1 Introduction
Modal logics of , an acronym for ‘seeing to it that’, have a long tradition in the formal investigation of agency, starting with a series of papers by Belnap and Perloff in the 1980s and culminating in [4]. For the past decades, logic has continued to receive considerable attention, proving itself invaluable in a multitude of fields concerned with formal reasoning about agentive choice making. For example, the framework has been applied to epistemic logic [6], deontic logic [12, 14], and the formal analysis of legal reasoning [6, 13]. Surprisingly, investigations of the mathematical properties of logics are limited [3, 18] and its prooftheory has only been addressed recently [5, 21]. What is more, despite AIoriented
papers motivating the need of tools for automated reasoning about agentive choicemaking
[2, 3, 5], the envisaged automation results are still lacking. The present work will be the first to provide terminating, automated proofsearch for a class of logics, including countermodel extraction directly based on failed proofsearch.The sequent calculus [8] is an effective framework for proofsearch, suitable for automated deduction procedures. Given the metalogical property of analyticity, a sequent calculus allows for the construction of proofs by merely decomposing the formula in question. In the present work, we employ the labelled sequent calculus—a useful formalism for a large class of modal logics [15, 20]—and introduce labelled sequent calculi (with ) for multiagent logics containing limited choice axioms, discussed in [22].
In order to appropriate these calculi for automated proofsearch, we take up a refinement method presented in [19]—developed for the restricted setting of display logic—and adapt it to the more general setting of labelled calculi. In the refinement process the external character of labelled systems—namely, the explicit presence of the semantic structure—is made internal through the use of alternative, yet equivalent, propagation rules [19]. The tailored propagation rules restrict and simplify the sequential structures needed in derivations, producing, for example, shorter proofs. Moreover, one can show that through the use of propagation rules, the structural rules, capturing the behavior of the logic’s modal operators, are admissible. In our case, the resulting refined calculi derive theorems using only forestlike sequents, allowing us to adapt methods from [19] and provide correct and terminating proofsearch algorithms for this class of logics.
In short, the contribution of this paper will be threefold: First, we provide new sound and cutfree complete labelled sequent calculi for all multiagent logics (with ) discussed in [22]—thus extending the class of logics addressed in [5]. Second, we show how to refine these calculi to obtain new calculi , which are suitable for proofsearch. Last, for each , we provide a terminating proofsearch algorithm deciding singleagent logic . We conclude by discussing the prospects of generalizing the latter results to the multiagent setting.
2 Logical Preliminaries
logic refers to a group of modal logics using operators that capture agential choicemaking. The basic logic , and the one that will be used throughout this paper, employs two types of modal operators: First, it contains a settledness operator
which expresses which formulae are ‘settled true’ at the current moment. Second, it contains, for each agent
in the language, an atemporal—i.e., instantaneous—choiceoperator expressing that ‘agent sees to it that’. This basic choiceoperator is referred to as the Chellas [4]. With these two types of operators, one can define a more refined deliberative operator: i.e., iff . Intuitively, holds when ‘agent sees to it that and it is possible for not to hold’. The multiagent language for is defined accordingly:Definition 1 (The Language [11])
Let be a finite set of agent labels s.t. and let be a countable set of propositional variables. is defined via the following BNF grammar:
where and
Notice, the language consists of formulae in negation normal form. This notation allows us to reduce the number of rules in our calculi, enhancing the readability and simplicity of our proof theory. The negation of , written as , is obtained by replacing each operator with its dual and each positive atom with its negation , and each with its positive variant [5]. Consequently, we obtain the following abbreviations: , , , and . We will freely use these abbreviations throughout this paper. Since we are working in negation normal form, diamondmodalities are introduced as separate primitive operators. We take and as the duals of and , respectively.
Following [11], since we work with instantaneous, atemporal it suffices to regard only single choicemoments in our relational frames. This means that we can forgo the traditional branching time structures of basic, atemporal logic [4]. In what follows, we define frames as those frames in which limits the amount of choices available to each agent to at most many choices (imposing no limitation when ).^{1}^{1}1For a discussion of the philosophical utility of reasoning with limited choice see [22].
Definition 2 (Relational Frames and Models)
Let and let for . An frame is defined as a tuple where is a set of worlds and:

For each , is an equivalence relation;

For all , ;

Let and , then
For all ,
An model is a tuple where is an frame and is a valuation assigning propositional variables to subsets of , i.e. . Additionally, we stipulate that condition (C3) is omitted when .
As in [11], the set of worlds is taken to represent a single moment in which agents from are making their decision. Following (C1), for every agent , the relation is an equivalence relation; that is, functions as a partitioning of into what will be called choicecells for agent . Each choicecell represents a set of possible worlds that may be realized by a choice of the agent. The condition (C2) expresses the principle independence of agents, ensuring that any combination of choices, available to different agents, is consistent. The last condition (C3), represents the principle which limits the amount of choices available to an agent to a maximum of . For a philosophical discussion of these principles we refer to [4, Ch. 7C].
Definition 3 (Semantic Clauses for [5, 11])
Let be an model and let be a world in its domain . The satisfaction of a formula on at is inductively defined as follows:

iff

iff

iff and

iff or

iff ,

iff ,

iff ,

iff ,
A formula is globally true on (i.e. ) iff it is satisfied at every world in the domain of . A formula is valid (i.e. ) iff it is globally true on every model. Last, semantically implies , written , iff for all models and worlds in , if for all , then .
It is worth emphasizing, that the semantic interpretation of refers to the domain of the model in its entirety; i.e. is settled true iff is globally true. This is an immediate consequences of considering instantaneous in a singlemoment setting (cf. semantics where a relation is introduced for , e.g. [5]).
The Hilbert calculus for in Fig. 1 is taken from [22]. Apart from the propositional axioms, it consists of axiomatizations for and , for each . It contains the standard bridge axiom , linking to . Furthermore, it contains an independence of agents axiom , as well as a choice axiom for each . The rules are modus ponens and necessitation.
3 Refinement of the Calculus
In this section, we introduce labelled calculi for multiagent STIT logics (with limited choice). Our calculi are modified, extended versions of the labelled calculi for the logics (with ) proposed in [5] and cover a larger class of logics. The calculi possess fundamental prooftheoretic properties such as contraction and cutadmissibility which follow from the general results on labelled calculi established in [15]. The main goal of this section is to refine the calculi through the elimination of structural rules, resulting in new calculi that derive theorems within a restricted class of sequents. As a result of adopting the approach in [11], the omission of the relational structure corresponding to the modality offers a simpler approach to proving the admissibility of structural rules in the the presence of propagation rules (Sec. 3.2). Let us start by introducing the class of calculi.
3.1 The Calculi
We define labelled sequents via the following BNF grammar:
where , and are from a countable set of labels . Labelled sequents consist exclusively of labelled formulae of the form and relational atoms of the form . For this reason, sequents can be partitioned into two parts: we sometimes use the notation to denote labelled sequents, where is the part consisting of relational atoms and is the part consisting of labelled formulae. Last, we interpret the commas between relational atoms in conjunctively, the comma between and in implicationally, and the commas between labelled formulae in disjunctively (cf. Def.7).
The labelled calculi (where ) are shown in Fig. 2. Note that for each agent , we obtain a copy for each of the rules , , , , and . We refer to , , , and as the structural rules of . The rule captures the independence of agents principle. Furthermore, the rule schema , limiting the amount of choices available to agent , provides different rules depending on the value of in (again, we reserve to assert that the rule does not appear). When , the rule contains premises, where each sequent (for and ) represents a different premise of the rule. As an example, for and the rules for agent are:











Theorem 3.1
The calculi have the following properties:

All sequents of the form are derivable;

Variablesubstitution is heightpreserving admissible;

All inference rules are heightpreserving invertible;

Weakening and contraction are heightpreserving admissible:

The cut rule is admissible:

For every formula , is derivable in if and only if , i.e. is sound and complete relative to .
Prooftheoretic properties like those expressed in (4) and (5) of Thm. 3.1 are essential when designing decidability procedures via proofsearch. In constructing a proof of a sequent, proofsearch algorithms proceed by applying inference rules of a calculus bottomup. A bottomup application of the rule in a proofsearch procedure, however, requires one to guess the cut formula , and thus risks nontermination in the algorithm. (One can think of similar arguments why and risk nontermination.) It is thus crucial that such rules are admissible; i.e. everything derivable with these rules, is derivable without them.
Remark 1
To obtain contraction admissibility (Thm. 3.1(4)) labelled calculi must satisfy the closure condition [15]: if a substitution of variables in a structural rule brings about a duplication of relational atoms in the conclusion, then the calculus must contain another instance of the rule with this duplication contracted.
We observe that if we substitute variable for in the structural rule (below left), we obtain the rule (below right), when the atom is contracted:


Thus, following the closure condition, we must also add to our calculus. However, is a special instance of the rule, and hence it is admissible; therefore, we can omit its inclusion in our calculi. None of the other structural rules possess duplicate relational atoms in their conclusions under a substitution of variables, and so, each calculus satisfies the closure condition.
3.2 Extracting the Calculi
We now refine the calculi, extracting new calculi to which proofsearch techniques from [19] may be adapted. In short, we introduce new rules to our calculi, called propagation rules, which are wellsuited for proofsearch and imply the admissibility of the less suitable structural rules and .
Propagation rules are special sequent rules that possess a nonstandard side condition, consisting of two components. For the first component (1), we transform the sequent occurring in the premise of the rule into an automaton. The labels appearing in the sequent determine the states of the automaton, whereas the relational atoms of the sequent determine the transitions between these states. The following definition, based on [19, Def. 4.1], makes this notion precise:
Definition 4 (Propagation Automaton)
Let be a labelled sequent, be the set of labels occurring in , and . We define a propagation automaton to be the tuple s.t. (i) is the automaton’s alphabet, (ii) is the set of states, (iii) is the initial state, (iv) is the accepting state, and (v) is the transition function where and iff .
We will often write instead of to denote a transition between states. A string is a, possibly empty, concatenation of symbols from (where indicates the empty string). We say that an automaton accepts a string iff there exists a transition sequence from the initial state to the accepting state . Last, we will abuse notation and use equivocally to represent both the automaton and the set of strings accepted by the automaton, i.e. . The use of notation can be determined from the context.
The second component (2) of the rule’s side condition restricts the application of the rule to a particular language that specifies and determines which types of strings occurring in the automaton allow for a correct application of the propagation rule. We define this language accordingly:
Definition 5 (Agent Application Language)
For each , we define the application language to be the language generated from the regular expression , that is, with the empty string.^{2}^{2}2For further information on regular languages and expressions, consult [17].
Bringing the two components together, a propagation rule is only applied when both the associated propagation automaton accepts a certain string—corresponding to a path of relational atoms in the premise of the rule—and the string is in the application language.
Definition 6 (Propagation Rule)
Let , , and . The propagation rule is defined as follows:
The superscript indicates that for .^{3}^{3}3Observe that . Hence, deciding which automaton to employ in determining the side condition is inconsequential: when applying the rule topdown we may consult , whereas during bottomup proofsearch we may regard .
We use the represent the set of all propagation rules.
The underlying intuition of the rule (applied bottomup) is that, given some labelled sequent , a formula is propagated from to another label , if and are connected by a sequence of relational atoms in (with fixed). In the corresponding propagation automaton , this amounts to the existence of a string which represents a sequence of transitions from to , such that all transitions are solely labelled with . To see how the language secures the soundness of the rule, we refer to Thm. 3.3 of App. 0.A. For a general introduction to propagation rules and propagation automata, see [19].
Let us make the introduced notions more concrete by providing an example:
Example 1
Let . The propagation automaton is depicted graphically as (where the singleboxed node designates the initial state and a doubleboxed node represents the accepting state):
@/^1pc/@.¿[rr]—⟨1 ⟩ u@/^1pc/@.¿[rr]—⟨2 ⟩@/^1pc/@.¿[ll]—⟨1 ⟩v @/^1pc/@.¿[rr]—⟨1 ⟩@/^1pc/@.¿[ll]—⟨2 ⟩@/^1pc/@.¿[ll]—⟨1 ⟩ 
Observe that every string the automaton accepts must contain at least one symbol. Since no string of this form exists in , it is not valid to propagate the formula to . That is, the sequent does not follow from applying the propagation rule to the premise.
On the other hand, consider the propagation automaton :
@/^1pc/@.¿[rr]—⟨1 ⟩ @/^1pc/@.¿[rr]—⟨2 ⟩@/^1pc/@.¿[ll]—⟨1 ⟩v @/^1pc/@.¿[rr]—⟨1 ⟩@/^1pc/@.¿[ll]—⟨2 ⟩z@/^1pc/@.¿[ll]—⟨1 ⟩ 
The automaton accepts the simple string , which is included in the language . Therefore, it is permissible to apply the propagation rule and derive from the premise .
Remark 2
We observe that both of the languages and are regular and, thus, the problem of determining whether , is decidable [19]. Consequently, the propagation rules in may be integrated into our proofsearch algorithm without risking nontermination.









The proof theoretic properties of are preserved when extended with the set of propagation rules (Lem. 1). Moreover, the nature of our propagation rules allows us to prove the admissibility of the structural rules and , for each (resp. Lem. 2 and 3), which results in the refined calculi (shown in Fig. 3). The proofs of Lem. 1 and 2 are present in App. 0.A (the latter is similar to the proof of Lem. 3 presented here).
Lemma 1
The calculi have the following properties: (i) all sequents of the form are derivable; (ii) variablesubstitution is heightpreserving admissible; (iii) all inference rules are heightpreserving invertible; (iv) the , and rules are heightpreserving admissible.
Lemma 2 (Elimination)
Every sequent derivable in is derivable without the use of .
Lemma 3 (Elimination)
Every sequent derivable in is derivable without the use of .
Proof
The result is proven by induction on the height of the given derivation. Also, we evoke Lem. 2 (whose detailed proof is provided in App. 0.A) and assume that all instances of have been eliminated from the given derivation.
Base Case. An application of on an initial sequent (below left) can be rewritten as an instance of the rule (below right).


Inductive step. We show the inductive step for the nontrivial cases: and (case (i) and (ii), respectively). All other cases are resolved by applying IH to the premise followed by an application of the corresponding rule.
(i). Let be auxiliary in the inference of the initial derivation (below (1)). Observe that when we apply the rule first (below (2)), the atom is no longer present in , and so, the rule is not necessarily applicable. Nevertheless, we may apply the rule to derive the desired conclusion since . Namely, the fact that only relies on the presence of in .

(1) 

(2) 
(ii). Let be the first premise of the initial derivation (below (3)). In the inference of the top derivation, we assume that is auxiliary, that is, the side condition of is satisfied because some string with . (NB. For the nontrivial case, we assume that relies on the presence of , that is, the automaton makes use of transitions or defined relative to .) When we apply the rule first in our derivation (below (4)), we can no longer rely on the relational atom to apply the rule. However, due to the presence of in we may still apply the rule. Namely, since , we know there is a sequence of transitions from to . We replace each occurrence of with and each occurrence of with . There will thus be a string in , and so, the rule may be applied.

(3) 

(4) 
Theorem 3.2 (Cutfree Completeness of )
For any formula , if , then is cutfree derivable in .
Proof
Last, we must ensure that is sound. To prove this, we need to stipulate how to interpret sequents on models. Our definition is based on [5]:
Definition 7 (Interpretation, Satisfaction, Validity)
Let be an model with domain , a labelled sequent, and Lab the set of labels. Let be an interpretation function mapping labels to worlds: i.e. .
is satisfied in with iff for all relational atoms , if holds in , then there must exist some in such that .
is valid iff it is satisfiable in every with any interpretation function .
Theorem 3.3 ( Soundness)
Every sequent derivable in is valid.
4 ProofSearch and Decidability
In this section, we provide a class of proofsearch algorithms, each deciding a logic (with ). (We use to denote the agent in the singleagent setting.) We end the section by commenting on the more complicated multiagent setting.
4.1 The SingleAgent Setting
In the singleagent case, the independence of agents condition is trivially satisfied, meaning we can omit the rule from each calculus and from consideration during proofsearch.
In what follows, we prove that derivations in need only use forestlike sequents. The forestlike structure of a sequent refers to a graph corresponding to the sequent. This control in sequential structure is what allows us to adapt methods from [19] to , and produce a proofsearch algorithm that decides , for each . Let us start by making the aforementioned notions precise.
Definition 8 (Sequent Graph)
We define a graph to be a tuple , where is the nonempty set of vertices, the set of edges , and is the labelling function that maps edges from into some nonempty set and vertices from into some nonempty set .
Let be a labelled sequent and let be the set of labels in . The graph of , denoted , is the tuple , where (i) , (ii) iff , and (iii) and iff .
Example 2
The sequent graph corresponding to the labelled sequent = is shown below:
y x[ll]—1z[ll]—1 
Definition 9 (Tree, Forest, Forestlike Sequent, Choicetree)
We say that a graph is a tree iff there exists a node , called the root, such that there is exactly one directed path from to any other node in the graph. We say that a graph is a forest iff it consists of a disjoint union of trees.
A sequent is forestlike iff its graph is a forest. We refer to each disjoint tree in the graph of a forestlike sequent as a choicetree and for any label in , we let represent the choicetree that belongs to.
The above notions will be significant for proofsearch algorithms, for example:
Remark 3
When interpreting a sequent, each choicetree that occurs in the graph of the sequent is a syntactic representation of an equivalence class of (i.e., a choicecell for agent ). Using this insight, we know that if agent is restricted to many choices, then if there are choicetrees in the sequent, at least two choicetrees must correspond to the same equivalence class in . We use this observation to specify how is applied in the algorithm.
The following definitions introduce the necessary tools for the algorithms:
Definition 10 (Saturation, realization, , propagated)
Let be a forestlike sequent and let w be a label in .
The label is saturated iff the following hold: (i) If , then , (ii) if , then and , (iii) if , then or .
A label in is realized iff for every , there exists a label such that . A label in is realized iff for every
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