A functional approach for temporal à modal logics - Semantic Scholar
Acta Informatica 39, 71–96 (2003) Digital Object Identifier (DOI) 10.1007/s00236-002-0098-z
c Springer-Verlag 2003
A functional approach for temporal × modal logics Alfredo Burrieza, Inma P. de Guzm´an Dept. de Matem´atica Aplicada, Univ. de M´alaga, Campus de Teatinos, 29071 M´alaga, Espa˜na Received: 12 November 2001 / 18 September 2002
Abstract. This work is focused on temporal × modal logics. We study the representation of properties of functions of interest because of their possible computational interpretations. The semantics is exposed in an algebraic style, and the definability of the basic properties of the functions is analysed. We introduce minimal systems for linear time with total functions as well as with a class of partial functions (those with uniform domain). Moreover, 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 the T × W -logics) have been introduced. The main interest of this investigation is focused on fields such as causation, the theory of action 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 of combinations is in the field of Mathematics and Computer Science. In this paper we present a new type of frames, which we call functional frames. 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 by W.
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3. A set of non empty partial functions (called accessibility functions), with the characteristic that at most one accessibility function is defined between two of the arbitraryorders. An example follows:
w • 1
?
? •
? w • 3
•? 4 @ I
? w •
• 7
2
6
6@
• 5@ I @ •?
@
8
@•
9
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: ww
−→: Tw → Tw ; w w
w w
−→: Tw → Tw ;
w w
−→ : Tw → Tw
and −→ : Tw → Tw , defined as follows: ww w w w w −→= idTw (identity on Tw ); −→ (1w ) = 3w , −→ (2w ) = 4w ; w w
w w
−→ (3w ) = 6w , −→ (5w ) = 8w ;
w w
w w
w w
w w
−→ (6w ) = −→ (7w ) = −→ (8w ) = 4w , −→ (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 definability of basic properties of the theory of functions (such as being injective, surjective, increasing, decreasing, etc.); concretely, we study properties of classes of functions without the need to resort to second order theories and, in addition, it is a general purpose tool for studying logics containing modal and temporal operators. The practical motivation is their use in computational applications. To this respect, the combinations of modal and tense operators is mentioned, for example, in [Reynolds(1997)] as a suitable tool to treat parallel processes, distributed systems and multiagents. Specifically, we can consider time-flows as memory of computers connected in a net, each computer with its own clock. Traditional approaches in this field are Thomason’s T × W -frames and also Kamp-frames, and both are special cases of the approach we introduce here, in the sense that, if we consider the restrictions imposed on the T × W (Kamp)-models in [Thomason(1984)], then every T × W (Kamp)-model has an equivalent functional model (i.e. the same formulas are valid in both models). If those restrictions are ignored, as in [Zanardo(1996)], then for all T × W (Kamp)-frame there exists an equivalent functional frame. The most remarkable differences between the functional approach and the previous ones are commented on below:
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– In a T × W -frame, the flow of time is shared by all the worlds. In a Kamp-frame each world has its own flow of time, although for two different worlds the time can coincide up to an instant. In our functional frames each world has its own flow of time and it is not required that flows of time for different worlds have to be neither totally nor partially isomorphic. – In both T × W -frames and Kamp-frames the flow of time is strictly linear. 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 are connected by equivalence relations defined for their respective temporal orders and, as a consequence, the axioms of the modal basis of S5 are valid for our operators of necessity and possibility. In our approach there is more flexibility, the inter-world accessibility is defined by partial functions (accessibility functions). These functions allow us to connect the worlds and so, to compare the measure of different courses of time in several ways. – The equivalence relations are used in previous approaches to establish which segments of different temporal flows are to be considered as the same history. This way, it is possible to define the notion of historical necessity. This notion can be defined using functional frames too, as it will be shown, but in addition, functions allow to define situations which do not require, neither partial nor total, isomorphisms between temporal flows (thinking again of flows of time as interconnected computer memories, 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 called LT ×W and its semantics is a functional semantics. In Sect. 3 we analyze the definability of the basic properties of the functions. In Sect. 4 we introduce minimal axiomatic systems for LT ×W and their completeness proofs are given. In Sect. 5 we prove that the T × W -frames and the Kamp-frames are particular cases of our approach. Finally, in Sect. 6, future work is outlined. 2 The Logic LF T ×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
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The well-formed formulae (wffs) are generated by the construction rules of classical propositional logic adding the following rule: If A is a wff, then GA, HA and 2A are wffs. We consider, as usual, the connectives ∧, ∨ F , P and to be defined connectives. The connectives G, H, F , and P have their usual readings, but 2 has the following meaning: 2A is read “A is true at every accessible present” (in the above example, the accessible present for 1w is {1w , 3w }, etc.). On the other hand, the notion of a mirror image of a formula is considered in the usual way. Definition 1. We define a functional frame for LT ×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, and if w = w , then Tw ∩ Tw = ∅. – F is a set of non-empty functions, called accessibility functions, such that: a) each function in F is a partial function from Tw to Tw , for some w, w ∈ W. b) for an arbitrary pair w, w ∈ W , there is (in F) at most one accessibility w w
function from Tw to Tw , denoted by −→. w w
We will denote Fw = { −→ ∈ F | w ∈ W }. Then F =
w∈W
Fw .
Let Σ = (W, T , F) be a functional frame. The elements tw of the disjoint union 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 w w −→ (X) is disjoint. nates X ⊆ Tw , the union Fw (X) = w w
−→ ∈Fw
(ii) The set of accessibility functions, F, is not necessarily closed under composition, 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 ].
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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 then we 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 ), we have: f ((a, →)) ⊆ [f (a), →) ( resp. f ((a, →)) ⊆ (←, f (a)] ). 4. f is strictly increasing (resp. strictly decreasing) if and only if, for all a ∈ 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. w w
Lemma 1. Let Σ = (W, T , F) be a functional frame and −→∈ Fw . Then, w w
if tw ∈ Tw we have tw ∈ Dom(−→) if and only if the following inclusion (∗1 ) is satisfied: w w
(∗1 )
w w
w w
w w
−→ ((←, tw )) ∪ −→ ((tw , →)) ⊆ (←, −→ ({tw })) ∪ [−→ ({tw }), → ) w w
Proof. If tw ∈ Dom(−→), (∗1 ) is obviously true, because: w w
w w
(←, −→ ({tw })) ∪ [ −→ ({tw }), → ) = Tw w w
Reciprocally, if tw ∈ Dom(−→), we obtain w w
w w
(←, −→ ({tw })) ∪ [−→ ({tw }), → ) = ∅ w w
w w
Now, since −→∈ Fw is non empty, then there exists tw ∈ Dom(−→) and, w w
w w
as a consequence, −→ ((←, tw )) ∪ −→ ((tw , →)) = ∅, thus, (∗1 ) fails to be true. Notation 2 Given a functional frame (W, T , F), for all w ∈ W , we shall denote w w Dom(−→) Dom(Fw ) = w w
−→ ∈Fw
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}) = ∅. 1
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Definition 2. Let (W, T , F) be a functional frame. We say that F satisfies w w
w w
the (U-Dom) condition if for all w ∈ W and all −→, −→ ∈ Fw we have that: w w w w (U-Dom) Dom(−→) = Dom( −→ ) This common domain is denoted by DomU (Fw ). Theorem 2. Let Σ = (W, T , F) be a functional frame. Then, F is a class of total functions (resp. (U-Dom) functions) if and only if for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), the following inclusion (Fw -∗1 ) is obtained: (Fw -∗1 ) Fw ((←, tw )) ∪ Fw ((tw , →)) ⊆ Fw ({tw })↓∗ ∪ Fw ({tw }) ↑. Proof. We consider the case of total functions. The other case is similar. w w
Let F be a class of total functions. By Lemma 1, for all −→ ∈ Fw and tw ∈ CoordΣ , the following relation holds: w w
w w
(∗1 )
w w
w w
−→ ((←, tw )) ∪ −→ ((tw , →)) ⊆ (←, −→ ({tw })) ∪ [−→ ({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 disjoint union. For the converse we reason in a similar way from inclusion (∗1 ). As an immediate consequence of Theorems 1 and 2, and Remark 1 we obtain 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 obtain the following inclusion (inj): inj
Fw ((←, tw )) ∪ Fw ((tw , →)) ⊆ 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): surj
Fw ({tw }) ↓∗ ∪ Fw ({tw }) ↑∗ ⊆ Fw ((←, tw )) ∪ Fw ((tw , →)) 3. F is a class of total increasing functions (resp. (U-Dom) increasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (inc+ ) and (inc− ): inc+
inc−
Fw ((tw , →)) ⊆ Fw ({tw }) ↑ and Fw ((←, tw )) ⊆ Fw ({tw }) ↓
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4. F is a class of total strictly increasing functions (resp. (U-Dom) strictly increasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (str-inc+ ) and (str-inc− ): str-inc+ str-inc− Fw ((tw , →)) ⊆ Fw ({tw })↑∗ and Fw ((← ,tw )) ⊆ Fw ({tw })↓∗
5. F is a class of total decreasing functions (resp. (U-Dom) decreasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (dec+ ) and (dec− ): dec+
dec−
Fw ((tw , →)) ⊆ Fw ({tw }) ↓ and Fw ((← ,tw )) ⊆ Fw ({tw }) ↑ 6. F is a class of total strictly decreasing functions (resp. (U-Dom) strictly decreasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (str-dec+ ) and (str-dec− ): str-dec+ str-dec− Fw ((tw , →)) ⊆ Fw ({tw })↓∗ and Fw ((← ,tw )) ⊆ 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 obtain the following inclusion (con): con
Fw ((←, tw )) ∪ Fw ((tw , →)) ⊆ Fw ({tw }) Definition 3. A functional model on Σ is a tuple M = (Σ, h), where Σ is a functional frame and h : LT ×W −→ 2CoordΣ is a function, called a functional 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(2A) = {tw ∈ CoordΣ | Fw ({tw }) ⊆ h(A)}. Definition 4. Let A be a formula in LT ×W . Let (Σ, h) be a functional model and tw ∈ CoordΣ , then, A is true at tw if tw ∈ h(A). A is said to be valid in the functional model (Σ, h) if h(A) = CoordΣ . If A is valid in every functional model on Σ, then A is said to be valid in the functional frame Σ, and denote it by |=Σ A. If A is valid in every functional frame, then A is said 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 to be valid in K.
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3 Definability in LF T ×W Definition 5. Let J be a class of functional frames and let K ⊆ J. We say that K is LF T ×W -definable in (or relative to) J if there exists a set Γ of formulas in LT ×W such that for every frame Σ ∈ J, we have that Σ ∈ K if and only if every formula of Γ is valid in Σ. If J is the class of all functional frames, 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 is F LF T ×W -definable if K is LT ×W -definable. In the rest of this paper, “definable” means “LF T ×W -definable”. Theorem 4. K = {(W, T , F) | F is a class of total functions } is definable. Proof. We will prove that the formula (T ot) 2(HA ∧ A ∧ GA) → (H2A ∧ G2A) defines K. To this end, let (Σ, h) be such that Σ ∈ K and tw ∈ CoordΣ . Then: iff tw ∈ h(2(HA ∧ A ∧ GA)) tw ∈ h(2HA) ∩ h(2A) ∩ h(2GA) iff Fw ({tw }) ↓∗ ∪ Fw ({tw }) ∪ Fw ({tw }) ↑∗ ⊆ h(A) iff Fw ({tw }) ↓∗ ∪ Fw ({tw }) ↑ ⊆ h(A)) On the other hand, tw ∈ h(H2A) iff Fw ((←, tw )) ⊆ h(A) and tw ∈ h(G2A) iff Fw ((tw , →)) ⊆ h(A). Now, by the inclusion (Fw -∗1 ) in Theorem 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 where h(p) = Fw ({tw })↓∗ ∪ Fw ({tw }) ↑. Then, we obtain tw ∈ h(2(Hp ∧ p ∧ Gp)) and Fw ((←, tw )) ∪ Fw ((tw , →)) h(p) Therefore tw ∈ h(H2p ∧ G2p) 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 (U D) 2(HA ∧ A ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)) defines K. The validity of (U D) in K is an immediate consequence of:
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i) if tw ∈ DomU (Fw ), then tw ∈ h(2⊥), and ii) 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 (T ot) in the theorem above and the proof is complete. Theorem 6. K = {(W, T , F) functions} is definable.
|
F is a class of (U-Dom)-non-total
Proof. We will prove that the following set defines K: {2(HA ∧ A ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)), 2P ⊥ ∨ 2⊥ ∨ F 2⊥} The validity of 2(HA ∧ A ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)) in K is guaranteed by Theorem 5, since the class of (U-Dom)-non-total functions is a subclass of the class of (U-Dom)-partial functions. On the other hand, for every frame in K, we have that for all Tw there is at least one tw ∈ Tw such that tw ∈ DomU (Fw ). Clearly, tw ∈ h(2⊥) if and only if tw ∈ DomU (Fw ). Thus, from the linearity of <w , we have that the formula (N on-T ot) : 2P ⊥ ∨ 2⊥ ∨ F 2⊥ holds at every coordinate in Tw . For the reverse, let (W, T , F) be a functional frame not in K. We consider the following two possible cases: w w
(i) There exists some total function −→∈ F. Then 2⊥ fails to be true at every coordinate in Tw (whatever the model based on that frame) and therefore 2P ⊥ ∨ 2⊥ ∨ F 2⊥ is not valid in that frame. (ii) F does not satisfy the (U-Dom) condition. In this case, we work as in Theorem 5. Theorem 7. The following classes of functional frames are definable: (1) (2) (3) (4) (5) (6) (7)
K1 = {(W, T , F) | F is a class of total injective functions }. K2 = {(W, T , F) | F is a class of surjective functions }. K3 = {(W, T , F) | F is a class of total increasing functions }. K4 = {(W, T , F) | F is a class of total strictly increasing functions }. K5 = {(W, T , F) | F is a class of total decreasing functions }. K6 = {(W, T , F) | F is a class of total strictly decreasing functions }. K7 = {(W, T , F) | F is a class of total constant functions }.
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Proof. (1) We will prove that 2(HA ∧ GA) → (H2A ∧ G2A) defines K1 . For this, let (Σ, h) be such that Σ ∈ K1 , then for any tw ∈ CoordΣ we have that: tw ∈ h(2(HA ∧ GA)) tw ∈ h(2HA) ∩ h(2GA) Fw ({tw }) ↓∗ ∪ Fw ({tw }) ↑∗ ⊆ h(A)
iff iff
On the other hand, tw ∈ h(H2A) if and only if Fw ((←, tw )) ⊆ h(A) and tw ∈ h(G2A) if and only if Fw ((tw , →)) ⊆ h(A). Now, by (inj) in Theorem 3, the proof of validity is complete. Reciprocally, if Σ ∈ K1 , a countermodel for the formula 2(HA ∧ GA) → (H2A ∧ G2A) can be defined in a similar way as for (T ot) in theorem 4, and the proof is complete. (2) The formula (H2A ∧ G2A) → 2(HA ∧ GA) defines K2 . This is a consequence of (surj) in Theorem 3. (3) The set {2(A ∧ GA) → G2A, 2(HA ∧ A) → H2A} defines K3 . This is a consequence of (inc+ ) and (inc− ) in Theorem 3. (4) The set {2GA → G2A, 2HA → H2A} defines K4 . This is a consequence of (str-inc+ ) and (str-inc− ) in Theorem 3. (5) The set {2(A ∧ GA) → H2A, 2(HA ∧ A) → G2A} defines K5 . This is a consequence of (dec+ ) and (dec− ) in Theorem 3. (6) The set {2GA → H2A, 2HA → G2A} defines K6 . This is a consequence of (str-dec+ ) and (str-dec− ) in Theorem 3. (7) The formula 2A → (H2A ∧ G2A) defines K7 . This is a consequence of (con) in Theorem 3. Theorem 8. Let K be a class of functional frames, (W, T , F), where F is a class of functions that satisfies the (U-Dom) condition and in which all functions 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 their corresponding formulas as follows: (U D-Inj): 2(HA ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)) (U D-Inc): {2(A ∧ GA) → (2⊥ ∨ G2A), 2(HA ∧ A) → (2⊥ ∨ H2A)} (U D-Str-Inc): {2GA → (2⊥ ∨ G2A), 2HA → (2⊥ ∨ H2A)} (U D-Dec): {2(A∧GA) → (2⊥∨H2A), 2(HA∧A) → (2⊥∨G2A)} (U D-Str-Dec): {2GA → (2⊥ ∨ H2A), 2HA → (2⊥ ∨ G2A)} (U D-Con): 2A → (2⊥ ∨ (H2A ∧ G2A))
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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 → GP A (H2) A → HF A (T R) 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 logic K: (K) 2(A → B) → (2A → 2B) 4. The characteristic axiom schema: (T ot) 2(HA ∧ A ∧ GA) → (H2A ∧ G2A) The rules of inference are those of Kl + K: (M P ): A, A → B B; (RG): A GA; (RH): A HA and (RN ) A 2A The concepts of proof and theorem are defined as usual. Theorem 9. The following formulas are theorems in ST ×W -Tot, T1: F 2A → 2(P A ∨ A ∨ F A).
T2: P 2A → 2(P A ∨ A ∨ F A).
Proof. We prove the theorem T1 (T2 is its mirror image). Let ψ = H¬A ∧ ¬A ∧ G¬A 1. 2(H¬ψ ∧ ¬ψ ∧ G¬ψ) → H2¬ψ 2. P ♦ψ → (♦P ψ ∨ ♦ψ ∨ ♦F ψ) 3. ψ → G¬A 4. ♦P ψ → ♦P G¬A 5. ♦P G¬A → ♦¬A 6. ♦P ψ → ♦¬A 7. ♦ψ → ♦¬A 8. ♦F ψ → ♦¬A 9. P ♦ψ → ♦¬A 10. 2A → H2¬ψ 11. ¬ψ → (P A ∨ A ∨ F A) 12. H2 ¬ψ → H2(P A ∨ A ∨ F A) 13. 2A → H2(P A ∨ A ∨ F A) 14. F 2A → 2(P A ∨ A ∨ F A)
from (T ot) and PL from 1, by K and definitions of ♦, F, P by PL from 3, by Kl and K by Kl and K from 4, 5 by PL by K [similar to 3-6] from 2, 6, 7, 8 by PL from 9 by PL and definitions of ♦ and P by PL and definitions of F and P from 11 by K and Kl from 10, 12 by PL from 13 by Kl
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Completeness theorem for ST ×W -Tot The proof of soundness is standard. We focus our attention on the completeness proof and adopt the usual definitions of a consistent (maximal) set of formulas. In the following, we abbreviate maximal consistent set as mc-set. On the other hand, by S we mean any axiomatic system for LT ×W which is an extension of Kl + K (in particular, the system ST ×W -Tot). Familiarity with the basic properties of mc-sets in classical propositional systems is assumed. Definition 6. Let Γ1 and Γ2 be mc-sets in S, then we define: Γ1 ≺T Γ2 Γ1 ≺W Γ2
if and only if {A | GA ∈ Γ1 } ⊆ Γ2 . if and only if {A | 2A ∈ 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 {F A | A ∈ Γ2 } ⊆ Γ1 iff {P A | A ∈ Γ1 } ⊆ Γ2 iff {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 F A ∈ Γ1 , exists an mc-set Γ2 ∈ S such that Γ1 ≺T Γ2 and A ∈ Γ2 . (b) If P A ∈ 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 that Γ2 ≺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 that Γ1 ≺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 conditions is satisfied: (i)
Γ2 ≺W Γ3 . In this case, we obtain the result taking Γ4 = Γ3 .
A functional approach for temporal × modal logics
83 ≺T
Γ1
≺W@
R @
Γ4 = Γ3
- Γ2
≺W
(ii) {A | 2A ∈ Γ2 } ∪ {A | GA ∈ Γ3 } is consistent. In this case, Lindenbaum’s lemma guarantees that there exists at least one mc extension, Γ4 , of the set {A | 2A ∈ Γ2 } ∪ {A | GA ∈ Γ3 } which trivially satisfies Γ2 ≺W Γ4 and Γ3 ≺T Γ4 . ≺T
Γ1 ≺W ?
≺T
Γ3
- Γ2 ≺W ? -Γ 4
(iii) {A | 2A ∈ Γ2 } ∪ {A | HA ∈ Γ3 } is consistent. Now, Lindenbaum’s lemma again guarantees that there exists at least one mc extension, Γ4 , of the set {A | 2A ∈ Γ2 }∪{A | HA ∈ Γ3 } which satisfies Γ2 ≺W Γ4 and Γ4 ≺T Γ3 . ≺T Γ1 Γ2
≺W ≺W
) ≺T - ?
Γ4 Γ3 If we assume that none of the conditions (i)–(iii) holds, we have: a) there exists a formula 2A ∈ Γ2 such that A ∈ Γ3 ; B1 , . . . , Bn1 ∈ {A | 2A ∈ Γ2 }, b) there are C1 , . . . , Cm1 ∈ {A | GA ∈ Γ3 } such that ¬(B1 ∧ . . . ∧ Bn1 ∧ C1 ∧ . . . ∧ Cm1 ); D1 , . . . , Dn2 ∈ {A | 2A ∈ Γ2 }, c) there are E1 , . . . , Em2 ∈ {A | HA ∈ Γ3 } such that ¬(D1 ∧ . . . ∧ Dn2 ∧ E1 ∧ . . . ∧ Em2 ). B = B1 ∧ . . . ∧ Bn1 , with 2B ∈ Γ2 ; C = C1 ∧ . . . ∧ Cm1 , with GC ∈ Γ3 ; Thus, we have: D = D1 ∧ . . . ∧ Dn2 , with 2D ∈ Γ2 ; E = E1 ∧ . . . ∧ Em2 , with HE ∈ Γ3 Now, from ¬(B ∧ C), we obtain 2B → 2¬C and (since 2B ∈ Γ2 ) we have 2¬C ∈ Γ2 . Similarly, from ¬(D ∧ E), we obtain 2¬E ∈ Γ2 . Thus, 2(A ∧ ¬C ∧ ¬E) ∈ Γ2 and, since Γ1 ≺T Γ2 , we obtain F 2(A ∧ ¬C ∧ ¬E) ∈ Γ1 . Therefore, by (T 1) in Theorem 9, we obtain: 2(P (A ∧ ¬C ∧ ¬E) ∨ (A ∧ ¬C ∧ ¬E) ∨ F (A ∧ ¬C ∧ ¬E)) ∈ Γ1 Henceforth, by the definition of ≺W , either
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(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), and GC ∈ 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 is any function ΦΣ : CoordΣ −→ 2LT ×W such that, for all tw ∈ CoordΣ , the set 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 , tw ∈ CoordΣ : if tw ∈ (tw , →), then ΦΣ (tw ) ≺T ΦΣ (tw ) modally coherent if, for all tw , tw ∈ CoordΣ : if tw ∈ Fw ({tw }), then ΦΣ (tw ) ≺W ΦΣ (tw ) coherent if it is temporally coherent and modally coherent. prophetic if it is temporally coherent and, moreover, for all wff, A, and all tw ∈ CoordΣ : (1) if F A ∈ ΦΣ (tw ), there exists a tw ∈ (tw , →) such that A ∈ ΦΣ (tw ) historic if it is temporally coherent and, moreover, for all wff, A, and all tw ∈ CoordΣ : (2) if P A ∈ ΦΣ (tw ), there exists a tw ∈ (←, tw ) such that A ∈ ΦΣ (tw ) possibilistic if it is modally coherent and, moreover, for all wff, A, and all tw ∈ CoordΣ : (3) if ♦A ∈ ΦΣ (tw ), there exists a tw ∈ Fw ({tw }) such that A ∈ ΦΣ (tw ). The conditional (1) (resp., (2) or (3)) is called a prophetic (historic or possibilistic) conditional for ΦΣ with respect to F A, (P A or ♦A) and tw . Definition 9. Let ΦΣ be a trace of Σ = (W, T , F). ΦΣ is called total if it is coherent and Σ satisfies the following property: for all tw , tw , tw ∈ CoordΣ (4) if tw ∈ Fw ({tw }) and tw = tw , then there exists tw ∈ Fw ({tw }) 2 (4) is called a total conditional for ΦΣ with respect to tw , tw and tw . ΦΣ is called total-full if it is prophetic, historic, possibilistic, and total. Definition 10. Let WΞ be a denumerable infinite set and TΞ = Ξ Ξ w∈WΞ Tw where Tw is a denumerable infinite set for each w ∈ WΞ and if w = w , then TwΞ ∩ TwΞ = ∅. We consider the class, Ξ, of functional frames, (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.
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85
– T = {(Tw ,
c Springer-Verlag 2003
A functional approach for temporal × modal logics Alfredo Burrieza, Inma P. de Guzm´an Dept. de Matem´atica Aplicada, Univ. de M´alaga, Campus de Teatinos, 29071 M´alaga, Espa˜na Received: 12 November 2001 / 18 September 2002
Abstract. This work is focused on temporal × modal logics. We study the representation of properties of functions of interest because of their possible computational interpretations. The semantics is exposed in an algebraic style, and the definability of the basic properties of the functions is analysed. We introduce minimal systems for linear time with total functions as well as with a class of partial functions (those with uniform domain). Moreover, 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 the T × W -logics) have been introduced. The main interest of this investigation is focused on fields such as causation, the theory of action 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 of combinations is in the field of Mathematics and Computer Science. In this paper we present a new type of frames, which we call functional frames. 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 by W.
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3. A set of non empty partial functions (called accessibility functions), with the characteristic that at most one accessibility function is defined between two of the arbitraryorders. An example follows:
w • 1
?
? •
? w • 3
•? 4 @ I
? w •
• 7
2
6
6@
• 5@ I @ •?
@
8
@•
9
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: ww
−→: Tw → Tw ; w w
w w
−→: Tw → Tw ;
w w
−→ : Tw → Tw
and −→ : Tw → Tw , defined as follows: ww w w w w −→= idTw (identity on Tw ); −→ (1w ) = 3w , −→ (2w ) = 4w ; w w
w w
−→ (3w ) = 6w , −→ (5w ) = 8w ;
w w
w w
w w
w w
−→ (6w ) = −→ (7w ) = −→ (8w ) = 4w , −→ (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 definability of basic properties of the theory of functions (such as being injective, surjective, increasing, decreasing, etc.); concretely, we study properties of classes of functions without the need to resort to second order theories and, in addition, it is a general purpose tool for studying logics containing modal and temporal operators. The practical motivation is their use in computational applications. To this respect, the combinations of modal and tense operators is mentioned, for example, in [Reynolds(1997)] as a suitable tool to treat parallel processes, distributed systems and multiagents. Specifically, we can consider time-flows as memory of computers connected in a net, each computer with its own clock. Traditional approaches in this field are Thomason’s T × W -frames and also Kamp-frames, and both are special cases of the approach we introduce here, in the sense that, if we consider the restrictions imposed on the T × W (Kamp)-models in [Thomason(1984)], then every T × W (Kamp)-model has an equivalent functional model (i.e. the same formulas are valid in both models). If those restrictions are ignored, as in [Zanardo(1996)], then for all T × W (Kamp)-frame there exists an equivalent functional frame. The most remarkable differences between the functional approach and the previous ones are commented on below:
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– In a T × W -frame, the flow of time is shared by all the worlds. In a Kamp-frame each world has its own flow of time, although for two different worlds the time can coincide up to an instant. In our functional frames each world has its own flow of time and it is not required that flows of time for different worlds have to be neither totally nor partially isomorphic. – In both T × W -frames and Kamp-frames the flow of time is strictly linear. 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 are connected by equivalence relations defined for their respective temporal orders and, as a consequence, the axioms of the modal basis of S5 are valid for our operators of necessity and possibility. In our approach there is more flexibility, the inter-world accessibility is defined by partial functions (accessibility functions). These functions allow us to connect the worlds and so, to compare the measure of different courses of time in several ways. – The equivalence relations are used in previous approaches to establish which segments of different temporal flows are to be considered as the same history. This way, it is possible to define the notion of historical necessity. This notion can be defined using functional frames too, as it will be shown, but in addition, functions allow to define situations which do not require, neither partial nor total, isomorphisms between temporal flows (thinking again of flows of time as interconnected computer memories, 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 called LT ×W and its semantics is a functional semantics. In Sect. 3 we analyze the definability of the basic properties of the functions. In Sect. 4 we introduce minimal axiomatic systems for LT ×W and their completeness proofs are given. In Sect. 5 we prove that the T × W -frames and the Kamp-frames are particular cases of our approach. Finally, in Sect. 6, future work is outlined. 2 The Logic LF T ×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
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The well-formed formulae (wffs) are generated by the construction rules of classical propositional logic adding the following rule: If A is a wff, then GA, HA and 2A are wffs. We consider, as usual, the connectives ∧, ∨ F , P and to be defined connectives. The connectives G, H, F , and P have their usual readings, but 2 has the following meaning: 2A is read “A is true at every accessible present” (in the above example, the accessible present for 1w is {1w , 3w }, etc.). On the other hand, the notion of a mirror image of a formula is considered in the usual way. Definition 1. We define a functional frame for LT ×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, and if w = w , then Tw ∩ Tw = ∅. – F is a set of non-empty functions, called accessibility functions, such that: a) each function in F is a partial function from Tw to Tw , for some w, w ∈ W. b) for an arbitrary pair w, w ∈ W , there is (in F) at most one accessibility w w
function from Tw to Tw , denoted by −→. w w
We will denote Fw = { −→ ∈ F | w ∈ W }. Then F =
w∈W
Fw .
Let Σ = (W, T , F) be a functional frame. The elements tw of the disjoint union 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 w w −→ (X) is disjoint. nates X ⊆ Tw , the union Fw (X) = w w
−→ ∈Fw
(ii) The set of accessibility functions, F, is not necessarily closed under composition, 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 ].
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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 then we 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 ), we have: f ((a, →)) ⊆ [f (a), →) ( resp. f ((a, →)) ⊆ (←, f (a)] ). 4. f is strictly increasing (resp. strictly decreasing) if and only if, for all a ∈ 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. w w
Lemma 1. Let Σ = (W, T , F) be a functional frame and −→∈ Fw . Then, w w
if tw ∈ Tw we have tw ∈ Dom(−→) if and only if the following inclusion (∗1 ) is satisfied: w w
(∗1 )
w w
w w
w w
−→ ((←, tw )) ∪ −→ ((tw , →)) ⊆ (←, −→ ({tw })) ∪ [−→ ({tw }), → ) w w
Proof. If tw ∈ Dom(−→), (∗1 ) is obviously true, because: w w
w w
(←, −→ ({tw })) ∪ [ −→ ({tw }), → ) = Tw w w
Reciprocally, if tw ∈ Dom(−→), we obtain w w
w w
(←, −→ ({tw })) ∪ [−→ ({tw }), → ) = ∅ w w
w w
Now, since −→∈ Fw is non empty, then there exists tw ∈ Dom(−→) and, w w
w w
as a consequence, −→ ((←, tw )) ∪ −→ ((tw , →)) = ∅, thus, (∗1 ) fails to be true. Notation 2 Given a functional frame (W, T , F), for all w ∈ W , we shall denote w w Dom(−→) Dom(Fw ) = w w
−→ ∈Fw
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}) = ∅. 1
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Definition 2. Let (W, T , F) be a functional frame. We say that F satisfies w w
w w
the (U-Dom) condition if for all w ∈ W and all −→, −→ ∈ Fw we have that: w w w w (U-Dom) Dom(−→) = Dom( −→ ) This common domain is denoted by DomU (Fw ). Theorem 2. Let Σ = (W, T , F) be a functional frame. Then, F is a class of total functions (resp. (U-Dom) functions) if and only if for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), the following inclusion (Fw -∗1 ) is obtained: (Fw -∗1 ) Fw ((←, tw )) ∪ Fw ((tw , →)) ⊆ Fw ({tw })↓∗ ∪ Fw ({tw }) ↑. Proof. We consider the case of total functions. The other case is similar. w w
Let F be a class of total functions. By Lemma 1, for all −→ ∈ Fw and tw ∈ CoordΣ , the following relation holds: w w
w w
(∗1 )
w w
w w
−→ ((←, tw )) ∪ −→ ((tw , →)) ⊆ (←, −→ ({tw })) ∪ [−→ ({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 disjoint union. For the converse we reason in a similar way from inclusion (∗1 ). As an immediate consequence of Theorems 1 and 2, and Remark 1 we obtain 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 obtain the following inclusion (inj): inj
Fw ((←, tw )) ∪ Fw ((tw , →)) ⊆ 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): surj
Fw ({tw }) ↓∗ ∪ Fw ({tw }) ↑∗ ⊆ Fw ((←, tw )) ∪ Fw ((tw , →)) 3. F is a class of total increasing functions (resp. (U-Dom) increasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (inc+ ) and (inc− ): inc+
inc−
Fw ((tw , →)) ⊆ Fw ({tw }) ↑ and Fw ((←, tw )) ⊆ Fw ({tw }) ↓
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4. F is a class of total strictly increasing functions (resp. (U-Dom) strictly increasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (str-inc+ ) and (str-inc− ): str-inc+ str-inc− Fw ((tw , →)) ⊆ Fw ({tw })↑∗ and Fw ((← ,tw )) ⊆ Fw ({tw })↓∗
5. F is a class of total decreasing functions (resp. (U-Dom) decreasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (dec+ ) and (dec− ): dec+
dec−
Fw ((tw , →)) ⊆ Fw ({tw }) ↓ and Fw ((← ,tw )) ⊆ Fw ({tw }) ↑ 6. F is a class of total strictly decreasing functions (resp. (U-Dom) strictly decreasing functions) if and only if, for all tw ∈ CoordΣ (resp. tw ∈ Dom(Fw )), we obtain the following inclusions (str-dec+ ) and (str-dec− ): str-dec+ str-dec− Fw ((tw , →)) ⊆ Fw ({tw })↓∗ and Fw ((← ,tw )) ⊆ 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 obtain the following inclusion (con): con
Fw ((←, tw )) ∪ Fw ((tw , →)) ⊆ Fw ({tw }) Definition 3. A functional model on Σ is a tuple M = (Σ, h), where Σ is a functional frame and h : LT ×W −→ 2CoordΣ is a function, called a functional 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(2A) = {tw ∈ CoordΣ | Fw ({tw }) ⊆ h(A)}. Definition 4. Let A be a formula in LT ×W . Let (Σ, h) be a functional model and tw ∈ CoordΣ , then, A is true at tw if tw ∈ h(A). A is said to be valid in the functional model (Σ, h) if h(A) = CoordΣ . If A is valid in every functional model on Σ, then A is said to be valid in the functional frame Σ, and denote it by |=Σ A. If A is valid in every functional frame, then A is said 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 to be valid in K.
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3 Definability in LF T ×W Definition 5. Let J be a class of functional frames and let K ⊆ J. We say that K is LF T ×W -definable in (or relative to) J if there exists a set Γ of formulas in LT ×W such that for every frame Σ ∈ J, we have that Σ ∈ K if and only if every formula of Γ is valid in Σ. If J is the class of all functional frames, 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 is F LF T ×W -definable if K is LT ×W -definable. In the rest of this paper, “definable” means “LF T ×W -definable”. Theorem 4. K = {(W, T , F) | F is a class of total functions } is definable. Proof. We will prove that the formula (T ot) 2(HA ∧ A ∧ GA) → (H2A ∧ G2A) defines K. To this end, let (Σ, h) be such that Σ ∈ K and tw ∈ CoordΣ . Then: iff tw ∈ h(2(HA ∧ A ∧ GA)) tw ∈ h(2HA) ∩ h(2A) ∩ h(2GA) iff Fw ({tw }) ↓∗ ∪ Fw ({tw }) ∪ Fw ({tw }) ↑∗ ⊆ h(A) iff Fw ({tw }) ↓∗ ∪ Fw ({tw }) ↑ ⊆ h(A)) On the other hand, tw ∈ h(H2A) iff Fw ((←, tw )) ⊆ h(A) and tw ∈ h(G2A) iff Fw ((tw , →)) ⊆ h(A). Now, by the inclusion (Fw -∗1 ) in Theorem 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 where h(p) = Fw ({tw })↓∗ ∪ Fw ({tw }) ↑. Then, we obtain tw ∈ h(2(Hp ∧ p ∧ Gp)) and Fw ((←, tw )) ∪ Fw ((tw , →)) h(p) Therefore tw ∈ h(H2p ∧ G2p) 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 (U D) 2(HA ∧ A ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)) defines K. The validity of (U D) in K is an immediate consequence of:
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i) if tw ∈ DomU (Fw ), then tw ∈ h(2⊥), and ii) 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 (T ot) in the theorem above and the proof is complete. Theorem 6. K = {(W, T , F) functions} is definable.
|
F is a class of (U-Dom)-non-total
Proof. We will prove that the following set defines K: {2(HA ∧ A ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)), 2P ⊥ ∨ 2⊥ ∨ F 2⊥} The validity of 2(HA ∧ A ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)) in K is guaranteed by Theorem 5, since the class of (U-Dom)-non-total functions is a subclass of the class of (U-Dom)-partial functions. On the other hand, for every frame in K, we have that for all Tw there is at least one tw ∈ Tw such that tw ∈ DomU (Fw ). Clearly, tw ∈ h(2⊥) if and only if tw ∈ DomU (Fw ). Thus, from the linearity of <w , we have that the formula (N on-T ot) : 2P ⊥ ∨ 2⊥ ∨ F 2⊥ holds at every coordinate in Tw . For the reverse, let (W, T , F) be a functional frame not in K. We consider the following two possible cases: w w
(i) There exists some total function −→∈ F. Then 2⊥ fails to be true at every coordinate in Tw (whatever the model based on that frame) and therefore 2P ⊥ ∨ 2⊥ ∨ F 2⊥ is not valid in that frame. (ii) F does not satisfy the (U-Dom) condition. In this case, we work as in Theorem 5. Theorem 7. The following classes of functional frames are definable: (1) (2) (3) (4) (5) (6) (7)
K1 = {(W, T , F) | F is a class of total injective functions }. K2 = {(W, T , F) | F is a class of surjective functions }. K3 = {(W, T , F) | F is a class of total increasing functions }. K4 = {(W, T , F) | F is a class of total strictly increasing functions }. K5 = {(W, T , F) | F is a class of total decreasing functions }. K6 = {(W, T , F) | F is a class of total strictly decreasing functions }. K7 = {(W, T , F) | F is a class of total constant functions }.
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Proof. (1) We will prove that 2(HA ∧ GA) → (H2A ∧ G2A) defines K1 . For this, let (Σ, h) be such that Σ ∈ K1 , then for any tw ∈ CoordΣ we have that: tw ∈ h(2(HA ∧ GA)) tw ∈ h(2HA) ∩ h(2GA) Fw ({tw }) ↓∗ ∪ Fw ({tw }) ↑∗ ⊆ h(A)
iff iff
On the other hand, tw ∈ h(H2A) if and only if Fw ((←, tw )) ⊆ h(A) and tw ∈ h(G2A) if and only if Fw ((tw , →)) ⊆ h(A). Now, by (inj) in Theorem 3, the proof of validity is complete. Reciprocally, if Σ ∈ K1 , a countermodel for the formula 2(HA ∧ GA) → (H2A ∧ G2A) can be defined in a similar way as for (T ot) in theorem 4, and the proof is complete. (2) The formula (H2A ∧ G2A) → 2(HA ∧ GA) defines K2 . This is a consequence of (surj) in Theorem 3. (3) The set {2(A ∧ GA) → G2A, 2(HA ∧ A) → H2A} defines K3 . This is a consequence of (inc+ ) and (inc− ) in Theorem 3. (4) The set {2GA → G2A, 2HA → H2A} defines K4 . This is a consequence of (str-inc+ ) and (str-inc− ) in Theorem 3. (5) The set {2(A ∧ GA) → H2A, 2(HA ∧ A) → G2A} defines K5 . This is a consequence of (dec+ ) and (dec− ) in Theorem 3. (6) The set {2GA → H2A, 2HA → G2A} defines K6 . This is a consequence of (str-dec+ ) and (str-dec− ) in Theorem 3. (7) The formula 2A → (H2A ∧ G2A) defines K7 . This is a consequence of (con) in Theorem 3. Theorem 8. Let K be a class of functional frames, (W, T , F), where F is a class of functions that satisfies the (U-Dom) condition and in which all functions 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 their corresponding formulas as follows: (U D-Inj): 2(HA ∧ GA) → (2⊥ ∨ (H2A ∧ G2A)) (U D-Inc): {2(A ∧ GA) → (2⊥ ∨ G2A), 2(HA ∧ A) → (2⊥ ∨ H2A)} (U D-Str-Inc): {2GA → (2⊥ ∨ G2A), 2HA → (2⊥ ∨ H2A)} (U D-Dec): {2(A∧GA) → (2⊥∨H2A), 2(HA∧A) → (2⊥∨G2A)} (U D-Str-Dec): {2GA → (2⊥ ∨ H2A), 2HA → (2⊥ ∨ G2A)} (U D-Con): 2A → (2⊥ ∨ (H2A ∧ G2A))
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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 → GP A (H2) A → HF A (T R) 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 logic K: (K) 2(A → B) → (2A → 2B) 4. The characteristic axiom schema: (T ot) 2(HA ∧ A ∧ GA) → (H2A ∧ G2A) The rules of inference are those of Kl + K: (M P ): A, A → B B; (RG): A GA; (RH): A HA and (RN ) A 2A The concepts of proof and theorem are defined as usual. Theorem 9. The following formulas are theorems in ST ×W -Tot, T1: F 2A → 2(P A ∨ A ∨ F A).
T2: P 2A → 2(P A ∨ A ∨ F A).
Proof. We prove the theorem T1 (T2 is its mirror image). Let ψ = H¬A ∧ ¬A ∧ G¬A 1. 2(H¬ψ ∧ ¬ψ ∧ G¬ψ) → H2¬ψ 2. P ♦ψ → (♦P ψ ∨ ♦ψ ∨ ♦F ψ) 3. ψ → G¬A 4. ♦P ψ → ♦P G¬A 5. ♦P G¬A → ♦¬A 6. ♦P ψ → ♦¬A 7. ♦ψ → ♦¬A 8. ♦F ψ → ♦¬A 9. P ♦ψ → ♦¬A 10. 2A → H2¬ψ 11. ¬ψ → (P A ∨ A ∨ F A) 12. H2 ¬ψ → H2(P A ∨ A ∨ F A) 13. 2A → H2(P A ∨ A ∨ F A) 14. F 2A → 2(P A ∨ A ∨ F A)
from (T ot) and PL from 1, by K and definitions of ♦, F, P by PL from 3, by Kl and K by Kl and K from 4, 5 by PL by K [similar to 3-6] from 2, 6, 7, 8 by PL from 9 by PL and definitions of ♦ and P by PL and definitions of F and P from 11 by K and Kl from 10, 12 by PL from 13 by Kl
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Completeness theorem for ST ×W -Tot The proof of soundness is standard. We focus our attention on the completeness proof and adopt the usual definitions of a consistent (maximal) set of formulas. In the following, we abbreviate maximal consistent set as mc-set. On the other hand, by S we mean any axiomatic system for LT ×W which is an extension of Kl + K (in particular, the system ST ×W -Tot). Familiarity with the basic properties of mc-sets in classical propositional systems is assumed. Definition 6. Let Γ1 and Γ2 be mc-sets in S, then we define: Γ1 ≺T Γ2 Γ1 ≺W Γ2
if and only if {A | GA ∈ Γ1 } ⊆ Γ2 . if and only if {A | 2A ∈ 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 {F A | A ∈ Γ2 } ⊆ Γ1 iff {P A | A ∈ Γ1 } ⊆ Γ2 iff {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 F A ∈ Γ1 , exists an mc-set Γ2 ∈ S such that Γ1 ≺T Γ2 and A ∈ Γ2 . (b) If P A ∈ 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 that Γ2 ≺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 that Γ1 ≺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 conditions is satisfied: (i)
Γ2 ≺W Γ3 . In this case, we obtain the result taking Γ4 = Γ3 .
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Γ1
≺W@
R @
Γ4 = Γ3
- Γ2
≺W
(ii) {A | 2A ∈ Γ2 } ∪ {A | GA ∈ Γ3 } is consistent. In this case, Lindenbaum’s lemma guarantees that there exists at least one mc extension, Γ4 , of the set {A | 2A ∈ Γ2 } ∪ {A | GA ∈ Γ3 } which trivially satisfies Γ2 ≺W Γ4 and Γ3 ≺T Γ4 . ≺T
Γ1 ≺W ?
≺T
Γ3
- Γ2 ≺W ? -Γ 4
(iii) {A | 2A ∈ Γ2 } ∪ {A | HA ∈ Γ3 } is consistent. Now, Lindenbaum’s lemma again guarantees that there exists at least one mc extension, Γ4 , of the set {A | 2A ∈ Γ2 }∪{A | HA ∈ Γ3 } which satisfies Γ2 ≺W Γ4 and Γ4 ≺T Γ3 . ≺T Γ1 Γ2
≺W ≺W
) ≺T - ?
Γ4 Γ3 If we assume that none of the conditions (i)–(iii) holds, we have: a) there exists a formula 2A ∈ Γ2 such that A ∈ Γ3 ; B1 , . . . , Bn1 ∈ {A | 2A ∈ Γ2 }, b) there are C1 , . . . , Cm1 ∈ {A | GA ∈ Γ3 } such that ¬(B1 ∧ . . . ∧ Bn1 ∧ C1 ∧ . . . ∧ Cm1 ); D1 , . . . , Dn2 ∈ {A | 2A ∈ Γ2 }, c) there are E1 , . . . , Em2 ∈ {A | HA ∈ Γ3 } such that ¬(D1 ∧ . . . ∧ Dn2 ∧ E1 ∧ . . . ∧ Em2 ). B = B1 ∧ . . . ∧ Bn1 , with 2B ∈ Γ2 ; C = C1 ∧ . . . ∧ Cm1 , with GC ∈ Γ3 ; Thus, we have: D = D1 ∧ . . . ∧ Dn2 , with 2D ∈ Γ2 ; E = E1 ∧ . . . ∧ Em2 , with HE ∈ Γ3 Now, from ¬(B ∧ C), we obtain 2B → 2¬C and (since 2B ∈ Γ2 ) we have 2¬C ∈ Γ2 . Similarly, from ¬(D ∧ E), we obtain 2¬E ∈ Γ2 . Thus, 2(A ∧ ¬C ∧ ¬E) ∈ Γ2 and, since Γ1 ≺T Γ2 , we obtain F 2(A ∧ ¬C ∧ ¬E) ∈ Γ1 . Therefore, by (T 1) in Theorem 9, we obtain: 2(P (A ∧ ¬C ∧ ¬E) ∨ (A ∧ ¬C ∧ ¬E) ∨ F (A ∧ ¬C ∧ ¬E)) ∈ Γ1 Henceforth, by the definition of ≺W , either
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(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), and GC ∈ 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 is any function ΦΣ : CoordΣ −→ 2LT ×W such that, for all tw ∈ CoordΣ , the set 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 , tw ∈ CoordΣ : if tw ∈ (tw , →), then ΦΣ (tw ) ≺T ΦΣ (tw ) modally coherent if, for all tw , tw ∈ CoordΣ : if tw ∈ Fw ({tw }), then ΦΣ (tw ) ≺W ΦΣ (tw ) coherent if it is temporally coherent and modally coherent. prophetic if it is temporally coherent and, moreover, for all wff, A, and all tw ∈ CoordΣ : (1) if F A ∈ ΦΣ (tw ), there exists a tw ∈ (tw , →) such that A ∈ ΦΣ (tw ) historic if it is temporally coherent and, moreover, for all wff, A, and all tw ∈ CoordΣ : (2) if P A ∈ ΦΣ (tw ), there exists a tw ∈ (←, tw ) such that A ∈ ΦΣ (tw ) possibilistic if it is modally coherent and, moreover, for all wff, A, and all tw ∈ CoordΣ : (3) if ♦A ∈ ΦΣ (tw ), there exists a tw ∈ Fw ({tw }) such that A ∈ ΦΣ (tw ). The conditional (1) (resp., (2) or (3)) is called a prophetic (historic or possibilistic) conditional for ΦΣ with respect to F A, (P A or ♦A) and tw . Definition 9. Let ΦΣ be a trace of Σ = (W, T , F). ΦΣ is called total if it is coherent and Σ satisfies the following property: for all tw , tw , tw ∈ CoordΣ (4) if tw ∈ Fw ({tw }) and tw = tw , then there exists tw ∈ Fw ({tw }) 2 (4) is called a total conditional for ΦΣ with respect to tw , tw and tw . ΦΣ is called total-full if it is prophetic, historic, possibilistic, and total. Definition 10. Let WΞ be a denumerable infinite set and TΞ = Ξ Ξ w∈WΞ Tw where Tw is a denumerable infinite set for each w ∈ WΞ and if w = w , then TwΞ ∩ TwΞ = ∅. We consider the class, Ξ, of functional frames, (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.
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– T = {(Tw ,
