Digital Object Identifier (DOI) 10.1007/s00236-002-0098-zActa Informatica 39, 71–96 (2003)

c© Springer-Verlag 2003

A functional approach for temporal × modal logics

Alfredo Burrieza, Inma P. de Guzman

Dept. de Matematica Aplicada, Univ. de Malaga, Campus de Teatinos, 29071 Malaga, Espana

Received: 12 November 2001 / 18 September 2002

Abstract. This work is focused on temporal × modal logics. We study therepresentation of properties of functions of interest because of their possi-ble computational interpretations. The semantics is exposed in an algebraicstyle, and the definability of the basic properties of the functions is anal-ysed. We introduce minimal systems for linear time with total functions aswell as with a class of partial functions (those with uniform domain). More-over, completeness proofs are offered for these minimal systems. Finally,functional-validity is compared with T ×W -validity and Kamp-validity.

1 Introduction

In recent years, several combinations of tense and modality (i.e theT × W -logics) have been introduced. The main interest of this in-vestigation is focused on fields such as causation, the theory of ac-tion and others (see [Van Fraassen(1981)], [ThomasonGupta(1981)],[Thomason(1984)], [BelPer(1990)], [Chellas(1992)], [Kuschera(1993)],[Belnap(1996)], [Reynolds(1997)]). However, our interest for this type ofcombinations is in the field of Mathematics and Computer Science.

In this paper we present a new type of frames, which we call functionalframes. Concretely, a functional frame is characterized by the following:

1. A nonempty set W of worlds.2. A nonempty set of pairwise disjoint strict linear orderings, indexed byW .

72 A. Burrieza, I.P. de Guzman

3. A set of non empty partial functions (called accessibility functions),with the characteristic that at most one accessibility function is definedbetween two of the arbitrary orders.

An example follows:

w′′ •6�

����

•�

7•�

����

8•�

����

9

w′ •

�

3•

4•

�

5

w

�

��•

�

1•

�

��2

�

W = {w,w′, w′′}, the linear temporal flows, (Tw, <w), (Tw′ , <w′),(Tw′′ , <w′′), where Tw = {1w, 2w}, Tw′ = {3w′ , 4w′ , 5w′} and Tw′′ ={6w′′ , 7w′′ , 8w′′ , 9w′′}. There are four accessibility functions:

w w−→: Tw → Tw; w w′−→: Tw → Tw′ ; w′ w′′

−→ : Tw′ → Tw′′

andw′′ w′−→ : Tw′′ → Tw′ , defined as follows:

w w−→= idTw (identity on Tw);w w′−→ (1w) = 3w′ ,

w w′−→ (2w) = 4w′ ;

w′ w′′−→ (3w′) = 6w′′ ,

w′ w′′−→ (5w′) = 8w′′ ;

w′′ w′−→ (6w′′) =w′′ w′

−→ (7w′′) =w′′ w′−→ (8w′′) = 4w′ ,

w′′ w′−→ (9w′′) = 5w′ .

The semantics based on this type of frame is called functional semantics.The theoretical interest of this approach is that it allows to study the defin-ability of basic properties of the theory of functions (such as being injective,surjective, increasing, decreasing, etc.); concretely, we study properties ofclasses of functions without the need to resort to second order theories and,in addition, it is a general purpose tool for studying logics containing modaland temporal operators. The practical motivation is their use in computa-tional applications. To this respect, the combinations of modal and tenseoperators is mentioned, for example, in [Reynolds(1997)] as a suitable toolto treat parallel processes, distributed systems and multiagents. Specifically,we can consider time-flows as memory of computers connected in a net, eachcomputer with its own clock.

Traditional approaches in this field are Thomason’s T ×W -frames andalso Kamp-frames, and both are special cases of the approach we introducehere, in the sense that, if we consider the restrictions imposed on the T ×W(Kamp)-models in [Thomason(1984)], then every T × W (Kamp)-modelhas an equivalent functional model (i.e. the same formulas are valid in bothmodels). If those restrictions are ignored, as in [Zanardo(1996)], then for allT ×W (Kamp)-frame there exists an equivalent functional frame.

The most remarkable differences between the functional approach andthe previous ones are commented on below:

A functional approach for temporal × modal logics 73

– In a T × W -frame, the flow of time is shared by all the worlds. Ina Kamp-frame each world has its own flow of time, although for twodifferent worlds the time can coincide up to an instant. In our functionalframes each world has its own flow of time and it is not required thatflows of time for different worlds have to be neither totally nor partiallyisomorphic.

– In both T × W -frames and Kamp-frames the flow of time is strictlylinear. In this paper, we will also use linear time in our functional frames.Nevertheless, this is not a mandatory requirement.

– It is typical in both, T ×W -frames and Kamp-frames, that worlds areconnected by equivalence relations defined for their respective tempo-ral orders and, as a consequence, the axioms of the modal basis of S5are valid for our operators of necessity and possibility. In our approachthere is more flexibility, the inter-world accessibility is defined by partialfunctions (accessibility functions). These functions allow us to connectthe worlds and so, to compare the measure of different courses of timein several ways.

– The equivalence relations are used in previous approaches to establishwhich segments of different temporal flows are to be considered as thesame history. This way, it is possible to define the notion of historicalnecessity. This notion can be defined using functional frames too, as itwill be shown, but in addition, functions allow to define situations whichdo not require, neither partial nor total, isomorphisms between tempo-ral flows (thinking again of flows of time as interconnected computermemories, that is not a requirement of their respective clocks).

The paper is structured as follows: in Sect. 2 we define the temporal× modal functional logic (LF

T×W ). The language of this logic is calledLT×W and its semantics is a functional semantics. In Sect. 3 we analyze thedefinability of the basic properties of the functions. In Sect. 4 we introduceminimal axiomatic systems for LT×W and their completeness proofs aregiven. In Sect. 5 we prove that the T ×W -frames and the Kamp-frames areparticular cases of our approach. Finally, in Sect. 6, future work is outlined.

2 The Logic LFT ×W

In this section we define the language LT×W and the functional semantics.

The alphabet of LT×W is defined as follows:

– an denumerable set, V , of propositional variables;– the constants � and ⊥, and the classical connectives ¬ and→;– the temporal connectives G and H , and the modal connective ��

74 A. Burrieza, I.P. de Guzman

The well-formed formulae (wffs) are generated by the construction rules ofclassical propositional logic adding the following rule: If A is a wff, thenGA, HA and �A are wffs.

We consider, as usual, the connectives∧,∨F ,P and � to be defined con-nectives. The connectives G, H , F , and P have their usual readings, but �

has the following meaning: �A is read “A is true at every accessible present”(in the above example, the accessible present for 1w is {1w, 3w′}, etc.). Onthe other hand, the notion of a mirror image of a formula is considered inthe usual way.

Definition 1. We define a functional frame forLT×W as a tuple (W, T ,F),where:

– W is a nonempty set (set of labels for a set of temporal flows).– T is a nonempty set of strict linear orders, indexed by W . Specifically:

T = {(Tw, <w) | w ∈W} such that Tw = ∅ for all w ∈W, andif w = w′, then Tw ∩ Tw′ = ∅.

– F is a set of non-empty functions, called accessibility functions, suchthat:

a) each function inF is a partial function from Tw to Tw′ , for somew,w′ ∈W .

b) for an arbitrary pairw,w′ ∈W , there is (inF) at most one accessibility

function from Tw to Tw′ , denoted byw w′−→ .

We will denote Fw = {w w′−→ ∈ F | w′ ∈W}. Then F =

⋃w∈W Fw.

LetΣ = (W, T ,F) be a functional frame. The elements tw of the disjointunion CoordΣ =

⊕w∈W Tw are called coordinates.

Remark 1. At this point it is worth to note the following:

(i) items a) and b) of Definition 1 ensures that for each subset of coordi-

nates X ⊆ Tw, the union Fw(X) =⋃

w w′−→ ∈Fw

w w′−→ (X) is disjoint.

(ii) The set of accessibility functions, F , is not necessarily closed undercomposition, as we can note in the example above.

Notation 1 If (A,≤) is a nonempty linearly ordered set and a ∈ A:

[a,→) = {a′ ∈ A | a ≤ a′}; (a,→)={a′ ∈ A | a < a′}.(←, a] = {a′ ∈ A | a′ ≤ a}; (←, a)={a′ ∈ A | a′ < a}.In particular, if tw ∈ CoorΣ and C ⊆ CoorΣ:

C ↑∗=⋃

tw∈ C(tw,→); C ↑=⋃

tw∈ C [tw,→).C ↓∗=

⋃tw∈ C(←, tw); C ↓=

⋃tw∈ C(←, tw].

A functional approach for temporal × modal logics 75

Theorem 1. If (A,≤A) and (B,≤B) are nonempty linearly ordered sets,f : A −→ B a nonempty partial function and Dom(f) its domain,1 thenwe have:

1. f is injective if and only if, for all a ∈ Dom(f), we have:f((←, a)) ∪ f((a,→)) ⊆ (←, f(a)) ∪ (f(a),→).

2. f is surjective if and only if, for all a ∈ A, we have:(←, f({a})) ∪ (f({a}),→) ⊆ f((←, a)) ∪ f((a,→)).

3. f is increasing (resp. decreasing) if and only if, for all a ∈ Dom(f), wehave:

f((a,→)) ⊆ [f(a),→) ( resp. f((a,→)) ⊆ (←, f(a)] ).4. f is strictly increasing (resp. strictly decreasing) if and only if, for alla ∈ Dom(f), we have:

f((a,→)) ⊆ (f(a),→) ( resp. f((a,→)) ⊆ (←, f(a)) ).5. f is constant if and only if, for all a ∈ Dom(f), we have:

f((←, a)) ∪ f((a,→)) ⊆ {f(a)}.

The following theorems characterize functional frames,Σ = (W, T ,F),in an algebraic style attending to the properties of the class of functions F .

Lemma 1. LetΣ = (W, T ,F) be a functional frame andw w′−→∈ Fw. Then,

if tw ∈ Tw we have tw ∈ Dom(w w′−→) if and only if the following inclusion

(∗1) is satisfied:

w w′−→ ((←, tw))∪ w w′

−→ ((tw,→))(∗1)⊆ (←,w w′

−→ ({tw}))∪ [w w′−→ ({tw}),→ )

Proof. If tw ∈ Dom(w w′−→), (∗1) is obviously true, because:

(←,w w′−→ ({tw})) ∪ [ w w′

−→ ({tw}),→ ) = Tw′

Reciprocally, if tw ∈ Dom(w w′−→), we obtain

(←,w w′−→ ({tw})) ∪ [w w′

−→ ({tw}),→ ) = ∅

Now, sincew w′−→∈ Fw is non empty, then there exists t′w ∈ Dom(w w′

−→) and,

as a consequence,w w′−→ ((←, tw)) ∪ w w′

−→ ((tw,→)) = ∅, thus, (∗1) failsto be true.

Notation 2 Given a functional frame (W, T ,F), for all w ∈ W , we shalldenote

Dom(Fw) =⋃

w w′−→∈Fw

Dom(w w′−→)

1 If f : A −→ B is a partial function from A to B and X ⊆ A, we define, as usual:f(X) = {f(x) | x ∈ X ∩Dom(f)}. Specifically, if a �∈ Dom(f), then f({a}) = ∅.

76 A. Burrieza, I.P. de Guzman

Definition 2. Let (W, T ,F) be a functional frame. We say that F satisfies

the (U-Dom) condition if for all w ∈ W and allw w′−→, w w′′

−→∈ Fw we havethat:

(U-Dom) Dom(w w′−→) = Dom(w w′′

−→ )

This common domain is denoted by DomU (Fw).

Theorem 2. Let Σ = (W, T ,F) be a functional frame. Then, F is aclass of total functions (resp. (U-Dom) functions) if and only if for alltw ∈ CoordΣ(resp. tw ∈ Dom(Fw)), the following inclusion (Fw-∗1)is obtained:

Fw((←, tw)) ∪ Fw((tw,→))(Fw-∗1)⊆ Fw({tw})↓∗ ∪Fw({tw}) ↑.

Proof. We consider the case of total functions. The other case is similar.

Let F be a class of total functions. By Lemma 1, for allw w′−→∈ Fw and

tw ∈ CoordΣ , the following relation holds:

w w′−→ ((←, tw))∪ w w′

−→ ((tw,→))(∗1)⊆ (←,w w′

−→ ({tw}))∪ [w w′−→ ({tw}),→ )

Now, (Fw-∗1) is true, since it is an immediate consequence of (∗1) and item(i) of Remark 1, which guarantees that the expression Fw(X) is a disjointunion. For the converse we reason in a similar way from inclusion (∗1).

As an immediate consequence of Theorems 1 and 2, and Remark 1 weobtain the following result.

Theorem 3. Let Σ = (W, T ,F) be a functional frame. Then,

1. F is a class of total injective functions (resp. (U-Dom) injective functions)if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw)), we obtainthe following inclusion (inj):

Fw((←, tw)) ∪ Fw((tw,→))inj⊆ Fw({tw})↓∗ ∪Fw({tw})↑∗

2. F is a class of surjective functions if and only if, for all tw ∈ CoordΣ ,we obtain the following inclusion (surj):

Fw({tw})↓∗ ∪Fw({tw})↑∗surj⊆ Fw((←, tw)) ∪ Fw((tw,→))

3. F is a class of total increasing functions (resp. (U-Dom) increasing func-tions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw)), weobtain the following inclusions (inc+) and (inc−):

Fw((tw,→))inc+

⊆ Fw({tw}) ↑ and Fw((←, tw))inc−⊆ Fw({tw}) ↓

A functional approach for temporal × modal logics 77

4. F is a class of total strictly increasing functions (resp. (U-Dom) strictlyincreasing functions) if and only if, for all tw ∈ CoordΣ (resp.tw ∈ Dom(Fw)), we obtain the following inclusions (str-inc+) and(str-inc−):

Fw((tw,→))str-inc+

⊆ Fw({tw})↑∗ and Fw((← ,tw))str-inc−⊆ Fw({tw})↓∗

5. F is a class of total decreasing functions (resp. (U-Dom) decreasingfunctions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw)),we obtain the following inclusions (dec+) and (dec−):

Fw((tw,→))dec+

⊆ Fw({tw})↓ and Fw((← ,tw))dec−⊆ Fw({tw})↑

6. F is a class of total strictly decreasing functions (resp. (U-Dom) strictlydecreasing functions) if and only if, for all tw ∈ CoordΣ (resp.tw ∈ Dom(Fw)), we obtain the following inclusions (str-dec+) and(str-dec−):

Fw((tw,→))str-dec+

⊆ Fw({tw})↓∗ and Fw((← ,tw))str-dec−⊆ Fw({tw})↑∗

7. F is a class of total constant functions (resp. (U-Dom) constant functions)if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw)), we obtainthe following inclusion (con):

Fw ((←, tw)) ∪ Fw((tw,→))con⊆ Fw({tw})

Definition 3. A functional model on Σ is a tupleM = (Σ, h), where Σis a functional frame and h : LT×W −→ 2CoordΣ is a function, called afunctional interpretation, satisfying:

h(⊥) = ∅;h(�) = CoordΣ; h(¬A) = CoordΣ − h(A);h(A→ B) = (CoordΣ − h(A)) ∪ h(B)h(GA) = {tw ∈ CoordΣ | (tw,→) ⊆ h(A)}h(HA) = {tw ∈ CoordΣ | (←, tw) ⊆ h(A)}h(�A) = {tw ∈ CoordΣ | Fw({tw}) ⊆ h(A)}.

Definition 4. LetA be a formula inLT×W . Let (Σ, h) be a functional modeland tw ∈ CoordΣ , then, A is true at tw if tw ∈ h(A). A is said to be validin the functional model (Σ, h) if h(A) = CoordΣ . If A is valid in everyfunctional model onΣ, thenA is said to be valid in the functional frameΣ,and denote it by |=Σ A. If A is valid in every functional frame, then A issaid to be valid, and denote it by |= A. Let K be a class of functional frames,if A is valid in every functional frame Σ such that Σ ∈ K, then A is said tobe valid in K.

78 A. Burrieza, I.P. de Guzman

3 Definability in LFT ×W

Definition 5. Let J be a class of functional frames and let K ⊆ J. We saythat K is LF

T×W -definable in (or relative to) J if there exists a set Γ offormulas in LT×W such that for every frame Σ ∈ J, we have that Σ ∈ K ifand only if every formula of Γ is valid inΣ. If J is the class of all functionalframes, we say that K is LF

T×W -definable.Let P be a property of functions (injectivity, etc) and K the class of all

functional frames whose functions have the property P . We say that P isLF

T×W -definable if K is LFT×W -definable.

In the rest of this paper, “definable” means “LFT×W -definable”.

Theorem 4. K = {(W, T ,F) | F is a class of total functions } is defina-ble.

Proof. We will prove that the formula

(Tot) �(HA ∧A ∧GA)→ (H�A ∧G�A)

defines K. To this end, let (Σ, h) be such that Σ ∈ K and tw ∈ CoordΣ .Then:

tw ∈ h(�(HA ∧A ∧GA)) ifftw ∈ h(�HA) ∩ h(�A) ∩ h(�GA) iffFw({tw}) ↓∗ ∪Fw({tw}) ∪ Fw({tw}) ↑∗ ⊆ h(A) iffFw({tw}) ↓∗ ∪ Fw({tw}) ↑⊆ h(A))

On the other hand, tw ∈ h(H�A) iff Fw((←, tw)) ⊆ h(A) and tw ∈h(G�A) iff Fw((tw,→)) ⊆ h(A). Now, by the inclusion (Fw-∗1) in The-orem 2, the proof of validity is complete.

Conversely, assume Σ /∈ K, then there exists tw ∈ CoordΣ such that

Fw((←, tw)) ∪ Fw((tw,→)) � Fw({tw})↓∗ ∪Fw({tw}) ↑

Let (Σ, h) be a functional model whereh(p) = Fw({tw})↓∗ ∪Fw({tw}) ↑.Then, we obtain

tw ∈ h(�(Hp ∧ p ∧Gp)) and Fw((←, tw)) ∪ Fw((tw,→)) � h(p)

Therefore tw ∈ h(H�p ∧G�p) and the proof is complete.

Theorem 5. K={(W, T ,F) | F is a class of (U-Dom)-partial functions}is definable.

Proof. We will prove that

(UD) �(HA ∧A ∧GA)→ (�⊥ ∨ (H�A ∧G�A))

defines K. The validity of (UD) in K is an immediate consequence of:

A functional approach for temporal × modal logics 79

i) if tw ∈ DomU (Fw), then tw ∈ h(�⊥), andii) if tw ∈ DomU (Fw), then the inclusion (Fw-∗1), in Theorem 2, holds

for tw.

Reciprocally, if Σ = (W, T ,F) ∈ K, then there exists tw ∈ Dom(Fw)such that

Fw((←, tw)) ∪ Fw((tw,→)) � Fw({tw})↓∗ ∪ Fw({tw}) ↑

Now, a countermodel can be defined in a similar way as for (Tot) in thetheorem above and the proof is complete.

Theorem 6. K = {(W, T ,F) | F is a class of (U-Dom)-non-totalfunctions} is definable.

Proof. We will prove that the following set defines K:

{�(HA ∧A ∧GA)→ (�⊥ ∨ (H�A ∧G�A)),�P⊥ ∨�⊥ ∨ F�⊥}

The validity of �(HA ∧ A ∧ GA) → (�⊥ ∨ (H�A ∧ G�A)) in Kis guaranteed by Theorem 5, since the class of (U-Dom)-non-total func-tions is a subclass of the class of (U-Dom)-partial functions. On the otherhand, for every frame in K, we have that for all Tw there is at least onetw ∈ Tw such that tw ∈ DomU (Fw). Clearly, tw ∈ h(�⊥) if and only iftw ∈ DomU (Fw). Thus, from the linearity of<w, we have that the formula(Non-Tot) : �P⊥ ∨ �⊥ ∨ F�⊥ holds at every coordinate in Tw. Forthe reverse, let (W, T ,F) be a functional frame not in K. We consider thefollowing two possible cases:

(i) There exists some total functionw w′−→∈ F . Then �⊥ fails to be true at

every coordinate in Tw (whatever the model based on that frame) andtherefore �P⊥ ∨�⊥ ∨ F�⊥ is not valid in that frame.

(ii) F does not satisfy the (U-Dom) condition. In this case, we work as inTheorem 5.

Theorem 7. The following classes of functional frames are definable:

(1) K1 = {(W, T ,F) | F is a class of total injective functions }.(2) K2 = {(W, T ,F) | F is a class of surjective functions }.(3) K3 = {(W, T ,F) | F is a class of total increasing functions }.(4) K4 = {(W, T ,F) | F is a class of total strictly increasing functions }.(5) K5 = {(W, T ,F) | F is a class of total decreasing functions }.(6) K6 = {(W, T ,F) | F is a class of total strictly decreasing functions }.(7) K7 = {(W, T ,F) | F is a class of total constant functions }.

80 A. Burrieza, I.P. de Guzman

Proof. (1) We will prove that �(HA ∧ GA) → (H�A ∧ G�A) definesK1. For this, let (Σ, h) be such that Σ ∈ K1, then for any tw ∈ CoordΣ wehave that:

tw ∈ h(�(HA ∧GA)) ifftw ∈ h(�HA) ∩ h(�GA) iffFw({tw}) ↓∗ ∪Fw({tw}) ↑∗⊆ h(A)

On the other hand, tw ∈ h(H�A) if and only if Fw((←, tw)) ⊆ h(A)and tw ∈ h(G�A) if and only if Fw((tw,→)) ⊆ h(A). Now, by (inj) inTheorem 3, the proof of validity is complete.

Reciprocally, if Σ ∈ K1, a countermodel for the formula

�(HA ∧GA)→ (H�A ∧G�A)

can be defined in a similar way as for (Tot) in theorem 4, and the proof iscomplete.(2) The formula (H�A ∧ G�A) → �(HA ∧ GA) defines K2. This is aconsequence of (surj) in Theorem 3.(3) The set {�(A∧GA)→ G�A,�(HA∧A)→ H�A} defines K3. Thisis a consequence of (inc+) and (inc−) in Theorem 3.(4) The set {�GA → G�A, �HA → H�A} defines K4. This is a con-sequence of (str-inc+) and (str-inc−) in Theorem 3.(5) The set {�(A ∧ GA) → H�A, �(HA ∧ A) → G�A} defines K5.This is a consequence of (dec+) and (dec−) in Theorem 3.(6) The set {�GA→ H�A, �HA→ G�A} defines K6. This is a conse-quence of (str-dec+) and (str-dec−) in Theorem 3.(7) The formula �A→ (H�A∧G�A) defines K7. This is a consequenceof (con) in Theorem 3.

Theorem 8. Let K be a class of functional frames, (W, T ,F), whereF is aclass of functions that satisfies the (U-Dom) condition and in which all func-tions are injective (respectively: constant, increasing, strongly increasing,decreasing, strongly decreasing). Then K is definable.

Proof. It is sufficient to enumerate the properties of functions and theircorresponding formulas as follows:

(UD-Inj): �(HA ∧GA)→ (�⊥ ∨ (H�A ∧G�A))(UD-Inc): {�(A∧GA)→ (�⊥∨G�A),�(HA∧A)→ (�⊥∨H�A)}(UD-Str-Inc): {�GA→ (�⊥ ∨G�A),�HA→ (�⊥ ∨H�A)}(UD-Dec): {�(A∧GA)→ (�⊥∨H�A),�(HA∧A)→ (�⊥∨G�A)}(UD-Str-Dec): {�GA→ (�⊥ ∨H�A),�HA→ (�⊥ ∨G�A)}(UD-Con): �A→ (�⊥ ∨ (H�A ∧G�A))

A functional approach for temporal × modal logics 81

4 Minimal axiomatic systems for LT ×W

4.1 The system ST×W -Tot

This system has the following axiom schemes:

1. All tautologies of the classical propositional logic, PL.2. Those of the minimal system of propositional linear temporal logic Kl:

(G1) G(A→ B)→ (GA→ GB)(H1) H(A→ B)→ (HA→ HB)(G2) A→ GPA(H2) A→ HFA(TR) GA→ GGA(L+) (G(A ∨B) ∧G(A ∨GB) ∧G(GA ∨B))→ (GA ∨GB)(L−) (H(A ∨B) ∧H(A ∨HB) ∧H(HA ∨B))→ (HA ∨HB)

3. The characteristic axiom schema of normal modal propositional logicK: (K) �(A→ B)→ (�A→ �B)

4. The characteristic axiom schema:(Tot) �(HA ∧A ∧GA)→ (H�A ∧G�A)

The rules of inference are those of Kl +K:

(MP ): A,A→ B � B; (RG): A � GA; (RH): A � HA and(RN ) A � �A

The concepts of proof and theorem are defined as usual.

Theorem 9. The following formulas are theorems in ST×W -Tot,

T1: F�A→ �(PA ∨A ∨ FA). T2: P�A→ �(PA ∨A ∨ FA).

Proof. We prove the theorem T1 (T2 is its mirror image).

Let ψ = H¬A ∧ ¬A ∧G¬A1. �(H¬ψ ∧ ¬ψ ∧G¬ψ) → H�¬ψ from (Tot) and PL2. P♦ψ → (♦Pψ ∨ ♦ψ ∨ ♦Fψ) from 1, by K and definitions of ♦, F, P3. ψ → G¬A by PL4. ♦Pψ → ♦PG¬A from 3, by Kl and K5. ♦PG¬A → ♦¬A by Kl and K6. ♦Pψ → ♦¬A from 4, 5 by PL7. ♦ψ → ♦¬A by K8. ♦Fψ → ♦¬A [similar to 3-6]9. P♦ψ → ♦¬A from 2, 6, 7, 8 by PL10. �A → H�¬ψ from 9 by PL and definitions of ♦ and P11. ¬ψ → (PA ∨A ∨ FA) by PL and definitions of F and P12. H� ¬ψ → H�(PA ∨A ∨ FA) from 11 by K and Kl

13. �A → H�(PA ∨A ∨ FA) from 10, 12 by PL14. F�A → �(PA ∨A ∨ FA) from 13 by Kl

82 A. Burrieza, I.P. de Guzman

Completeness theorem for ST×W -Tot

The proof of soundness is standard. We focus our attention on the complete-ness proof and adopt the usual definitions of a consistent (maximal) set offormulas. In the following, we abbreviate maximal consistent set asmc-set.On the other hand, by S we mean any axiomatic system for LT×W which isan extension of Kl + K (in particular, the system ST×W -Tot). Familiaritywith the basic properties of mc-sets in classical propositional systems isassumed.

Definition 6. Let Γ1 and Γ2 be mc-sets in S, then we define:

Γ1 ≺T Γ2 if and only if {A | GA ∈ Γ1} ⊆ Γ2.Γ1 ≺W Γ2 if and only if {A | �A ∈ Γ1} ⊆ Γ2

The following lemmas 2–4 are standard in modal and tense logic.

Lemma 2. Let Γ1 and Γ2 be mc-sets in S, then we have:

(a) Γ1 ≺T Γ2 iff {FA | A ∈ Γ2} ⊆ Γ1 iff {PA | A ∈ Γ1} ⊆ Γ2iff {A | HA ∈ Γ2} ⊆ Γ1.

(b) Γ1 ≺W Γ2 iff {♦A | A ∈ Γ2} ⊆ Γ1.(c) (Lindenbaum’s Lemma) Any consistent set of formulas in S can be

extended to an mc-set in S.

Lemma 3. Let Γ1 be an mc-set in S, then we have:

(a) If FA∈Γ1, exists an mc-set Γ2∈S such that Γ1≺T Γ2 and A∈Γ2.(b) If PA∈Γ1, exists an mc-set Γ2∈S such that Γ2≺T Γ1 and A∈Γ2.(c) If ♦A∈Γ1, exists an mc-set Γ2∈S such that Γ1≺W Γ2 and A∈Γ2.

Lemma 4. Let Γ1, Γ2, Γ3 be mc-sets in S.

(a) If Γ1 ≺T Γ2 and Γ2 ≺T Γ3, then Γ1≺T Γ3.(b) If Γ1 ≺T Γ2 and Γ1 ≺T Γ3, then Γ2 =Γ3 or Γ2≺T Γ3 or Γ3 ≺T Γ2.

The following lemma is specific to our system for total functions.

Lemma 5. Let Γ1, Γ2, Γ3 be mc-sets in ST×W -Tot, then we have:

(a) If Γ1 ≺T Γ2 and Γ1 ≺W Γ3, then there exists an mc-set, Γ4, such thatΓ2 ≺W Γ4 and, either Γ3 = Γ4 or Γ3 ≺T Γ4 or Γ4 ≺T Γ3.

(b) If Γ1 ≺T Γ2 and Γ2 ≺W Γ3, then there exists an mc-set, Γ4, such thatΓ1 ≺W Γ4 and, either Γ3 = Γ4 or Γ3 ≺T Γ4 or Γ4 ≺T Γ3.

Proof. We prove (a). The proof of (b) is similar. Let Γ1 ≺T Γ2 and Γ1 ≺W

Γ3. To construct Γ4, it suffices to prove that one of the following conditionsis satisfied:

(i) Γ2 ≺W Γ3. In this case, we obtain the result taking Γ4 = Γ3.

A functional approach for temporal × modal logics 83

≺W���

Γ2≺T �

��

≺W

Γ1

Γ4 = Γ3

(ii) {A | �A ∈ Γ2} ∪ {A | GA ∈ Γ3} is consistent. In this case, Linden-baum’s lemma guarantees that there exists at least one mc extension,Γ4, of the set {A | �A ∈ Γ2} ∪ {A | GA ∈ Γ3} which triviallysatisfies Γ2 ≺W Γ4 and Γ3 ≺T Γ4.

�≺T

≺W�

Γ2≺T �

�≺W

Γ1

Γ4Γ3

(iii) {A | �A ∈ Γ2} ∪ {A | HA ∈ Γ3} is consistent. Now, Lindenbaum’slemma again guarantees that there exists at least one mc extension, Γ4,of the set {A | �A ∈ Γ2}∪{A | HA ∈ Γ3}which satisfiesΓ2 ≺W Γ4and Γ4 ≺T Γ3.

≺W

≺T

� �

≺TΓ2

�

�

≺W

Γ1

Γ4 Γ3

If we assume that none of the conditions (i)–(iii) holds, we have:a) there exists a formula �A ∈ Γ2 such that A ∈ Γ3;

b) there are

{B1, . . . , Bn1 ∈ {A | �A ∈ Γ2},C1, . . . , Cm1 ∈ {A | GA ∈ Γ3}

such that � ¬(B1 ∧ . . . ∧Bn1 ∧ C1 ∧ . . . ∧ Cm1);

c) there are

{D1, . . . , Dn2 ∈ {A | �A ∈ Γ2},E1, . . . , Em2 ∈ {A | HA ∈ Γ3}

such that � ¬(D1 ∧ . . . ∧Dn2 ∧ E1 ∧ . . . ∧ Em2).

Thus, we have:

B = B1 ∧ . . . ∧Bn1 , with �B ∈ Γ2;C = C1 ∧ . . . ∧ Cm1 , with GC ∈ Γ3;D = D1 ∧ . . . ∧Dn2 , with �D ∈ Γ2;E = E1 ∧ . . . ∧ Em2 , with HE ∈ Γ3

Now, from � ¬(B∧C), we obtain � �B → �¬C and (since �B ∈ Γ2)we have �¬C ∈ Γ2. Similarly, from � ¬(D ∧ E), we obtain �¬E ∈ Γ2.Thus, �(A∧¬C ∧¬E)∈Γ2 and, since Γ1≺T Γ2, we obtain F�(A∧¬C ∧¬E) ∈ Γ1. Therefore, by (T1) in Theorem 9, we obtain:

�(P (A ∧ ¬C ∧ ¬E) ∨ (A ∧ ¬C ∧ ¬E) ∨ F (A ∧ ¬C ∧ ¬E)) ∈ Γ1

Henceforth, by the definition of ≺W , either

84 A. Burrieza, I.P. de Guzman

(1) P (A ∧ ¬C ∧ ¬E) ∈ Γ3, or(2) (A ∧ ¬C ∧ ¬E) ∈ Γ3, or(3) F (A ∧ ¬C ∧ ¬E) ∈ Γ3.

However, HE ∈ Γ3 is contrary to (1); A ∈ Γ3 is contrary to (2), andGC ∈ Γ3 is contrary to (3). Since we obtain a contradiction in any case,one of these conditions (i)–(iii) is satisfied.

Definition 7. Let Σ = (W, T ,F) be a functional frame. A trace of Σ isany function ΦΣ : CoordΣ −→ 2LT×W such that, for all tw ∈ CoordΣ , theset ΦΣ(tw) is an mc-set.

Definition 8. Let ΦΣ be a trace of Σ = (W, T ,F). ΦΣ is called:temporally coherent if, for all tw, t′w ∈ CoordΣ:

if t′w ∈ (tw,→), then ΦΣ(tw) ≺T ΦΣ(t′w)modally coherent if, for all tw, t′w′ ∈ CoordΣ:

if t′w′ ∈ Fw({tw}), then ΦΣ(tw) ≺W ΦΣ(t′w′)coherent if it is temporally coherent and modally coherent.prophetic if it is temporally coherent and, moreover, for all wff, A, and alltw ∈ CoordΣ :(1) if FA ∈ ΦΣ(tw), there exists a t′w ∈ (tw,→) such that A ∈ ΦΣ(t′w)historic if it is temporally coherent and, moreover, for all wff, A, and alltw ∈ CoordΣ :(2) if PA ∈ ΦΣ(tw), there exists a t′w ∈ (←, tw) such that A ∈ ΦΣ(t′w)possibilistic if it is modally coherent and, moreover, for all wff, A, and alltw ∈ CoordΣ :(3) if ♦A∈ΦΣ(tw), there exists a t′w′ ∈Fw({tw}) such that A∈ΦΣ(t′w′).

The conditional (1) (resp., (2) or (3)) is called a prophetic (historic orpossibilistic) conditional for ΦΣ with respect to FA, (PA or ♦A) and tw.

Definition 9. LetΦΣ be a trace ofΣ = (W, T ,F).ΦΣ is called total if it iscoherent andΣ satisfies the following property: for all tw, t′w, t′w′ ∈ CoordΣ

(4) if t′w′ ∈Fw({tw}) and t′w = tw, then there exists t′′w′ ∈Fw({t′w}) 2

(4) is called a total conditional for ΦΣ with respect to tw, t′w and t′w′ . ΦΣ

is called total-full if it is prophetic, historic, possibilistic, and total.

Definition 10. Let WΞ be a denumerable infinite set and TΞ =⋃w∈WΞ

TΞw where TΞ

w is a denumerable infinite set for each w ∈ WΞ

and if w = w′, then TΞw ∩ TΞ

w′ = ∅. We consider the class, Ξ , of functionalframes, (W ′, T ′,F ′), such that:

– W ′ is a nonempty finite subset of WΞ .

2 That is, ΦΣ is called total when is a coherent trace of a total frame.

A functional approach for temporal × modal logics 85

– T ′ = {(T ′w, <

′w) | w ∈ W ′}, where T ′

w is a nonempty finite subset ofTΞ

w .

If Σ1 = (W1, T1,F1), Σ2 = (W2, T2,F2) ∈ Ξ , we say that Σ2 is anextension of Σ1 if the following conditions are satisfied:

•W1 ⊆W2;• either T1 ⊂ T2, or for every (Tw, <w) ∈ T1, the set T2 contains anextension of (Tw, <w);

• either F1 ⊂ F2 or, for everyw w′−→∈ F1, the set F2 contains a function

which extendsw w′−→.

Definition 11. Let Ξ be as in Definition 10 and let ΦΣ′ be a trace of afunctional frame Σ′ = (W ′, T ′,F ′) ∈ Ξ .

I) Let the prophetic conditional be:

(1) if FA ∈ ΦΣ′(tw), there exists a t′w ∈ (tw,→) such that A ∈ΦΣ′(t′w)

We say that the (1) is active, if its antecedent is fulfilled but its con-sequent is not, that is, tw ∈ CoordΣ′ , and FA ∈ ΦΣ′(tw), but there is not′w ∈ (tw,→) such thatA ∈ ΦΣ′(t′w). We say that (1) is exhausted, if its con-sequent is fulfilled, i.e. there exists a t′w ∈ (tw,→) such that A ∈ ΦΣ′(t′w).

The case for a historic conditional is defined in the same way as propheticconditional, by means of the obvious substitutions.

II) Let the possibilistic conditional be:

(2) if ♦A ∈ ΦΣ′(tw), there exists a t′w′ ∈ F ′w({tw}) such that

A ∈ ΦΣ′(t′w′)

We say that (2) is active, if its antecedent is fulfilled but its consequentis not, that is, tw ∈ CoordΣ′ and ♦A ∈ ΦΣ′(tw), but there is no t′w′ ∈F ′

w({tw}) such that A ∈ ΦΣ′(t′w′). We say that (3) is exhausted, if itsconsequent is fulfilled, that is, there exists a t′w′ ∈ F ′

w({tw}) such that A ∈ΦΣ′(t′w′).

III) Let the total conditional be:

(3) if t′w′ ∈ F ′w({tw}) and t′w = tw, then there exists t′′w′ ∈ F ′

w({t′w})

We say that (3) is active, if its antecedent is fulfilled but its consequentis not, that is, tw, t′w, t′w′ ∈ CoordΣ′ , t′w′ ∈ F ′

w({tw}), t′w = tw, butt′′w′ ∈ F ′

w({t′w}) does not exist. We say that (3) is exhausted, if its consequentis fulfilled, i.e. t′′w′ ∈ F ′

w({t′w}) exists.

86 A. Burrieza, I.P. de Guzman

Lemma 6 (trace lemma). LetΦΣ be a total-full trace of a functional frameΣ. Let h be a functional interpretation assigning each propositional vari-able, p, the set h(p) = {tw ∈ CoordΣ | p ∈ ΦΣ(tw)}. Then, for any wff,A, we have h(A) = {tw ∈ CoordΣ | A ∈ ΦΣ(tw)}.

Proof. By induction on the complexity of A.

In order to prove the completeness theorem, for each consistent formula,A, we will construct (using the class, Ξ , of functional frames in Defini-tion 10) a total functional frame Σ = (W, T ,F) and a total-full trace ΦΣ ,so that A ∈ ΦΣ(tw) for some tw ∈ CoordΣ . To do this, we will define:

– an enumeration of elements in WΞ : WΞ = {wi | i ∈ N}.– an enumeration of elements in TΞ =

⋃w∈WΞ

TΞw :

TΞ =⋃i∈N

TΞwi

; TΞwi

= {t(i,j) | j ∈ N}

– an enumeration of wffs of LT×W : A0, A1, . . . , An, . . .

Therefore, we can also assign a code number for each prophetic conditional(historic conditional, etc.) in the usual way.

Remark 2. Given a conditional for ΦΣ , if we simply replace the label Σwith Σ′ so that Σ ⊆ Σ′, we have a conditional for ΦΣ′ but with the samecode number as the conditional for ΦΣ . Then we can say that in both caseswe refer to the same conditional.

Now, given a consistent formula, A, the construction of Σ and ΦΣ gostep by step as follows:

We begin with a finite frame Σ0 = (W0, T0,F0) ∈ Ξ and a trace ΦΣ0 ,where:

W0 = {w0},T0 = {({t(0,0)},∅)},F0 = ∅,ΦΣ0(t(0,0)) = Γ0, where Γ0 is an mc-consistent set containing A.

Assume that Σn = (Wn, Tn,Fn) and ΦΣn are defined. Then, Σn+1 andΦΣn+1 are defined as follows:

• If all conditionals are not active, then Σn+1 = Σn, ΦΣn+1 = ΦΣn andthe construction is finished.• In other case, i.e., if there are prophetic conditionals (or historic, pos-

sibilistic) for ΦΣn with respect to FA (respectively, PA or ♦A) andtw, or there is a total conditional for ΦΣn with respect to tw, t′w, andt′w′ which are active, then we choose the conditional with the lowest

A functional approach for temporal × modal logics 87

code number and the lemma below ensures that there exists an extensionΣn+1 = (Wn+1, Tn+1,Fn+1) ∈ Ξ of Σn and an extension ΦΣn+1 ofΦΣn so that this conditional for ΦΣn+1 is exhausted.

The result is a denumerable sequence of finite functional frames (in Ξ)

Σ0, Σ1, ..., Σn, ...

whose union is Σ , and a denumerable sequence of corresponding traces

ΦΣ0 , ΦΣ1 , ..., ΦΣn , ...

whose union is ΦΣ .Each finite frame of the above sequence satisfies the condition of linearity

of orders and each trace of it is coherent, but in general, it fails to be prophetic,historic, possibilistic or total. However, as we shall show,Σ is such that ΦΣ

is total-full. Thus, A is verified by the trace lemma.

Lemma 7 (Exhausting lemma). Let Ξ be as in Definition 10, ΦΣn a co-herent trace of Σn ∈ Ξ , and suppose that there is a prophetic (historic,possibilistic or total) conditional, (α), for ΦΣn which is active. Then thereexists a coherent trace ΦΣn+1 , an extension of ΦΣn , such that (α) is a con-ditional for ΦΣn+1 which is exhausted.

Proof. Let ΦΣn be a coherent trace of Σn = (Wn, Tn,Fn) in Ξ , and let:

(1) if FA ∈ ΦΣn(tw), there exists t′w ∈ (tw,→) such thatA ∈ ΦΣn(t′w)

a prophetic conditional for ΦΣn with respect to FA and tw, which is active.In this case, the proof is a simple adaptation of the one offered in (Burgess,1984), by induction on the number l of successors of tw ∈ Tw:

Since (1) is active, we obtain that FA ∈ ΦΣn(tw), but there is no coor-dinate t′w ∈ (tw,→) such that A ∈ ΦΣn(t′w). Then, by Lemma 3, item (a),we have that there is some mc-set Γ such that ΦΣn(tw) ≺T Γ and A ∈ Γ .We want to extend the function ΦΣn , assign Γ to a new coordinate t′w andpreserve the linear order and coherence of the trace. The result will be a newframe Σn+1 = (Wn+1, Tn+1,Fn+1) ∈ Ξ and a trace ΦΣn+1 . For this, weconsider the number l of successors of tw in Tw. Thus, if l = 0:

– Wn+1 = Wn.– Tn+1 = (Tn − {(Tw, <w)}) ∪ {(T ′

w, <′w)}), where T ′

w = Tw ∪ {t′w},and <′

w =<w ∪ {(tw, t′w)} ∪ {(t1w , t′w) | t1w <w tw}

– Fn+1 = Fn and ΦΣn+1 = ΦΣn ∪ {(t′w, Γ )}If l > 0, we assume that for any natural number m < l the constructionis well-defined and consider the case l. We reason as follows: let t∗w be thesuccessor of tw in Tw.

88 A. Burrieza, I.P. de Guzman

Clearly, ¬A ∈ ΦΣn(t∗w); in other case, the conditional should be ex-hausted. If FA ∈ ΦΣn(t∗w), the case is resolved by inductive hypothesis. Ifnot, that is, if ¬FA ∈ ΦΣn(t∗w), then by (b) in Lemma 4, we obtain thatΓ ≺T ΦΣn(t∗w). Thus, we have ΦΣn(tw) ≺T Γ ≺T ΦΣn(t∗w). Then:

– Wn+1 = Wn

– Tn+1 = (Tn−{(Tw, <w)})∪{(T ′w, <

′w)}, where T ′

w = Tw ∪{t′w} and<′

w=<w ∪ {(tw, t′w), (t′w, t∗w)} ∪ {(t1w , t′w) | t1w <w tw)} ∪

{(t′w, t1w) | t∗w <w t1w}– Fn+1 = Fn and ΦΣn+1 = ΦΣn ∪ {(t′w, Γ )}.

In both cases, l = 0 and l > 0, the item (a) in Lemma 4 allows us to concludethat ΦΣn+1 is coherent and the proof is complete.

The proof for a historic conditional is similar.

Assume a possibilistic conditional for ΦΣn with respect to ♦A, tw:

(2) if ♦A∈ΦΣn(tw), there exists t′w′ ∈Fw({tw}) such that A∈ ΦΣn(t′w′)

is active. Then, ♦A ∈ ΦΣn(tw) but there is no t′w′ ∈ Fw({tw}) with A ∈ΦΣn(t′w′). We know – by item (c) in Lemma 3 – that there exists an mc-set,Γ , such that ΦΣn(tw) ≺W Γ and A ∈ Γ ; then we require a new finite flowof time Tw′ with a coordinate t′w′ associated with Γ so that t′w′ ∈ Fw({tw}).That is:

– Wn+1 = Wn ∪ {w′}– Tn+1 = Tn ∪ {(Tw′ , <w′)}, where Tw′ = {t′w′} and <w′= ∅

– Fn+1 = Fn ∪ {w w′−→}, where

w w′−→= {(tw, t′w′)} and

– ΦΣn+1 = ΦΣn ∪ {(t′w′ , Γ )}.It is easy to see that ΦΣn+1 is coherent.

Now, assume a total conditional for ΦΣn with respect to tw, t′w and t′w′ :

(3) if t′w′ ∈ Fw({tw}) and t′w = tw, then there exists t′′w′ ∈ Fw({t′w})is active. Then, t′w′ ∈ Fw({tw}) and t′w = tw , but t′′w′ ∈ Fw({t′w}) doesnot exist. Therefore, we must fix an image for t′w. We have either t′w ∈(tw,→) or t′w ∈ (←, tw). Now, by hypothesis with respect to ΦΣn , we haveΦΣn(tw) ≺W ΦΣn(t′w′) and either ΦΣn(tw) ≺T ΦΣn(t′w) or ΦΣn(t′w) ≺T

ΦΣn(tw).In any case, by Lemma 5, there is an mc-set, Γ , such that ΦΣn(t′w) ≺W Γand one of the following conditions is satisfied:

(1) ΦΣn(t′w′) = Γ ; (2) ΦΣn(t′w′) ≺T Γ ; (3) Γ ≺T ΦΣn(t′w′)

In all cases, we have Wn+1 = Wn. Moreover:

a) If condition 1 above holds, then

A functional approach for temporal × modal logics 89

– Tn+1 = Tn;

– Fn+1 = (Fn−{w w′−→}) ∪ {

w w′

−→′}, wherew w′

−→′ = w w′−→ ∪{(t′w, t′w′)}

– ΦΣn+1 = ΦΣn

Clearly, ΦΣn+1 is coherent.b) If condition 2 above holds, then we consider {tiw′ | 1 ≤ i ≤ n} =

(t′w′ ,→) i.e., the number n of successors of t′w′ in the flow Tw′ .If this set is empty, then we choose a new coordinate t′′w′ to be associatedto Γ and we have:– Tn+1 = (Tn − {(Tw′ , <w′)}) ∪ {(T ′

w′ , <′w′)}, where T ′

w′ = Tw′ ∪{t′′w′} and <′

w′ =<w′ ∪{(t′w′ , t′′w′)} ∪ {(t∗w′ , t′′w′) | t∗w′ <w′ t′w′}.

– Fn+1 = (Fn − {w w′−→}) ∪ {

w w′

−→′}), wherew w′

−→′ = w w′−→ ∪{(t′w, t′′w′)}

– ΦΣn+1 = ΦΣn ∪ {(t′′w′ , Γ )}. (†)Now, using Lemma 4 again, the proof of the coherence of ΦΣn+1 iscomplete.

If {tiw′ | 1 ≤ i ≤ n} = (t′w′ ,→) is not empty, we take into accountthe immediate successor of t′w′ , namely t1w′ . Now, as ΦΣn(t′w′) ≺T

ΦΣn(t1w′ ), by item (b) of Lemma 4, we have three cases:(i) ΦΣn(t1w′ ) = Γ ;(ii) Γ ≺T ΦΣn(t1w′ );(iii) ΦΣn(t1w′ ) ≺T Γ .

Case (i) is the same as condition 1 above but with t1w′ instead of t′w′ .Case (ii) means that ΦΣn(t′w′) ≺T Γ ≺T ΦΣn(t1w′ ). We select a newcoordinate t′′w′ to be associated with Γ . Therefore:– Tn+1 = (Tn − {(Tw′ , <w′)}) ∪ {(T ′

w′ , <′w′)}, where T ′

w′ =Tw′ ∪ {t′′w′} and <′

w′=<w′ ∪ {(t′w′ , t′′w′), (t′′w′ , t1w′ )} ∪ {(t∗w′ , t′′w′) |t∗w′ <w′ t′w′} ∪ {(t′′w′ , t∗w′) | t1w′ <w′ t∗w′}

– Fn+1 and ΦΣn+1 are defined as in (†) above.Again item (a) in Lemma 4 completes the proof of the coherence ofΦΣn+1 . Case (iii) lead us to consider the immediate successor of t1w′ ,namely t2w′ , and we proceed similarly.

By iterating this operation at most n times, we fix the image of t′wassociating an mc-set to it, preserving coherence and linearity.

c) If the condition 3 above holds, the treatment is similar to condition 2.This completes the proof of the exhausting lemma.

Finally we can enunciate the following theorem.

Theorem 10 (Completeness theorem for ST×W -Tot). If a formula,A, isvalid in the class of functional frames

{(W, T ,F) | F is a class of total functions}then A is a theorem of ST×W -Tot.

90 A. Burrieza, I.P. de Guzman

4.2 System ST×W -UD

In this section we consider the minimal system for the class of partial func-tions satisfying the (U-Dom)-condition. The system ST×W -UD has the fol-lowing axiom schemes:

1. All tautologies of the classical propositional logic, PL.2. Those of the minimal system of propositional linear temporal logic Kl.3. The characteristic axiom schema of the modal propositional logic K.4. The characteristic axiom schema.

(UD) � (HA ∧A ∧GA)→ (�⊥ ∨ (H�A ∧G�A))

The rules of inference are those of ST×W -Tot.

In order to prove the completeness of this system we need the followinglemma:

Lemma 8. Let Γ1, Γ2, Γ3 be mc-sets in ST×W -UD, then:

a) If Γ1 ≺T Γ2, Γ1 ≺W Γ3 and �⊥ ∈ Γ2, then there exists an mc-set Γ4such that Γ2 ≺W Γ4 and either Γ3 = Γ4 or Γ3 ≺T Γ4 or Γ4 ≺T Γ3.

b) If Γ1 ≺T Γ2, Γ2 ≺W Γ3 and �⊥ ∈ Γ1, then there exists an mc-set Γ4such that Γ1 ≺W Γ4 and either Γ3 = Γ4 or Γ3 ≺T Γ4 or Γ4 ≺T Γ3.

Proof. This is similar to the proof of Lemma 5. However, in this case wetake into account theorem F (�A ∧ ¬�⊥) → �(PA ∨ A ∨ FA) and itsmirror image, which are easy to prove following a pattern of proof similarto that of Theorem 9.

Now, we have a new condition on traces:

Definition 12. LetΦΣ be a trace of a functional frameΣ = (W, T ,F).ΦΣ

is called UD if it is coherent and, moreover, for all tw, t′w, t′w′ ∈ CoordΣ wehave:

(5) if t′w′ ∈ Fw({tw}), t′w = tw and �⊥ ∈ ΦΣ(t′w),there exists t′′w′ ∈ Fw({t′w})

(5) is called a UD-conditional for ΦΣ with respect tw, t′w, and t′w′ . ΦΣ

is called UD-full if it is prophetic, historic, possibilistic, and UD.

The exhausting lemma must be reformulated to treat ST×W -UD. To dothis, we simply take into account the active UD-conditionals instead of thetotal conditionals for a given ΦΣ . The modifications of the given proof forST×W -Tot are trivial using Lemma 8 instead of Lemma 5. Thus, we canformulate:

A functional approach for temporal × modal logics 91

Theorem 11 (Completeness theorem for ST×W -UD). If a wff A is validin the class

{(W, T ,F) | F is a class of (U-Dom )-partial functions }

then A is a theorem of ST×W -UD.

4.3 System ST×W -Non -Tot

The minimal system for the class of (U-Dom)-non-total functions, denotedST×W -Non -Tot, is an extension of ST×W -UD by adding the axiomschema:

(Non-Tot) P�⊥ ∨�⊥ ∨ F�⊥The exhausting lemma for ST×W -Non -Tot is formulated in a similar

way to ST×W –UD. Thus, we have:

Theorem 12 (Completeness theorem for ST×W -Non -Tot). If a formulaA is valid in the class

{(W, T ,F) | F is a class of (U-Dom)-non total functions}

then A is a theorem of ST×W -Non -Tot.

5 T × W -validity, Kamp-validity and functional-validity

In this section, we analyze the relation of T×W -validity and Kamp-validityto functional-validity. As we shall see, the functional context introduced inthis paper is a generalization of T ×W and Kamp contexts. Specifically,we can generate a functional model,MFunc, from a T ×W -model (resp.Kamp-model),MT×W (resp.MKamp) and prove the equivalence betweenMFunc andMT×W (resp,MKamp).

5.1 T ×W -validity and functional-validity

Definition 13. A T ×W -frame is a quadruple (T,<,W,≈), consisting of:

1. A nonempty set T (“time-points”) and a strict linear order < on T .2. A nonempty set W (“worlds” or “histories”).3. A family ≈= {≈t}t∈T of equivalence relations ≈t on W such that the

following condition is satisfied:“if w ≈t1 w

′ and t2 < t1, then w ≈t2 w′”.

The expression w ≈t w′ is read “w and w′ share the same history up to

(and including) t”.

92 A. Burrieza, I.P. de Guzman

Definition 14. A T × W -model is a tuple MT×W = (T,<,W,≈, h†)where (T,<,W,≈) is a T ×W -frame and h† is a function assigning eachatomic formula, p, a subset of T ×W and satisfying:

if w ≈t w′, then (t, w) ∈ h†(p) iff (t, w′) ∈ h†(p) (∗)

h† is recursively extended to LT×W treating truth-functional connectives inthe usual way. The modal and temporal connectives run as follows:

h†(GA) ={(t, w) ∈ T ×W | (t,→)× {w} ⊆ h†(A)};h†(HA) ={(t, w) ∈ T ×W | (←, t)× {w} ⊆ h†(A)};h†(�A) ={(t, w) ∈ T ×W | {t} × [w]≈t ⊆ h†(A)};

where [w]≈t denotes the equivalence class of w for the relation ≈t

5.1.1 Generating a functional model from a T ×W -model.A functional frame, (W ∗, T ,F), is generated from a T × W -frame,

(T,<,W,≈), as follows:

1. W ∗ = W .2. T is the set of all (Tw, <w) such that for each w ∈ W , (Tw, <w) is

an isomorphic copy of (T,<). The copy of t in Tw will be denoted tw.Thus, if w = w′, tw and tw′ are considered different.

3. F is the class {w w′−→| w,w′ ∈W ∗}, where

w w′−→ is defined as follows:

if there exists t0 ∈ T such that w ≈t0 w′, then the domain of

w w′−→ is

{tw ∈ Tw | w ≈t w′} and, for each tw ∈ Tw,

w w′−→ (tw) = tw′ .

As a consequence of the properties of the equivalence relations, we have:(a) idTw ∈ Fw for all w ∈W ∗.

(b) ifw w′→ ∈ Fw, then �w w′

−1 ∈ Fw′ .(c) F is closed under composition of functions.

The following lemma (whose proof is trivial) ensures that, in the aboveconstruction, ≈ is determined by F :

Lemma 9. Let (T,<,W,≈) be aT×W -frame and letΣ = (W ∗, T ,F) bea functional frame generated from (T,<,W,≈) as defined. Then, ifw,w′ ∈W and t ∈ T , we have that w ≈t w

′ if and only if tw′ ∈ Fw({tw}) 3.

Now we restrict the notion of a model given in Definition 3.

Definition 15. Let Σ = (W, T ,F) the functional frame generated from aT ×W -frame. We define a functional model on Σ as a tupleM = (Σ, h)as in Definition 3 with the following restriction for h: for all atoms p ∈ Vand all tw, t′w′ ∈ CoordΣ such that t′w′ ∈ Fw({tw}), we have:

tw ∈ h(p) if and only if t′w′ ∈ h(p) (∗∗)3 Where tw is the copy of t at Tw and tw′ is the copy of t at Tw′

A functional approach for temporal × modal logics 93

Lemma 10. LetMT×W = (T,<,W,≈, h†) aT×W -model on theT×W -frame, U = (T,<,W,≈), Σ = (W, T ,F) the functional frame generatedfrom U, and let M = (Σ, h) as in Definition 15. Then, if the followingcondition is satisfied

tw ∈ h(p) if and only if (t, w) ∈ h†(p)

we have for all formula A:

tw ∈ h(A) if and only if (t, w) ∈ h†(A).

Proof. By structural induction on A.

Now, the following result is immediate:

Theorem 13. A is T ×W -valid in a T ×W -frame, U , if and only if Ais valid in the functional frame generated from U , according to restriction(∗∗) established in Definition 15.

Remark 3. If we eliminate the restriction (∗) on the Definition 14, asin [Zanardo(1996)], we obtain the same result established by Theorem 13but without the corresponding restriction on the functional frames in Defi-nition 15.

5.2 Kamp-validity and functional-validity

Definition 16. A Kamp-frame is a quadruple (T,W,<,≈), consisting of:

1. A nonempty set T (“time-points”).2. A nonempty set W (“worlds” or “histories”).3. A family < = {<w| w ∈ W} of binary relations on T , where each <w

is a strict linear order on a subset Tw ⊆ T and⋃

w∈W

Tw = T .

4. A family ≈ = {≈t}t∈T of equivalence relations ≈t on W such that thefollowing conditions are satisfied:

(a) if w ≈t w′ then:

{t ∈ Tw ∩ Tw′

{t1 ∈ Tw | t1 <w t} = {t1 ∈ Tw′ | t1 <w′ t}(b) if w ≈t w

′ and t′ <w t, then w ≈t′ w′

As in the T ×W -frames, the expression w ≈t w′ is read “w and w′

share the same history up to (and including) t”.

Definition 17. A Kamp-model is MKamp = (T,W,<,≈, h†) where(T,W,<,≈) is a Kamp-frame and h† is a function assigning each atomicformula, p, a subset of {(t, w) ∈ T ×W | t ∈ Tw} and satisfying:

if w ≈t w′, then (t, w) ∈ h†(p) if and only if (t, w′) ∈ h†(p)

94 A. Burrieza, I.P. de Guzman

h† is recursively extended toLT×W treating truth-functional connectivesin the usual way. The modal and temporal connectives are defined in a similarway as for T ×W frames.

5.2.1 Generating a functional model from a Kamp-modelWe generate a functional frame,Σ = (W ∗, T ,F), from a Kamp-frame,

ΣKamp = (T,W,<,≈), as follows:

1. W ∗ = W .2. T =

⋃w∈W

(T ∗w, <

∗w), where for each (Tw, <w) in ΣKamp, (T ∗

w, <∗w) is a

copy of (Tw, <w), so that ifw,w′ ∈W andw = w′, then T ∗w∩T ∗

w′ = ∅.The copy of t ∈ T =

⋃w∈W Tw is denoted t∗w. Thus, if t ∈ Tw ∩ Tw′ ,

then there is a copy t∗w of t in T ∗w and a copy t∗w′ of t in T ∗

w′ .

3. F is the class {w w′−→| w,w′ ∈W ∗}, where

w w′−→ is defined as follows:

if there exists t0 ∈ T such that w ≈t0 w′, then the domain of

w w′−→ is

{t∗w ∈ T ∗w | w ≈t w

′} and, for each t∗w ∈ T ∗w,

w w′−→ (t∗w) = t∗w′ .

As a consequence of the properties of the equivalence relations, we have:(a) idT ∗

w∈ Fw for all w ∈W ∗.

(b) ifw w′→ ∈ Fw, then �w w′

−1 ∈ Fw′ .(c) F is closed under composition of functions.The following lemma ensures that, in the above construction, ≈ is de-termined by F :

Lemma 11. Let (T,W,<,≈) be a Kamp-frame and let Σ = (W ∗, T ,F)a functional frame generated from (T,W,<,≈) as defined. Then, ifw,w′ ∈W and t ∈ T , we have that:

w ≈t w′ if and only if t∗w′ ∈ Fw({t∗w}).

The rest follows step-by-step as in T ×W -frames.

6 Future work: Some remarks about incompleteness

In this section, we show that if we want to obtain complete minimal systemswith respect to the classes of frames with injective (surjective, etc.) functions,then, in general, we cannot follow the method given in Sect. 4, that is, it isnot sufficient (as a standard generalization) to consider the method of addingsuccessively to the basis of Kl + K the formulae introduced in the sectiondefining these classes. In consequence, deeper study is required which weshall consider as future work. As an example, we shall show that adding theformula

A functional approach for temporal × modal logics 95

(Tot-Inj) �(HA ∧GA)→ (H�A ∧G�A)

as axiom schema to the basis ofKl +K is not sufficient to obtain a completesystem with respect to the class of frames with total injective functions and,indeed, taking into account that this class is defined by (Tot-Inj), the systemKl +K+(Tot-Inj) is incomplete in a wider sense, that is, there is no classK of functional frames such that the theorems of that system are preciselythe valid formulas in K. For the incompleteness ofKl +K+ (Tot-Inj) weshall prove that there is a formula valid in the class of functional frames withtotal injective functions which is not a theorem of it. Let X be the formula

F (�A ∧ F�B)→ �[F (A ∧ FB) ∨ F (FA ∧B) ∨ (FA ∧ PB)∨(PA ∧ FB) ∨ P (A ∧ PB) ∨ P (PA ∧B)]

Now, it is sufficient to prove the following conditions:

1. The formula X is valid in the class of all functional frames with totalinjective functions.

2. There is a model of Kl +K + (Tot-Inj) in which X is not valid.

The proof of 1 is easy. For 2, we define a new type of frame for LT×W .Let Ψ = (W, T ,F•) where W and T are as in Definition 1 but F• is a setof correspondences, called accessibility correspondences, such that:

a) each correspondence inF• is a correspondence from Tw to Tw′ for somew,w′ ∈W .

b) for an arbitrary pairw,w′ ∈W , there is (inF•) at most one accessibilitycorrespondence from Tw to Tw′ , denoted φw w′ . We will denote F•

w ={φw w′ ∈ F• | w′ ∈W}. Then, F• =

⋃w∈W F•

w.

The set of coordinates of a frame Ψ = (W, T ,F•), denoted CoordΨ , isdefined as in Definition 1.

A correspondence model on a frame Ψ = (W, T ,F•) is a tuple (Ψ, h•),where h• is a function: h• : LT×W −→ 2CoordΨ , satisfying the usual condi-tions for the boolean and temporal connectives, and the specific condition:

h•(�A) = {tw ∈ CoordΨ | F•w({tw}) ⊆ h•(A)}

The notions of valid formula in a correspondence model, valid formulain a correspondence frame, and valid formula are defined in a standard way.

Now, let the following correspondence frame be:

– W = {w,w′};– T = {(Tw, <w), (Tw′ , <w′)}, where

– Tw = {1w, 2w, 3w}, <w= {(1w, 2w), (1w, 3w), (2w, 3w)};– Tw′ = {4w′ , 5w′}, <w′= {(4w′ , 5w′)};

– F• = {φww′}, where φww′(1w) = φww′(2w) = φww′(3w) = Tw′

96 A. Burrieza, I.P. de Guzman

Now we consider an arbitrary model, M, on this frame, i.e., an arbitraryinterpretation function h : V −→ 2{1w,2w,3w,4w′ ,5w′}.

To verify thatM is a model of Kl +K+ (Tot-Inj) it suffices to showthe validity of its axioms inM. Details of proofs are omitted. On the otherhand,M falsifies X at 1w. Thus, we conclude that X is not a theorem ofKl +K + (Tot-Inj) as required.

Acknowledgements. We would like acknowledge the very interesting comments receivedfrom the referees which have helped to improve the results of the paper.

References

[Belnap(1996)] Belnap, N.: Agents in branching time, In Logic and Reality: Essays in Pureand Applied Logic, in Memory of Arthur Prior, edited by B. J. Copeland, pp. 239-71Oxford: Oxford University Press 1996

[BelPer(1990)] Belnap, N. and Perloff, M.: Seeing to it that: A canonical form of agentives,In Knowledge Representation and Defeasible Reasonings, ed. by H.E. Kyburg, R.P.Loui, and G.N. Carlson, Dordrecht: Kluwer Academic Publishers 1990

[Burgess(1984)] Burgess, J.P.: Basic Tense Logic, In Handbook of Philosophical Logic, vol2: Extensions of Classical Logic, edited by D. Gabbay and F. Guenthner, pp. 89-133.Dordrecht: Reidel 1984

[Chellas(1992)] Chellas, B.: Time and Modality in the Logic of Agency, Studia Logica,51: 485-517, 1992

[Kuschera(1993)] Kuschera, F.v.: Causation, Journal of Philosophical Logic 22: 563–88,1993

[Reynolds(1997)] Reynolds, M.: A decidable temporal logic of parallelism, Notre DameJournal of Formal Logic 38: 419–436, 1997

[Thomason(1984)] Thomason, R.H.: Combinations of tense and modality, In Handbookof Philosophical Logic, Vol.2: Extensions of Classical Logic, edited by D. Gabbay andF. Guenthner, pp. 135–65. Dordrecht: Reidel 1984

[ThomasonGupta(1981)] Thomason, R.H., Gupta, A.: A theory of conditionals in the con-text of branching time, in Ifs: Conditionals, Belief, Decision, Chance and Time, editedby W. Harper, R. Stalnaker, and G. Pearce, pp. 299–322. Dordrecht: Reidel 1981

[Van Fraassen(1981)] Van Fraassen, B.: A temporal framework for conditional and chance,In Ifs: Conditionals, Belief, Decision, Chance and Time, edited by W. Harper, R.Stalnaker, and G. Pearce, pp. 323–340. Dordrecht: Reidel 1981

[Wolfl(1999)] Wolfl, S.: Combinations of Tense and Modality for Predicate Logic, Journalof Philosophical Logic 28: 371–398, 1999

[Zanardo(1986)] Zanardo, A.: On the Characterizability of the frames for the Unpreventabil-ity of the Present and the Past, Notre Dame Journal of Formal Logic 27: 556–564,1986

[Zanardo(1996)] Zanardo, A.: Branching-time logic with quantification over branches: thepoint of view of modal logic, The Journal of Symbolic Logic 61: 1–39, 1996