# Cut-free Calculi and Relational Semantics for Temporal STIT Logics

We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm, Tstit and Xstit. All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi G3Ldm and G3TSTIT are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also XSTIT can be characterized through relational frames, omitting the use of BT+AC frames.

## Authors

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02/18/2019

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## 1 Introduction

Various autonomous machines are developed with the aim of performing particular human tasks. Human acting, however, is inevitably connected to legal and moral decision making–sometimes more than we think. Hence, such machines will eventually be found in difficult scenarios in which normatively acceptable actions must be generated [12]. What is more, these decisions can quickly turn into complex (technical) problems [13]. The above stresses the need for formal tools that allow for reasoning about agents, the choices they have, and the actions they are able and allowed to perform. Implementable logics of agency can play an important role in the development of such automated systems: they can provide explicit proofs that can be checked and which, more importantly, can be understood by humans (e.g. [1]). The present work takes a first step in this direction by providing cut-free sequent calculi for one of the central formalisms of agency: STIT logic with temporal operators.

The logic of STIT, which is an acronym for ‘Seeing To It That’, is a prominent modal framework for the formal analysis of multi-agent interaction and reasoning about choices.111For an introduction to STIT logic and a historical overview we refer to [3, 4, 16]. In short, STIT logics contain modal formulae of the form , capturing the notion that “the agent sees to it that the state of affairs is brought about”. STIT logic knows many fruitful extensions and its recent application to legal theory, deontic reasoning, and epistemics shows that issues of agency are essentially tied to temporal aspects of choice: for example, consider issues in legal responsibility [18]; social commitment [17]; knowledge-based obligations [7]; agent-bound instrumentality [5]; and actions as events [28].

Unfortunately, nearly all available proof systems for STIT logics are Hilbert-style systems, which are known to be cumbersome for proof search and not suitable for proving metalogical properties of the intended formalisms. To this purpose, a renowned alternative proof framework is Gentzen’s sequent calculus [11]. It allows one to construct proofs that decompose the formulae to be proven in a stepwise manner; making it an effective tool for proof search and a good candidate for automated deduction procedures. However, this framework is not strong enough to design cut-free analytic calculi for many modal logics of interest [20]; including STIT logic. In this work, we will treat several STIT logics through a more expressive extension of this formalism: Labelled Sequent Calculi [20, 26].

The aim of the present paper is to provide labelled calculi for several central temporal STIT logics: , and . To our knowledge, there have only been three attempts to capture STIT logics in alternative proof systems: in [1] a natural deduction system for a deontic STIT logic is proposed and in [24, 27] tableaux systems for multi-agent deliberative STIT logics are presented.

On the one hand, the novelty of the present contribution compared to previous works, is that all presented calculi (i) possess useful proof-theoretic properties such as contraction- and cut-admissibility and (ii) are modular and extend to several temporal STIT-logics, including both temporal operators and inherently temporal STIT-operators (in a multi-agent, as well as a group setting). In doing so, we answer an open question in [27] regarding the construction of a rule-based proof system for temporal extensions of . On the other hand, the investigation of STIT has been with an essential focus on its intuitive semantics: branching time structures, extended with histories as paths and agential choice-functions (BT+AC-frames). Recent work [2, 14, 17], however, shows that the basic atemporal STIT logic and its temporal extension are characterizable through simpler relational frames. The current work extends these results by showing that also the logic can be semantically characterized without using BT+AC structures.

In section 2 we will introduce the base logic and its corresponding labelled calculus. Thereafter, in section 3, we provide a cut-free calculus for the temporal STIT logic , introduced in [17], which exploits a temporal irreflexivity rule based on [10]. Last, in section 4, we provide a labelled calculus for the inherently temporal STIT logic from [7, 8]. Here we show that the independence of agents principle of STIT logic can be captured using systems of rules from [22]. We conclude and highlight some envisaged future work in section 5.

## 2 The Logic Ldm

### 2.1 Axioms and Relational Semantics for Ldm

The basic STIT logic

offers a framework for reasoning about individual agents realizing propositions via the choices available to them at particular moments in time. In the semantics of

, each moment can be formalized as an equivalence class of worlds, where each world sits in a linear chain (referred to as a history) extending to the future and (possibly to) the past. Therefore, each world contained in a particular moment can be thought of as an alternative state of affairs that evolves along a different timeline. Moreover, for each agent, moments are further partitioned into equivalence classes, where each class represents a possible choice available to the agent for realizing a set of potential outcomes. Hence, if a proposition holds true in every world of a particular choice for an agent , then we claim that “ sees to it that ” (written formally as ) at each world of that choice; i.e. ’s committal to the choice ensures regardless of which world in the choice set is actual.

The above STIT operator is referred to as the Chellas-STIT (i.e. cstit) [4]. It is often distinguished from the deliberative STIT (i.e. dstit) which consists of cstit together with a negative condition: we say that “agent deliberatively sees to it that ” (written formally as ) when (i) “ sees to it that ” and (ii) “ is currently not settled true” [15, 16]. The second condition ensures that the realization of depends on the choice made by the agent; i.e. might not have been case had the agent chosen to act differently. By making use of the settledness operator , which is prefixed to a formula when the formula holds true at every world in a moment, cstit and dstit become inter-definable: namely, iff . As an example of a STIT formula, the formula must be interpreted as follows: at the current moment, agent has a possible choice available that allows to see to it that is guaranteed, and there is an alternative choice present to that does not guarantee . In this paper, we introduce and as primitive and take as defined.

In this section, we make all of the aforementioned notions formally precise and provide a relational semantics for along with a corresponding cut-free labelled calculus. In section 3, we will extend with temporal operators, obtaining the logic . Since both logics rely on the same semantics, we introduce their languages and semantics simultaneously, avoiding unnecessary repetition. Lastly, in what follows we give all formulae of the associated logics in negation normal form. This reduces the number of rules in the associated calculi and offers a simpler presentation of the proof theory. The languages for and are given below:

###### Definition 1 (The Languages LLdm and LTstit)

Let be a finite set of agent labels and let be a countable set of propositional variables. The language is given by the following BNF grammar:

 ϕ::=p | ¯¯¯p | ϕ∧ϕ | ϕ∨ϕ | □ϕ | ◊ϕ | [i]ϕ | ⟨i⟩ϕ

The language is defined accordingly:

 ϕ::=p | ¯¯¯p | ϕ∧ϕ | ϕ∨ϕ | □ϕ | ◊ϕ | [i]ϕ | ⟨i⟩ϕ | [Ag]ϕ | ⟨Ag⟩ϕ | Gϕ | Fϕ | Hϕ | Pϕ

where and .

The language extends through the incorporation of the tense modalities , , , and and the modalities and for the grand coalition of agents. and are duals and read, respectively, as ‘always will be in the future’ and ‘somewhere in the future’. are are also dual and are interpreted, respectively, as ‘always has been in the past’ and ‘somewhere in the past’ (cf. [17, 25]). The operator captures the notion that ‘the grand coalition of agents sees to it that’. Note that the negation of a formula , written , is obtained in the usual way by replacing each operator with its dual, each positive propositional atom with its negation , and each negative propositional atom with its positive version . We may therefore define as , as , as , and as . We will use these abbreviations throughout the paper.

At present, we are principally interested in and temporal frames: in particular, since will be introduced as the temporal extension of and, more generally, because the logic of STIT has an implicit temporal intuition underlying choice-making (cf. original branching-time frames employed for [4, 15, 16]). We will prove that is strongly complete with respect to these more elaborate irreflexive Temporal Kripke STIT frames.

###### Definition 2 (Relational Tstit Frames and Models [17])

Let for . A relational Temporal STIT frame (-frame) is defined as a tuple where is a non-empty set of worlds and:

• For all , , , are equivalence relations where:

• For each , ;

• For all , if for all , then ;

• For all , ;

• is a transitive and serial binary relation and is the converse of , and the following conditions hold:

• For all , if and , then , , or ;

• For all , if and , then , , or ;

• ; (Relation composition is defined as usual.)

• For all , if , then ;

A -model is defined as a tuple where is a -frame and is a valuation function assigning propositional variables to subsets of ; that is, .

The property expressed in C2 corresponds to the familiar independence of agents principle of STIT logic, which states that if it is currently possible for each distinct agent to make a certain choice, then it is possible for all such choices to be made simultaneously. Condition C6 captures the STIT principle of no choice between undivided histories, which ensures that if two time-lines remain undivided at some future moment, then no agent can currently make a choice realizing one time-line without the other. (This principle is inexpressible in the atemporal language of the base logic .) For a philosophical discussion of these principles see [4]. Last, condition C7 ensures that the temporal frames under consideration are irreflexive, which means that the future is a strict future (excluding the present). For a discussion of the other frame properties we refer to [17].

###### Definition 3 (Semantics for LLdm and LTstit)

Let be a -model and let be a world in its domain . The satisfaction of a formula on at is inductively defined as follows (in clauses 1-14 we omit explicit mention of ):

• iff

• iff

• iff and

• iff or

• iff ,

• iff ,

• iff ,

• iff ,

• iff ,

• iff ,

• 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.

###### Definition 4 (The Logic Ldm[4])

The Hilbert system of consists of the following axioms and inference rules:

 ϕ→(ψ→ϕ) (¯¯¯¯ψ→¯¯¯ϕ)→(ϕ→ψ) (ϕ→(ψ→χ))→((ϕ→ψ)→(ϕ→χ))

 □ϕ→ϕ ◊ϕ→□◊ϕ □(ϕ→ψ)→(□ϕ→□ψ) [i]ϕ→ϕ ⟨i⟩ϕ→[i]⟨i⟩ϕ
 □ϕ∨◊¯¯¯ϕ [i]ϕ∨⟨i⟩¯¯¯ϕ ⋀i∈Ag◊[i]ϕi→◊(⋀i∈Ag[i]ϕi)
 [i](ϕ→ψ)→([i]ϕ→[i]ψ) □ϕ→[i]ϕ ϕ □ϕ ϕ ϕ→ψ ψ

A derivation of in from a set of premises , is written as . When is the empty set, we refer to as a theorem and write .

The axiomatization contains duality-axioms and which ensure the usual interaction between the box and diamond modalities. Furthermore, the axiom is the independence of agents (IOA) axiom.

###### Theorem 2.1 (Soundness [17])

For any formula , if , then .

Observe that all axioms of are within the Sahlqvist class. Therefore, we know that is already strongly complete relative to the simpler class of frames defined by the first-order properties corresponding to its axioms [6] (cf. [2, 14] for alternative completeness proofs of relative to this class of relational frames). As mentioned previously, we are interested in relative to the more involved temporal frames. The usual canonical model construction from [6] cannot be applied to obtain completeness of in relation to -frames. This follows from the fact that the axioms of do not impose any temporal structure on the canonical model of , and hence, we are not ensured that the resulting model qualifies as a -model. Theorem 2.2 is therefore proved via an alternative canonical model construction, which can be found in appendix 0.A.

###### Theorem 2.2 (Completeness)

Any consistent set is satisfiable.

### 2.2 A Cut-free Labelled Calculus for Ldm

We now provide a cut-free labelled calculus for , which can be seen as a simplification of the tableaux calculus in [27]. Labelled sequents are defined through the following BNF grammar:

 Γ::=x:ϕ | Γ,Γ | Rαxy,Γ

where is from a countable set of labels , , and . Note that commas are used equivocally in the interpretation of a labelled sequent: representing (i) a conjunction when occurring between relational atoms, (ii) a disjunction when occurring between labelled formulae, and (iii) an implication when binding the multiset of relational atoms to the multiset of labelled formulae, which comprise a sequent. Last, we use the notation (for ) to denote here and later that the labelled formula is derivable in the calculus .

The first order correspondents of all axioms are geometric axioms: that is, axioms of the form where each is atomic and does not contain free occurrences of (for ), and each is a conjunction of atomic formulae. The calculus is obtained by transforming all such correspondents into rules; i.e. geometric rules. (For further discussion on extracting rules from axioms, we refer to [20, 22].) Last, since our formulae are in negation normal form, we provide a one-sided version of the calculi introduced in [20]. This allows for a simpler formalism with fewer rules, but which is equivalent in expressivity.

###### Definition 5 (The Calculus G3Ldm)
 Γ,w:ϕ (id) Γ,w:ψ Γ,w:p,w:¯¯¯p (∧) Γ,w:ϕ∧ψ Γ,w:ϕ,w:ψ (∨) Γ,w:ϕ∨ψ Γ,R□wv,v:ϕ (□)∗ Γ,w:□ϕ
 Γ,R□wu,w:◊ϕ,u:ϕ (◊) Γ,R□wu,w:◊ϕ Γ,Riwv,v:ϕ ([i])∗ Γ,w:[i]ϕ Γ,Riwu,w:⟨i⟩ϕ,u:ϕ (⟨i⟩) Γ,Riwu,w:⟨i⟩ϕ
 R□ww,Γ (refl□) Γ Riww,Γ (refl[i]) Γ R□wu1,...,R□wun,R1u1v,...,Rnunv,Γ (IOA)∗ R□wu1,...,R□wun,Γ
 R□wu,R□wv,R□uv,Γ (eucl□) R□wu,R□wv,Γ R□wu,Riwu,Γ (br[i]) Riwu,Γ Riwu,Riwv,Riuv,Γ (eucl[i]) Riwu,Riwv,Γ

The ‘’ on the labels , , and indicates an eigenvariable condition for this rule: i.e. the label occurring in the premise of the rule cannot occur in the conclusion.

The rule is an initial sequent and the rules , , , , and allow us to decompose connectives. Furthermore, as indicated by the relational atoms, the rules capture the behavior of the corresponding modal operators, and the rule secures independence of agents in . In order to establish the intended soundness and completeness results, we need to formally interpret a labelled sequent relative to a given model. For the sake of brevity, we provide the semantics uniformly for all labelled sequent languages appearing in this paper:

###### Definition 6 (Interpretation, Satisfiability, Validity)

Let . Let be a model for with domain , the set of labels used in the labelled sequent language of , a sequent in and let be a relation of . (We have for , for , and , for all , when . We take as the complement of the relation .) Last, let be an interpretation function of on that maps labels to worlds; i.e. . We say that,

• a sequent is satisfied in with iff for all relational atoms and equalities in , if holds in , then there must exist some in such that .

A sequent is valid iff it is satisfiable in any model with any of on .

###### Theorem 2.3 (Soundness)

Every sequent derivable in is valid.

###### Proof

By induction on the height of the given derivation. For initial sequents of the form the claim is clear. The inductive step is argued by showing that each inference rule preserves validity (cf. theorem 5.3 in [21]).

###### Lemma 1

For all , if , then .

###### Proof

The derivation of each axiom and inference rule of , except for the IOA-axiom, is straightforward (See [20, 23]). For readability, we only present the derivation of the IOA-axiom for two agents; the general case is similar:

 R1vu,R1yv,R1yu,...,y:⟨1⟩¯¯¯ϕ1,u:¯¯¯ϕ1,u:ϕ1 R1vu,R1yv,Riyu,...,y:⟨1⟩¯¯¯ϕ1,u:ϕ1 R1vu,R1yv,...,y:⟨1⟩¯¯¯ϕ1,u:ϕ1 R1yv,...,y:⟨1⟩¯¯¯ϕ1,v:[1]ϕ1 R2vu,R2zv,Rizw,...,z:⟨2⟩¯¯¯ϕ2,w:¯¯¯ϕ2,w:ϕ2 R2vw,R2zv,R2zw,...,z:⟨2⟩¯¯¯ϕ2,w:ϕ2 R2vw,R2zv,...,z:⟨2⟩¯¯¯ϕ2,w:ϕ2 R2zv,...,z:⟨2⟩¯¯¯ϕ2,v:[2]ϕ2 R1yv,R2zv,R□xy,R□yv,R□xv,R□xz,y:⟨1⟩¯¯¯ϕ1,z:⟨2⟩¯¯¯ϕ2,x:◊([1]ϕ1∧[2]ϕ2),v:[1]ϕ1∧[2]ϕ2 R1yv,R2zv,R□xy,R□yv,R□xv,R□xz,y:⟨1⟩¯¯¯ϕ1,z:⟨2⟩¯¯¯ϕ2,x:◊([1]ϕ1∧[2]ϕ2) R1yv,R2zv,R□xy,R□yv,R□xz,y:⟨1⟩¯¯¯ϕ1,z:⟨2⟩¯¯¯ϕ2,x:◊([1]ϕ1∧[2]ϕ2) R1yv,R2zv,R□xy,R□xz,y:⟨1⟩¯¯¯ϕ1,z:⟨2⟩¯¯¯ϕ2,x:◊([1]ϕ1∧[2]ϕ2) R□xy,R□xz,y:⟨1⟩¯¯¯ϕ1,z:⟨2⟩¯¯¯ϕ2,x:◊([1]ϕ1∧[2]ϕ2) x:□⟨1⟩¯¯¯ϕ1,x:□⟨2⟩¯¯¯ϕ2,x:◊([1]ϕ1∧[2]ϕ2) x:□⟨1⟩¯¯¯ϕ1∨□⟨2⟩¯¯¯ϕ2∨◊([1]ϕ1∧[2]ϕ2)

The dashed lines in the above proof indicate the use of transitivity rules, which are derivable from the , , , and rules (see [20]).

###### Theorem 2.4 (Completeness)

For all , if , then .

###### Proof

Follows from theorem 2.2 and lemma 1.

Due to the fact that all labelled sequent calculi given in this paper fit within the scheme presented in [20, 22], we obtain the subsequent theorem specifying their proof-theoretic properties:

###### Theorem 2.5

Each calculus with has the following properties:

1. All sequents of the form are derivable in with in the language ;

2. All inference rules of are height-preserving invertible;

3. Weakening, contraction, and variable-substitution are height-preserving admissible;

###### Proof

See [20] and [22] for details.

In order to maintain the admissibility of contraction, our calculi must satisfy the closure condition [20, 22]. That is, the calculi and adhere to the following condition: For any generalized geometric rule in which a substitution of variables produces a duplication of relational atoms or equalities active in the rule, the instance of the rule with such duplicates contracted is added to the calculus. Since variable substitutions can only bring about a finite number of rule instances possessing duplications, the closure condition adds at most finitely many rules and is hence unproblematic. (Generalized geometric rules extend the class of geometric rules and can be extracted from generalized geometric axioms. In short, these are formulae of the form , where each (for ) stands for a conjunction of generalized geometric axioms, inductively constructed up to -depth with the base case being a geometric axiom. For a formal treatment of these axioms and rules see [22].)

## 3 The Logic Tstit

### 3.1 Axiomatization for Tstit

The logic extends the logic through the incorporation of tense modalities and the modality for the grand coalition of agents (see definition 1). This additional expressivity allows for the application of in settings where one wishes to reason about the joint action of all agents, or the consequences of choices over time. The logic was originally proposed in [17] as a Hilbert system, in this section we provide a corresponding cut-free calculus.

###### Definition 7 (The Logic Tstit[17])

The Hilbert system for the logic is defined as the logic extended with the following axioms and inference rules:

 [Ag]ϕ→ϕ ⟨Ag⟩ϕ→[Ag]⟨Ag⟩ϕ ⋀1≤i≤n[i]ϕi→[Ag]⋀1≤i≤nϕi ϕ→GPϕ
 ϕ→HFϕ Gϕ→Fϕ FFϕ→Fϕ FPϕ→Pϕ∨ϕ∨Fϕ PFϕ→Pϕ∨ϕ∨Fϕ
 Gϕ∨F¯¯¯ϕ Hϕ∨P¯¯¯ϕ [Ag]ϕ∨⟨Ag⟩¯¯¯ϕ α(ϕ→ψ)→(αϕ→αψ) for α∈{G,H,[Ag]}
 F◊ϕ→⟨Ag⟩Fϕ ϕ Gϕ ϕ Hϕ (□¬p∧□(Gp∧Hp))→ϕ with p∉ϕ ϕ

A derivation of in from a set of premises , is written as . When is the empty set, we refer to as a theorem and write .

Note that the axiom characterizes the no choice between undivided histories property (definition 2, C6). Furthermore, the last inference rule, a variation of Gabbay’s irreflexivity rule [10], characterizes the property of -irreflexivity (definition 2, C7). For a discussion of all axioms and rules see [17].

###### Theorem 3.1 (Soundness and Completeness [17])

For any formula , iff .

### 3.2 A Cut-free Labelled Calculus for Tstit

Let be a countable set of labels. The language of is defined as follows:

 Γ::=x:ϕ | Γ,Γ | Rαxy,Γ

where , , and . On the basis of this language, we construct the calculus as an extension of .

###### Definition 8 (The Calculus G3Tstit)

The labelled calculus consists of all the rules of extended with the following set of rules:

 RHwu,RGuw,Γ (convH) RHwu,Γ Γ,RHwu,w:Pϕ,u:ϕ (P) Γ,RHwu,w:Pϕ (compG1) RGwu,˘RGwu,Γ
 Γ,RGwv,v:ϕ (G)∗ Γ,w:Gϕ Γ,RGwu,w:Fϕ,u:ϕ (F) Γ,RGwu,w:Fϕ RGwu,RHuw,Γ (convG) RGwu,Γ
 Γ,RAgwu,w:⟨Ag⟩ϕ,u:ϕ (⟨Ag⟩) Γ,RAgwu,w:⟨Ag⟩ϕ RAgww,Γ (reflAg) Γ w=w,Γ (refl=) Γ
 RGuv,RGwu,RGwv,Γ u=v,RGwu,RGwv,Γ RGvu,RGwu,RGwv,Γ (connG) RGwu,RGwv,Γ
 RHuv,RHwu,RHwv,Γ u=v,RHwu,RHwv,Γ RHvu,RHwu,RHwv,Γ (connH) RHwu,RHwv,Γ
 RGwu,R□uz,RAgwv,RGvz,Γ (ncuh)∗ RGwu,R□uz,Γ RGwu,Γ ˘RGwu,Γ (compG2) Γ
 RGwu,RGuv,RGwv,Γ (transG) RGwu,RGuv,Γ RAgwu,Riwu,Γ (agd) RAgwu,Γ Γ,RHwv,v:ϕ (H)∗ Γ,w:Hϕ
 RGwv,Γ (serG)∗ Γ RAgwu,RAgwv,RAguv,Γ (euclAg) RAgwu,RAgwv,Γ R□wu,˘RGwu,Γ (irrG) R□wu,Γ
 w=u,Δ[w],Δ[u],Γ (sub=) w=u,Δ[w],Γ w=u,w=v,u=v,Γ (eucl=) w=u,w=v,Γ Γ,RAgwv,v:A ([Ag])∗ Γ,w:[Ag]A

For , , , , and the ‘’ states that must be an eigenvariable.

We note that the rules and express the converse relation between and , and the rules , , , and correspond to conditions (C3)-(C7) of definition 2, respectively. Furthermore, the notation in the substitution rule is used to express a collection of relational atoms and labelled formulae where all occurrences of the label in have been replaced by occurrences of . This notation uniformly captures all of the substitution rules given in [20].

###### Theorem 3.2 (Soundness)

Every sequent derivable in is valid.

###### Proof

Similar to theorem 2.3.

Unfortunately, with respect completeness, we cannot use the relatively simple strategy applied in proving completeness. This is because the irreflexivity rule (def. 7) does not readily lend itself to derivation in . Here we prove completeness relative to irreflexive -frames by leveraging the methods presented in [21]. (NB. For this reason, we introduced –the complement of –directly into the language of the proof system.)

###### Lemma 2

Let be a -sequent. Either, is -derivable, or it has a -countermodel.

###### Proof

We construct the Reduction Tree (RT) of a given sequent , following the method of [21]. If RT is finite, all leaves are initial sequents that are conclusions of or . If RT is infinite, by König’s lemma, there exists an infinite branch: , , …, ,… (with ). Let = . We define a -model as follows: Let iff . (Usage of the rules () and () in the infinite branch ensure is an equivalence relation.) Define to consist of all equivalence classes of labels in under . For each let (with ), and for each labelled propositional atom , let . It is a routine task to show that all relations and the valuation are well-defined. Last, let the interpretation map each label to the class of labels containing , and suppose maps all other labels not in arbitrarily. We show that: (i) is a model, and (ii) is a counter-model for .

(i) First, we assume w.l.o.g. that because the empty sequent is not satisfied on any model. Thus, there must exist at least one label in ; i.e. .

We argue that is an equivalence relation and omit the analogues proofs showing that and are equivalence relations. Suppose, for some in the infinite branch there occurs a label but . By definition of RT, at some later stage the rule will be applied; hence, . The argument is similar for the rule. Properties (C1) and (C2) follow from the rules and , respectively. Regarding (C3), we only obtain in via the rule. Using lemma 9 of [17], we can transform into a model where (i) and where (ii) the model satisfies the same formulae.

We obtain that is transitive and serial due to the and rules. is the converse of by and . The properties (C4), (C5) and (C6) follow from the rules , and , respectively.

(C7) follows from , , and the equality rules: these rules ensure that () if , then