \(s \in {\mathcal {S}eq}\), where \({\mathcal {S}eq}\) is a set of finite, possibly empty, sequences of action steps from \({\mathcal {S}tep}\), \(w_1\) is an initial state of the first transition, \(w_2\) is a final state of the last transition and each element of \(s\) starts at the state in which the previous one ends. We allow \(s\) to be empty and in that case \(w_1 = w_2\). The sequence \(s\), the third element of our triple, if nonempty, contains complete information about the transition, since \(w_1\) occurs in its first element and \(w_2\)—in its last element. Thus, the two remaining elements: \(w_1\) and \(w_2\) are redundant in that case. We keep them explicitly shown for the sake of readability. It is worth mentioning that Segerberg in (2009) builds his action theory for the purpose of deontic consideration in a similar way. The basic elements of his action theory are paths (which correspond to our sequences) being sequences of points. Each finite path has its first and last element. Two paths can be combined into one if the last element of the first path is the same as the last element of the second path. In Segerberg’s framework it is possible to extract all the steps a path is made of and two paths with the same first and last point are not necessarily the same.

For any \(w \in {\mathcal {W}}\) a triple \(\langle w, w, [] \rangle \)

^{2}is a transition.If \(\langle w_1, w_2, s\rangle \) (\(s \in {\mathcal {S}eq}\)) is a transition and \(\langle w_2, w_3, e \rangle \) is an action step from \({\mathcal {S}tep}\) then \(\langle w_1, w_3, s' \rangle \), where \(s'\) is a result of adding the action step \(\langle w_2, w_3, e \rangle \) at the end of the sequence \(s\), is also a transition.

^{3}

### 2.3 Identity of actions in the model

^{4}:

*basic actions*(or in other words

*action generators*) \(Act_0\), \(\mathbf {0}\) is the impossible action, \(\mathbf {skip}\) is a special action of doing nothing analogous to the

*skip*program used in the theory of programming, \(\alpha \sqcup \beta \)—\(\alpha \) or \(\beta \) is a free choice between \(\alpha \) and \(\beta \); \(\alpha \sqcap \beta \)—\(\alpha \) and \(\beta \) is a parallel execution of \(\alpha \) and \(\beta \). \(Act\) is a set of all actions which can be expressed in the language (\(Act_0\subseteq Act\)).

One-step actions (basic actions and their combinations achieved by the use of operators \(\sqcap \) and \(\sqcup \) or, in other words, actions which do not contain sequential composition “\(;\)” and \(\mathbf {skip}\)) are interpreted as non-empty sets of one-step transitions. By synchronicity assumption all one-step actions take the same amount of time. The special actions \(\mathbf {0}\) and \(\mathbf {skip}\) do not take any time—\(\mathbf {0}\) is impossible, so it cannot be executed, \(\mathbf {skip}\) can be understood as doing nothing in no time.

*successful transitions*, and to the latter as

*unsuccessful transitions*(see Fig. 1 and its caption).

### 2.4 Interpretation of successful and unsuccessful parts of actions

The interpretation for basic actions \(a_i\) are sets of one-step transitions—see (5). The intended interpretation for successful basic one-step action is such that (i) it is always successful, when it is in the model and (ii) one-step transitions with the same label belong to the interpretation of the same action generator. Condition (6) states that \(\mathbf {skip}\) can be carried out in any situation. It is worth stressing that its execution does not take any time. We do it without passing through any sequences. The impossible action \(\mathbf {0}\) can never be successfully executed.

For unsuccessful transitions the situation is more complicated since they behave less nicely in the context of the operations on actions. Thus, to define \(unsuc\) (for unsuccessful transitions) we need to introduce three auxiliary functions on sets of transitions.

*beginnings*):

*maximal*transitions, i.e., no proper initial fragment of a transition from \(T\) can be an element of the resulting set:

Condition (14) states that action generators (being one-step actions) and \(\mathbf {skip}\) do not have unsuccessful executions. Moreover, the impossible action \(\mathbf {0}\) cannot be executed in any way, so it has no unsuccessful transitions either. Condition (15) states that unsuccessful transitions of a free choice of actions \(\alpha \) and \(\beta \) consists of unsuccessful transitions of \(\alpha \) and \(\beta \) that are not the beginnings of successful transitions of \(\beta \) and \(\alpha \) respectively. Function \(max\) eliminates non-maximal elements from the resulting set to maintain uniqueness. Condition (16) constructs \(unsuc(\alpha \sqcap \beta )\) as a set of maximal common beginnings of successful and unsuccessful transitions connected with \(\alpha \) and \(\beta \) that are not successful for \(\alpha \sqcap \beta \). Condition (17) states that the set of unsuccessful transitions of \(\alpha ; \beta \) consists of unsuccessful transitions of \(\alpha \), elements of \(suc(\alpha )\) that do not have a continuation in \(\beta \) and unsuccessful transitions of \(\beta \) attached as a continuation to \(\alpha \).

### 2.5 Axiomatisation of action algebra

*dead end trap*. Formally we define the operator \(\mathbf {trap}\):

### 2.6 Soundness and completeness of the algebra with respect to the intended model

**Theorem 1**

For any action \(\alpha \) there exists an action \(\beta \) in the normal form such that \(\alpha = \beta \).

*Proof*

To prove the theorem it is enough to use distributivity laws (28), (34) and (36), absorption law (35), and (37).\(\square \)

**Theorem 2**

Axiomatisation is complete

*Proof*

The theorem is a consequence of the previous one. Let \(\alpha \) and \(\beta \) be actions which are not equal in the algebra. If so, the difference between their normal forms \(\alpha _1\) and \(\beta _1\) is other than the order of disjuncts and conjuncts or their repetitions. Let us now consider a model in which all basic actions occurring in \(\alpha _1\) and \(\beta _1\) are executable. In such a model the interpretations of \(\alpha _1\) and \(\beta _1\) differ, so \(\alpha \) and \(\beta \) are not equal in the model.\(\square \)

## 3 Deontic operators for sequentially composed actions

### 3.1 Intuitive introduction

We assume that deontic characterisation of sequentially composed actions is determined by deontic characterisation of one-step actions which constitute them. That excludes direct regulation of deontic values of multi-step actions. It is a simplification (one may even say a limitation), but it has the advantage of making our model more transparent. Still we can create or verify complex procedures by analysing what should be done in each state of their realisation. Then an agent can execute them being sure that it will comply with local regulations.

We assume that for each situation we know which one-step actions are permitted, forbidden and obligatory. On that basis we reconstruct deontic characterisation of sequentially composed actions. The difficulty lies in the fact that, after the execution of the first step of a sequentially composed action an agent is in a different situation, with different local *deontology* (i.e., description of the situation including norms). We must also remember that we may find nondeterministic choice at each step. To set out a deontic value of an action we have to look at its every possible execution.

Accepting such an approach, what remains to be done is to define how deontic operators interact with a sequential composition. A one-step deontic action logic which is the base for the realisation of purpose can be chosen from many existing ones, see e.g., Trypuz and Kulicki (2013) to review some of them. This makes our solution flexible enough to be adopted to different normative scenarios.

To define the deontic value of an action we need to take into account successful and unsuccessful executions of sequentially composed actions. It is especially important if we assume that an agent does not have to be aware of all the obstacles it can face and all the regulations which apply to its plan during its realisation.

Thus, taking into account what has been said above we say that a multi-step action is permitted in our theory if each action step making it up is permitted in a situation where it is planned to be triggered. Also unsuccessful attempts to perform a permitted action have to be permitted—an agent should not find itself in a situation when choosing a way to perform a permitted action ends up in a situation where following the plan further is not permitted, even if that attempt is at the end unsuccessful as a dead end trap. A multi-step action is prohibited if each possible successful or unsuccessful realisation of that action contains at least one prohibited action step.

As far as obligatory multi-step actions are concerned they are treated as good procedures and instructions. As such they should always lead to their final result. There should be no way to fail during their realisation; thus there should be no dead end traps in obligatory actions. Moreover, we believe that any initial fragment of an obligatory action should be obligatory. Thus, an action is obligatory only if every way of its realisation is required at each step.

Finally it seems reasonable to explain why the unsuccessful realisations of actions are taken into account when considering deontics. In our paper we tackle a situation in which one starts with deontic modelling of one-step action and then, on that basis, tries to do the same with complex actions, especially sequences. Unsuccessfulness of complex actions naturally emerges when one can freely combine basic actions to create a plan. That type of unsuccessfulness is not a result of “unexpected causes” but rather the nature of actions which are combined; some sequences are unsuccessful by nature, e.g., none can go to Warsaw and see La Gioconda (since the painting is not exhibited there).

### 3.2 Formalisation of deontology

Let us now put forward a formal model for the above intuitions. Let us notice that some intuitions will be expressed on the level of satisfaction conditions in the next section.

Thus, on the basis of the functions \({\mathcal {LEG}}\), \({\mathcal {ILL}}\) and \({\mathcal {REQ}}\) we define similar functions describing legal, illegal and required sets of transitions. Let us use symbols \({\mathcal {LEG}}^*\), \({\mathcal {ILL}}^*\) and \({\mathcal {REQ}}^*\) respectively for the new functions. They transform the states from \({\mathcal {W}}\) into \(2^{2^{{\mathcal {T}rans}}}\). We intend \({\mathcal {LEG}}^*(s)\) and \({\mathcal {REQ}}^*(s)\) to be defined in such a way that each step of each transition is legal (required) in an appropriate state and that \({\mathcal {ILL}}^*(s)\) must have some illegal steps in each transition.

For formal definitions of the sets we need the following auxiliary functions: \(ini\), \(fso\) and \(rem\). Let \(T\subseteq {\mathcal {T}rans}\) be a set of transitions and \(w\in {\mathcal {W}}\) a state.

Having in mind Fig. 1 one may think that \(T = \{\langle w_1, w_4, s_1\rangle , \langle w_1, w_5, s_2 \rangle \}\), where \(s_1 = [\langle w_1, w_2, a\rangle , \langle w_2, w_4, c\rangle ]\) and \(s_2 = [\langle w_1, w_3, b\rangle , \langle w_3, w_5, d\rangle ]\). We shall refer to that example below in order to explain the newly introduced functions.

*initial*steps:

*first step outcomes*) for a given set of transitions \(T\) and a given situation \(w_1\) returns a set of states that are reachable from \(w_1\) by the first step of any transition from \(T\):

*remainders*of a set of transitions \(T\) with respect to the initial state \(w_1\) and the first step outcome \(w_2\), returns a set of transitions starting from \(w_2\) obtained from transitions from \(T\) by removing their first steps:

It is worth mentioning that for the purposes of the recursive definition of the \({\mathcal {REQ}}^*\), \(\emptyset \) is an element of \({\mathcal {REQ}}^*(w)\) for any \(w\). That allows us also to treat the initial fragments of obligatory actions as obligatory. At the same time we do not want to treat the impossible action \(\mathbf {0}\) itself as obligatory. We shall prevent that by placing an appropriate requirement in satisfaction conditions for the operator “\(\mathtt {O}\)” (see below).

### 3.3 Satisfaction conditions

For a permitted (forbidden) action we require that both successful and unsuccessful transitions are legal (illegal). For obligatory actions we require that such an action cannot be unsuccessfully started, so the set of unsuccessful transitions has to be empty. Obligatory actions have to be possible, so the set of successful transitions has to be non-empty. Of course transition sets for all states have to be required.

The concepts of satisfiability, validity and tautology are defined in the standard way.

## 4 From one-step to multi-step deontic logic

As we have stated at the end of the previous section, satisfaction conditions for one-step actions can be expressed by using only sets \({\mathcal {LEG}}(w)\), \({\mathcal {ILL}}(w)\) and \({\mathcal {REQ}}(w)\) (without “\(*\)”). It means that a logic for sequential actions emerges from a logic for one-step actions. In the next section we use a fragment of a minimal one-step deontic action logic from Trypuz and Kulicki (2013) without action negation as an example. Then in Sects. 4.2 and 4.3 we extend it to a multi-step case.

### 4.1 A model and logic for one-step actions

In this system we connect deontic notions with actions both on the level of syntax and in formal semantics. That is why \({\mathcal {LEG}}(w)\), \({\mathcal {ILL}}(w)\) and \({\mathcal {REQ}}(w)\) have been defined as functions from \({\mathcal {W}}\) to \(2^{2^{{\mathcal {S}tep}}}\). Since in the present paper we use a nondeterministic model in which an action can have different outcomes even if it is executed twice in the same state, we have to modify slightly their definitions.

Now the functions \({\mathcal {LEG}}(w)\), \({\mathcal {ILL}}(w)\) and \({\mathcal {REQ}}(w)\) have values in \(2^{{\mathcal {S}tep}}\). We also assume that whenever \(t_1 = \langle w, w_1, e\rangle \) belongs to some \(X\) in \({\mathcal {LEG}}(w)\), \({\mathcal {ILL}}(w)\) or \({\mathcal {REQ}}(w)\), then for any \(w_2\), \(t_2 = \langle w, w_2, e\rangle \) also belongs to \(X\). The condition has to do with the fact that an action undertaken by an agent may have different outcomes independently of the agent’s will and behaviour and it states that the action (not the outcomes) are the subject of deontic qualification.

^{5}:

### 4.2 Multi-step model consequences

### 4.3 Multi-step logic

Let us start with the laws of multi-step logic with deontic operators without sequential composition and \(\mathbf {skip}\).

It is interesting to what extent the multi-step logic inherits the laws of the one-step logic. Formulas (55), (56), (57) and (59) are tautologies in the multi-step system.

To see that formula (55) is valid, it is enough to recall that \(suc(\alpha \sqcup \beta ) = suc(\alpha ) \cup suc(\beta )\) and notice that \(unsuc(\alpha \sqcup \beta ) \subseteq beg(all(\alpha ) \cup all(\beta ))\) and \(unsuc(\alpha ) \cup unsuc(\beta ) \subseteq beg(all(\alpha \sqcup \beta ))\). Thus, by the satisfaction condition for \(\mathtt {P}\) (55) is valid. The validity of (56) follows immediately from the satisfaction conditions for \(\mathtt {F}\), and properties (66). The validity of (57) follows immediately from the conditions for \(\mathtt {P}\) and \(\mathtt {F}\). To see that (59) is valid let us suppose \(\alpha \) is obligatory in a state \(w\). Then, for \(w\), \(suc(\alpha ) \ne \emptyset \) and \(suc(\alpha ,w) \in {\mathcal {REQ}}^*(w) \subseteq {\mathcal {LEG}}^*(w)\). Thus, for any \(u \in suc(\alpha )\) each action label must be an element of respective \({\mathcal {REQ}}(w') \subseteq {\mathcal {LEG}}(w')\). But \({\mathcal {LEG}}(w')\) and \({\mathcal {ILL}}(w')\) are disjoint for any state. Thus, \(u\) cannot contain an illegal fragment and consequently \(\lnot \mathtt {F}(\alpha )\).

Formula (77) is a derivative of formula \(\mathtt {P}(\mathbf {0})\) for one step actions (it follows directly from (57)), which we have accepted after Segerberg (1982). It states that an impossible action is permitted as it cannot have any bad effects. Any attempt to perform action \(a;\mathbf {0}\) can contain only fragments of \(a\). Thus, if \(a\) is permitted, then \(a;\mathbf {0}\) should also be permitted. The difference is that \(a\) is completely realised and action \(a;\mathbf {0}\) is not, since after \(a\) an agent should perform an impossible action (which of course is not possible) but it should not affect permissibility.

Formula (79) states that \(\mathbf {skip}\) understood as “no action in no time” is obligatory. Why should it be so? The technical reason is that \(\mathbf {skip}\) can be added at the beginning and at the end of any action \(\alpha \) without changing it. If \(\alpha \) is obligatory, to keep the consistency of the satisfaction conditions, \(\mathbf {skip}\) should also be obligatory. Intuitively it is not that obvious. To defend (79) we can only say that \(\mathbf {skip}\) is inevitable – in no time it is impossible to do anything else then \(\mathbf {skip}\). Thus, (79) corresponds to the law of standard deontic logic stating that tautology is obligatory.

^{6}(dead ends) for obligatory actions:

(80) was assumed in the satisfaction condition for obligation. In fact the requirement on the model level is even stronger—there can be no dead end transitions within the interpretation of obligatory action. However, in the language of logic we cannot express it fully since we do not have the access to single transitions.

We conjecture that formulas (23)–(80) with the rules of Modus Ponens and Substitution constitute a complete axiomatisation of the system, but we have not formulated the proof yet.

## 5 Conclusion and future perspectives

We have presented a deontic action framework taking into account unsuccessful sequences of actions. It has been founded on deontic action logic and its model for one-step actions. Within that setting it is possible to treat actions as instructions or procedures that an agent can realise step-by-step being sure that each step of a permitted action is permitted and each way of starting an obligatory action allows continuing such an action to its end. In our framework we have introduced the interpretation function for actions that takes into account their successful and unsuccessful executions.

The system can be extended in a straightforward way by imposing additional conditions on the model (e.g. determinism of action outcomes) or by enriching the language with modalities and PDL operators.

Another interesting extension of the system could be obtained by adding action negation (complement). Such an extension is, however, more challenging because of the well known technical problems occurring in systems in which sequential composition coexists with action negation.

## Footnotes

- 1.
See Sect. 5.2 in Trypuz (2007).

- 2.
The symbol \([]\) denotes an empty sequence of action types.

- 3.
The symbol \(\oplus \) stands for a concatenation of sequences from \({\mathcal {S}eq}\).

- 4.
We do not introduce action complement (negation) to our language. The operator is not essential for our task and in the context of sequential composition causes the well known intuitive and technical problems (see e.g. Broersen 2004).

- 5.
Henceforward universal quantifications over whole formulas are left implicit.

- 6.
\(\mathbf {trap}\) is defined by (41).

## Notes

### Acknowledgments

We would like to thank Prof. J.-J.Ch. Meyer and the two anonymous reviewers for valuable remarks on the earlier versions of this work. This research was supported by the National Science Centre of Poland (DEC-2011/01/D/HS1/04445).

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