A regular dynamic logic HIREAO of having executed actions on objects *

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1 A regular dynamic logic HIREAO of having executed actions on objects * Li Xiaowu (Institute of Logic and Cognition of Sun Yat-sen University, Philosophy Department of Sun Yat-sen University, , Guangdong, China) Abstract: Usual dynamic logics do not directly characterize the logical properties of having executed an action on objects from the aspect of syntax. We have presented a regular dynamic logic of having executed actions on objects in [1] of References in the end of this paper. Now, from another point of view, we will establish another regular dynamic logic directly characterizing having executed actions on objects by Half-infinitary method, and then present the corresponding semantics, and thus prove that the system is strongly sound and strongly complete with respect to the semantics, respectively. From the point of view of linguistic logics, to study the logical properties of having executed actions on objects is to study the logical properties of verb phrases made up of transitive verbs, so the results that we shall obtain below will make progress with the study of linguistic logics. Keywords: having executed actions on objects; regular dynamic logic; Half-infinitary method 1. Introduction Usual dynamic logics do not directly characterize the logical properties of having executed an action from the aspect of syntax (w.r.t an axiomatization system). For example, they do not directly characterize the sentences of the form He has gone. In other words, they just characterize the sentences of the form [α]ϕ w.r.t an action α (the intended interpretation of [α]ϕ is that it is necessary that after executing α, ϕ is true), but such sentences just indirectly characterize the action α itself. As for the sentences of the form He has gone form which we can generalize the sentences of the form some agent has executed action α, denoted as Eα, the logics do not reveal what are logical properties of them. But in my opinion, such sentences are very important in the way of investigating agent having executed actions. In [2] of References, we give out a regular dynamic logic REA directly characterizing having executed actions. But we cannot directly characterize the logical properties of having Fund: The paper is sponsored by a key project (No. 05JJD ) from key research institutes for humanities and social sciences of Chinese Ministry of Education. Introduction to the author: Li Xiaowu (1955- ), professor of Sun Yat-sen University and visiting researcher of Shusheng Research Center of Beijing Shusheng Company. LXW121@yahoo.com.cn LXWLYY@hotmail.com 1

2 A regular dynamic logic HIREAO of having executed actions on objects executed an action on objects in REA. For example, it does not directly characterize the sentences of the form I have updated this computer or He has pulled out a wagon (from this room). In p. 147 of [3], D. C. Dennett uses PULLOUT(WAGON, ROOM) to represent to pull out the wagon from the room. If we view ROOM as a state in which PULLOUT is an action acted on the wagon, then we can roughly view PULLOUT as a function acted on the wagon and other things such that PULLOUT(WAGON) corresponds to a relation between a state u and another state w, where u can be viewed as the room in which the wagon was still and w can be viewed as the room from which the wagon has been pulled out. Let α be an action and o an object. We will logically characterize that some agent has executed α acted on an object o. From the above view, we can introduce a binary accessibility relation R α(o). That some agent has executed α on o, denoted by Eαo, can be characterized by the following truth definition: (%) Eαo is true in a state w there exists a state u such that ur α(o) w. So, in [1], we gave out a regular dynamic system REAO characterizing having executed actions on objects, and then proved the strong soundness theorem and the strong completeness theorem w.r.t. a semantics for (%). Now we consider PULLOUT(WAGON, ROOM) from another point of view. If we view WAGON and ROOM as two objects, and PULLOUT as an action acted on the wagon such that after PULLOUT has been executed, a relation between WAGON and ROOM is changed. In other words, an original relation between WAGON and ROOM does not exist, so, at least a relation between WAGON and ROOM does not exist. If we analogously consider I have updated this computer, then after I have updated the computer, at least a property of the computer is changed. In other words, at least a property of the computer does not exist. In both of the cases, we can also say that after we have executed an action α acted on an object o, either at least a new relation between WAGON and ROOM appears or at least a new property of the computer appears. So after having been executed an action α acted on an object o, we have actually changed at least a property of o or a relation between o and other some objects, and thus turned a state into another state. Hence we have the following truth definition: (#) Eαo is true in a state w there exist a state u, an n-ary relation r and objects o 1,, o n 1 such that ur α w, and either o 1,, o,, o n 1 have r in u and o 1,, o,, o n 1 do not have r in w, or o 1,, o,, o n 1 do not have r in u and o 1,, o,, o n 1 have r in w. 1 From the point of view of linguistic logics, to study the logical properties of having executed actions on objects is to study the logical properties of verb phrases made up of transitive verbs, so the results that we shall obtain below will make progress with the study of linguistic logics. This paper is organized as follows. In Section 2, we give out a dynamic semantics with (#) and the corresponding half-infinitary inference system HIREAO describing Eαo, and then prove the strong soundness theorem w.r.t. the semantics. In Section 3, we prove the strong completeness theorem w.r.t. the semantics by Adapted Lindenbaum lemma. 2. System, Semantics and strong soundness theorem We first give the definition of language expressing the system HIREAO. From the (#) in Section 1 When n = 1, r can be seen as a property. 2

3 1, it is clear that in essence, we do not need quantifiers, so, for simplicity, we just introduce a kind of prepositional language below. Let PS = {P 1,, P n, } be a countable set of predicate symbols. Let f: PS (N {0}) be a mapping, where N is the set of all natural numbers, such that f assigns to each P PS a finite arity. Let OS = {o 1,, o n, } be a countable set of object symbols (individual constant symbols) such that OS PS =, and AA a countable set of atomic actions. Definition 2.1 (Language of HIREAO) (1) The language L of HIREAO consists of a set of formulas ϕ and a set of actions α, given by the following rules: α ::= a (α β) (α ; β) α *, where a AA ϕ ::= Pa 1 a f(p) ϕ (ϕ ψ) [α]ϕ Eαo, where P PS and a 1,, a f(p), o OS. Let Action be the class of all actions defined as above. (2) For all o OS, PO(o) is defined as the smallest set closed under the following condition: Given any P PS, if f(p) = 1, then Po PO(o); if f(p) > 1, then for all a 1,, a n 1, a n + 1,, a f(p) OS, Pa 1 a n 1 oa n + 1 a f(p) PO(o). (3) PO is defined as {PO(o) o OS}. 1 (4) Given any θ PO, define a pair of dual formulas θ # and θ #d as follows: θ # {θ, θ}, and if θ # = θ then θ #d = θ, if θ # = θ then θ #d = θ. (5) [α n ]ϕ is defined as follows: [α 0 ]ϕ = ϕ, [α n+1 ]ϕ = [α][α n ]ϕ. In above (2), Po intuitively represents that o has the property P; if f(p) > 1, Pa 1 a n 1 oa n + 1 a f(p) intuitively represents that there exists the relation P between a 1,, a n 1, o, a n + 1,, a f(p). Note that there are two sorts of atomic formulas here: PO, {Eαo α Action and o OS }, as all they cannot be separated into smaller formulas. The intended interpretation of [α]ϕ is that ϕ is a prerequisite to α has been executed. The intended interpretation of α β is that choose either α or β nondeterministically and execut it. The intended interpretation of α ; β is that execut α, then execut β. The intended interpretation of α* is that execut α a nondeterministically chosen finite number of times (zero or more). α β, α ; β and α* are called compound actions., ; and * of α β, α ; β and α* are called the choice operator, the composition operator and the iteration operator, respectively (cf. p. 166 in [4] of References). Formulas (ϕ ψ), (ϕ ψ) and (ϕ ψ) are defined as usual. For convenience, we usually abbreviate if then, if and only if, for all, there exist(s) by,, and, respectively. We will omit the outside parentheses of a formula. We also usually omit the inside parentheses subject to the convention that, [α],,,, are taken in this order of priority. We use T as P 1 o 1 o f(p1 ) P 1 o 1 o f(p1 ), as T, <α>ϕ as [α] ϕ, Φ and Ψ as formula sets, 1 Hence PO is the set of all the formulas of the form ra 1 a f(r). 3

4 A regular dynamic logic HIREAO of having executed actions on objects ϕ 1 ϕ 2 ϕ n as ϕ 1 (ϕ 2 ( ϕ n ) ). Definition 2.2 (Axioms for HIREAO) Here follows the set of axioms of HIREAO: (Taut) all instantiations of tautologies, (K α ) [α](ϕ ψ) [α]ϕ [α]ψ, (AX ) [α β]ϕ [α]ϕ [β]ϕ, (AX ; ) [α ; β]ϕ [β][α]ϕ, (AX * ) [α * ]ϕ ϕ [α][α * ]ϕ, (EAo) <α>θ # θ #d Eαo for all θ PO(o). Note that the when α is fixed, EAo is not an axiom schema because all the θ in EAo are restricted within PO(o). In the following we give the half-infinitary proof system HIREAO by inductively defining a derivation relation Φ ϕ. Besides the usual shorthand notation Φ, ϕ for Φ {ϕ}, Φ, Ψ for Φ Ψ, ϕ for ϕ, and ϕ 1,, ϕ n ψ for {ϕ 1,, ϕ n } ψ, we shall also write: Φ Ψ for Φ ψ for all ψ Ψ, [α]φ for {[α]ϕ ϕ Φ}, [α] Φ for {ϕ [α]ϕ Φ}, ϕ Φ for {ϕ ψ ψ Φ}. Definition 2.3 (Half-infinitary derivation relation for HIREAO) (1) is defined as the smallest relation between sets of formulas of L and formulas of L such that is closed under the following rules. By convention, we use Φ ϕ as (Φ, ϕ) below: (AX) ϕ is an axiom of HIREAO ϕ; (MP) ϕ, ϕ ψ ψ; (Inf * ) {[α n ]ϕ n N} [α * ]ϕ; (Inf EAo ) {<α>θ # θ # θ PO(o)} Eαo; (SN α ) Φ ϕ [α]φ [α]ϕ; (Ded) Φ, ϕ ψ Φ ϕ ψ; (Weak) Φ ϕ Φ, Ψ ϕ; (Cut) Φ Ψ and Φ, Ψ ϕ Φ ϕ. (2) Every Φ ϕ can be called a sequent, the relation can be called a sequent calculus or sequent system. Axiom EAo and Rule Inf EAo characterize the basic properties of having executed α acted on o, Axiom A, A ; and AX * characterize the basic property of the compound action operation α β and α ; β and α*, respectively. The dynamic logic (system) merely characterizing α β, α ; β and α* is called the regular logic in p. 22 and p. 240 in [5] of References, but the logic consisting of such a logic and Axiom [?ϕ]ψ (ϕ ψ) is called the regular logic in p. 203 in [4], where the axiom characterizes the test action operation?ϕ of testing ϕ (Note that the test action?ϕ is a 4

5 action acted on an object (proposition) ϕ). Here we follow the appellation of [5], because, by adding the test action operation and the above axiom, we cannot prove the frame soundness theorem and the frame completeness theorem of HIREAO w.r.t. the following semantics, but this is an important goal that logics pursue. On the other hand, from Axiom [?ϕ]ψ (ϕ ψ) and Rule Inf EAo, it is easy to obtain E(?ϕ)o. In my opinion, this is not natural. Note that strictly speaking, MP, Inf * and Inf EAo are sequents rather than rules. In the definition above, the language remains finitary (all formulas have finite length) and only the sequents Inf * and Inf EAo are non-standard (requiring infinitely many premises). In other words, Half-infinitary derivation relation defined in the previous definition actually give a half-infinitary (inference) system formed by finitary axioms, finitary rules and infinitary rules, because the intended interpretations of the sequents are the inferences that we want to describe. Such a method of constructing half-infinitary systems and then prove the strong soundness theorems and the strong completeness theorems for them is called Half-infinitary method. Lemma 2.4 (1) ϕ is an axiom of HIDE Φ ϕ. (by AX and Weak) (2) ϕ [α]ϕ. (by SN α ) (3) ϕ ψ ϕ ψ. (by Ded) (4) Φ Ψ and Ψ ϕ Φ ϕ; (by Weak and Cut) Φ ψ and ψ ϕ Φ ϕ. (5) Φ ϕ ψ Φ, ϕ ψ; ϕ ψ ϕ ψ; ϕ ψ ϕ ψ. (6) Φ ϕ and ϕ ψ Φ ψ; (by (5) and (4)) Φ ϕ and Φ ϕ ψ Φ ψ; θ ϕ and ϕ ψ θ ψ; θ ϕ and ϕ ψ θ ψ; ϕ and ϕ ψ ψ. (7) Φ ϕ ψ Φ ψ ϕ. (Conditionalization rule) (8) ϕ Φ Φ ϕ. (9) Φ, ϕ Φ ϕ. (by Ded, (1) and (6)) (10) [α * ]ϕ [α n ]ϕ for all n N. (11) ϕ [α * ](ϕ [α]ϕ) [α * ]ϕ. (Induction axiom) (12) Φ {[α n ]ϕ n N} Φ [α * ]ϕ. (13) Φ {<α>θ # θ # θ PO(o)} Φ Eαo. Proof. Proof of (5). Assume that Φ ϕ ψ. We have: Φ, ϕ ϕ ψ by Weak. On the other hand, we have: Φ, ϕ, ϕ ψ ψ by MP and Weak. Hence we have: Φ, ϕ ψ by Cut. So, let Φ =, we have: ϕ ψ ϕ ψ, hence ϕ ψ ϕ ψ by (3). Proof of (7). Assume that Φ ϕ. 1 ψ θ, ψ θ, θ Φ 2 ψ Φ, ψ Φ, by MP by 1, Weak 5

6 A regular dynamic logic HIREAO of having executed actions on objects 3 ψ Φ, ψ, Φ ϕ, 4 ψ Φ, ψ ϕ, 5 ψ Φ ψ ϕ. by the assumption, Weak by 2, 3, Cut by 4, Ded Proof of (8). By AX and (5), ϕ ϕ, so Φ ϕ by the assumption and Weak. Proof of (10). By AX *, 1 [α * ]ϕ ϕ, and 2 [α * ]ϕ [α][α * ]ϕ. [α][α * ]ϕ [α]ϕ by 1, (2), K α and (6), so [α * ]ϕ [α]ϕ by 2 and (6). Continue the process, we can finally get what we want to prove. Proof of (11). Using Inf * and AX * (cf. p. 5 in [5]). Proof of (12). : Using Inf * and (4). : Using (9) and (6). Proof of (13). : Using Inf EAo and (4). : Using EAo and (6). Definition 2.5 (1) A dynamic frame for L is a tuple (W, R) such that W is a set of states or possible worlds such that W, R is a mapping such that R(a) is a binary relation on W for all a AA. (2) A dynamic model for L is a tuple (W, R, V) such that (W, R) is a dynamic frame and V: PO (W) is a mapping such that V assigns a set of states to each θ PO, where (W) is the power set of W. V is also called a valuation on (W, R). (3) Let Frame be the class of dynamic frames and Model the class of dynamic models. Below R(a) is abbreviated as R a. For each w W, we introduce R a (w) ::= {u W: ur a w}. Definition 2.6 (Truth definition) Let M = (W, R, V) Model. For every formula ϕ PO, the truth set V(ϕ) of ϕ w.r.t. M is defined inductively as follows: for all w W, (1) w V( ϕ) w V(ϕ), (2) w V(ϕ ψ) w V(ϕ) and w V(ψ), (3) w V([α]ϕ) R α (w) V(ϕ) where R α (w) ::= {u W: ur α w}, (4) w V(Eαo) θ PO(o), w V(<α>θ # θ #d ); where R α is extended by the following frame conditions: (ch) R α β ::= R α R β = {(u, w) W 2 ur α w or ur β w}, (co) R α ; β ::= R α ο R β = {(u, w) W 2 v W (ur α v and vr β w)}, (it) R α * = {(u, w) W 2 n N v 0,, v n W (w = v 0, u = v n and for 0 m n 1, v m + 1R α v m )}. R α * is called the reflexive transitive closure of R α. Given any w, u W, ur α n w is defined as follows: ur α 0 w u = w, ur α n + 1 w v W (ur α n v and vr α w). So, by 2.6 (it), we have: ur α *w n N v 0,, v n W (w = v 0, u = v n and for 0 m n 1, v m + 1R α v m ) n N (ur α v n 1 and and v 1 R α w) n N (ur α n w). 6

7 Hence we have: ur α *w n N (ur n α w) for all u, w W. And thus R α * = {R n α n N}. It is easy to see that the semantics of the operator [α] is the same as one of the tense operator H by (3), so the semantics of the operator <α> is the same as one of the tense operator P. (cf. p. 93 in [7] of References.) By Truth definition, we have: w V(Eαo) θ PO(o), w V(<α>θ # θ #d ) θ PO(o), w V(<α>θ # ) and w V(θ #d ) θ PO(o), u W (ur α w and u V(θ # )) and w V(θ #d ) θ PO(o) u W (ur α w and u V(θ # ) and w V(θ #d )) by (5) in [8] (Note that here u w). This is just what (#) in Section 1 says. Definition 2.7 By Φ ϕ we mean the pointed consequence relation between Φ and ϕ: for any (W, R, V) Model and w W, w V(ψ) for every ψ Φ w V(ϕ). Below we shall also write w V(Φ) for w V(ψ) for every ψ Φ. Theorem 2.8 (Strong soundnesss theorem) Φ ϕ Φ ϕ for all Φ L and ϕ L. Proof. It suffices to verify that Φ ϕ Φ ϕ for every rule in 2.3 (1). Verify AX: Consider Axiom K α. Given any (W, R, V) Model and w W, assume that w V([α](ϕ ψ)) and w V([α]ϕ), hence, by 2.6 (3), R α (w) V(ϕ ψ), R α (w) V(ϕ). Obviously, R α (w) V(ψ), so w V([α]ψ) by 2.6 (3) again. Consider Axiom AX. Given any (W, R, V) Model and w W, we have: w V([α β]ϕ) R α β (w) V(ϕ) by 2.6 (3) u W (ur α β w u V(ϕ)) u W ((ur α w or ur β w) u V(ϕ)) by 2.6 (ch) u W (ur α w u V(ϕ)) and u W (ur β w u V(ϕ)) R α (w) V(ϕ) and R β (w) V(ϕ) w V([α]ϕ [β]ϕ) by 2.6 (3) (2) Consider Axiom AX ;. Given any (W, R, V) Model and w W, we have: w V([α ; β]ϕ) R α ; β (w) V(ϕ) by 2.6 (3) u W (ur α ; β w u V(ϕ)) u W ( v W (ur α v and vr β w) u V(ϕ))) by 2.6 (co) v, u W (vr β w (ur α v u V(ϕ))) v W (vr β w R α (v) V(ϕ)) v W (vr β w v V([α]ϕ)) by 2.6 (3) R β (w) V([α]ϕ) w V([β][α]ϕ) by 2.6 (3) 7

8 A regular dynamic logic HIREAO of having executed actions on objects Consider Axiom AX *. Given any (W, R, V) Model and w W, we have: w V([α * ]ϕ) R α *(w) V(ϕ) by 2.6 (3) u W (ur α *w u V(ϕ)) u W ( n N (ur n α w) u V(ϕ)) by (@) below 2.6 u W n N (ur n α w u V(ϕ)) n N (R n α (w) V(ϕ)) n N (w V([α n ]ϕ)) by 2.6 (3) w V([α 0 ]ϕ) and n N {0}(w V([α n ]ϕ)) w V(ϕ) and n N {0}(w V([α n ]ϕ)) by 2.1 (5) w V(ϕ) and u W (ur α w u V([α * ]ϕ)) w V(ϕ [α][α * ]ϕ) by 2.6 (3) As usual, it is easy to verify the remaining axioms in AX. Verify Rule Inf EAo : assume that {<α>θ # θ # θ PO(o)} Eαo. We will prove that {<α>θ # θ # θ PO(o)} Eαo. Assume that for any (W, R, V) Model and w W, we have: 1 w V({<α>θ # θ # θ PO(o)}). On the other hand, by Truth definition 2.6, we have: 2 w V( Eαo) θ PO(o), w V(<α>θ # θ #d ). By 2.1 (4), we have: 3 w V( Eαo) θ PO(o), w V(<α>θ # θ # ). So w V( Eαo) by 1 and 3. Verify Rule SN α : assume that [α]φ [α]ϕ such that [α]φ [α]ϕ is derived from Φ ϕ by SN α. By the induction hypothesis about Φ ϕ, we have: Φ ϕ. We will prove that [α]φ [α]ϕ. Assume that for any (W, R, V) Model and w W, we have: w V([α]Φ). We will prove that w V([α]ϕ). Given any u R α (w). Since w V([α]Φ), u V(Φ). Since Φ ϕ, u V(ϕ). As usual, it is easy to verify the remaining rules in 2.3 (1). 3. Strong completeness theorem We shall show that the system HIREAO is also strongly complete in the following sense: Φ ϕ Φ ϕ for all Φ L and ϕ L. To prove this, we first need the following definitions and lemmas. Definition 3.1 (1) Φ is consistent Φ. (2) Φ is maximal Φ contains exactly one from ϕ, ϕ for every formula ϕ L. (3) Φ is maximal consistent it is consistent and maximal. Lemma 3.2 Let Φ be a maximal consistent set. (1) ϕ Φ ϕ Φ. ϕ ψ Φ ϕ Φ and ψ Φ. 8

9 ϕ ψ Φ ϕ Φ or ψ Φ. ϕ, ϕ ψ Φ ψ Φ. (2) ϕ ϕ Φ. Proof. (1) As an example, we will prove that ϕ Φ and ϕ ψ Φ ψ Φ. Given any ϕ, ϕ ψ Φ. Assume that ψ Φ, ψ Φ by Maximality of Φ. Hence Φ ϕ, Φ ϕ ψ, Φ ψ by 2.4 (8), so it is easy to prove Φ by 2.4 (6), contradicting Consistency of Φ. (2) Assume that ϕ. Then Φ ϕ by Weak. If ϕ Φ, ϕ Φ by Maximality of Φ, Φ ϕ by 2.4 (8), so it is easy to prove Φ, contradicting Consistency of Φ. We shall prove strong completeness of HIREAO via the Henkin construction of a canonical model. In the infinitary context, the usual Lindenbaum lemma that allows for extending every consistent set to a maximal consistent set cannot be straightforwardly proved. The trick used by [6] of References is to use an adapted Lindenbaum lemma based on the seemingly weaker concept of saturation instead of maximal consistency, and then to show that the two concepts, in fact, coincide. There are two steps to the strong completeness proof: first we show that every consistent set can be extended to a saturated and thus maximal consistent set, and then we construct a canonical model consisting of maximal consistent sets. We first introduce a pseudo-modality s (cf. the proof of Lemma 9 in [6] of References): Definition 3.3 (1) s is a pseudo-modality if s = (s 1,, s n ) is a finite (possibly empty) sequence consisting of the modalities of {[α] α Action} and the formulas of L. (2) Let s be a pseudo-modality. The formula sϕ is inductively defined as follow: ( )ϕ = ϕ where ( ) is the empty sequence, (ψ, s 2,, s n )ϕ = ψ (s 2,, s n )ϕ, ([α], s 2,, s n )ϕ = [α](s 2,, s n )ϕ. (3) Below we shall write sφ for {sϕ ϕ Φ}. For example, ([β], ψ, [α])ϕ is an abbreviation of the formula [β](ψ [α]ϕ). Lemma 3.4 Let s be any pseudo-modality. (K s ) s(ϕ ψ) sϕ sψ. (SN s ) Φ ϕ sφ sϕ. (RN s ) ϕ sϕ. (RM s ) ϕ ψ sϕ sψ. Proof. With induction over the length of s, by SN α and 2.4 (7), it is easy to prove that K s and SN s hold. By SN s, it is easy to see that RN s holds. By RN s, K s and 2.4 (6), it is easy to prove that RM s holds. Definition 3.5 (Saturated sets) Φ is saturated if the following conditions hold: (1) Φ is maximal: ϕ Φ ϕ Φ for all ϕ L, (2) Φ is finitely -closed: Ψ Φ is finite and Ψ ϕ ϕ Φ, (3) Φ is s-inf * -closed: s{[α n ]ϕ n N} Φ s[α * ]ϕ Φ for any pseudo-modality s, 9

10 A regular dynamic logic HIREAO of having executed actions on objects (4) Φ is s-inf EAo -closed: s{<α>θ # θ # θ PO(o)} Φ s Eαo Φ Lemma 3.6 Let Φ be a saturated set. Then (1) ϕ ϕ Φ. (by 3.5 (2)) (2) Φ. (by (1) and 3.5 (1)) (3) ϕ ψ, ϕ Φ ψ Φ. (by MP and 3.5 (2)) (4) (ϕ Φ ψ Φ) ϕ ψ Φ. (5) ϕ ψ Φ (ϕ Φ ψ Φ), (by (3) and (4)) ϕ ψ Φ (ϕ Φ ψ Φ). (6) ϕ ψ and ϕ Φ ψ Φ. (by (1) and (3)) (7) ϕ ψ and ϕ Φ ψ Φ. (by 3.5 (2)) for any pseudo-modality s. Proof. The proof of (4). We argue via contradiction: assume that ϕ ψ Φ, we have: ϕ ψ Φ by 3.5 (1), 3.6 (1) and (3). On the other hand, it is easy to prove that ϕ ψ ϕ and ϕ ψ ψ. So, by ϕ ψ Φ and 3.5 (2) - (1), we have: ϕ Φ and ψ Φ. We now prove that the concepts of saturated and maximal consistent sets coincide. Lemma 3.7 (Saturation lemma) (1) Every maximal consistent set is saturated. (2) Every saturated set is maximal consistent. Proof. (1) Let Φ be maximal consistent. First, we prove that Φ is finitely -closed: assume that Ψ Φ is finite and Ψ ϕ. Without loss of generality, we can assume that Ψ = {ψ 1,, ψ n }. By Ψ ϕ and Ded, we have: ψ 1 ψ n ϕ, So, by 3.2 (2), we have: ψ 1 ψ n ϕ Φ. Hence ϕ Φ by Ψ Φ and 3.2 (1). Second, we prove that Φ is s-inf * -closed: assume that s{[α n ]ϕ n N} Φ. By 2.4 (8), we have: Φ s{[α n ]ϕ n N}. By Inf *, SN s of 3.4 and 2.4 (4), we have: (#) Φ s[α * ]ϕ. Assume that s[α * ]ϕ Φ, [α * ]ϕ Φ by Maximality of Φ, so Φ [α * ]ϕ by 2.4 (8). So, by (#), it is easy to prove that Φ, contradicting Consistency of Φ. Third, we prove that Φ is s-inf EAo -closed: assume that s{<α>θ # θ # θ PO(o)} Φ. By 2.4 (8), we have: Φ s{<α>θ # θ # θ PO(o)}. By Inf EAo, SN s and 2.4 (4), we have: (%) Φ s Eαo. Assume that s Eαo Φ, s Eαo Φ by Maximality of Φ, so Φ s Eαo by 2.4 (8). So, by (%), it is easy to prove that Φ, contradicting Consistency of Φ. (2) Let Φ be saturated. Maximality of Φ follows directly from 3.5 (1), so we only have to show Consistency of Φ. For this, it suffices to show that Φ contains all its consequences, i.e. (a) Φ ϕ ϕ Φ. For, Φ by (a) and 3.6 (2), so Φ is consistent. 10

11 We will prove a more general statement than (a), namely: (b) Ψ ϕ and sψ Φ sϕ Φ for all Ψ L, ϕ L and pseudo-modality s. Let Ψ be Φ and s the empty sequence in (b), it is easy to see that (b) implies (a). Now we prove (b) with induction over a derivation of Ψ ϕ. So we assume Ψ ϕ and sψ Φ, we want to show sϕ Φ, and we consider the last rule applied in the derivation of Ψ ϕ: AX: so Ψ = and ϕ is an axiom of HIREAO. We have: sϕ by RN s of 3.4, and thus sϕ Φ by 3.6 (1). MP: so Ψ = {ψ, ψ ϕ} for some ψ. Now sψ, s(ψ ϕ) Φ by Assumption sψ Φ. By the latter, K s of 3.4 and 3.6 (6), we have: sψ sϕ Φ, so sϕ Φ by the former and 3.6 (3). Inf * : so Ψ = {[α n ]ψ n N} and ϕ = [α * ]ψ for some α and ψ. By sψ Φ, namely, s{[α n ]ψ n N} Φ, we have: s[α * ]ψ = sϕ Φ by 3.5 (3). Inf EAo : so Ψ = {<α>θ # θ # θ PO(o)} and ϕ = Eαo for some α and o. By sψ Φ, namely, s{<α>θ # θ # θ PO(o)} Φ, we have: s Eαo = sϕ Φ by 3.5 (4). SN α : so Ψ = [α]θ and ϕ = [α]θ for some α, Θ, θ and Θ θ. Let s t be the concatenation of the sequences s and t. By the induction hypothesis to Θ θ, we have: s ([α])θ Φ s ([α])θ Φ. So s[α]θ Φ s[α]θ Φ, hence sψ Φ sϕ Φ. So sϕ Φ by Assumption sψ Φ. Ded: so ϕ = ψ θ for some ψ, θ, and Ψ, ψ θ. By the induction hypothesis to Ψ, ψ θ, we have: (c) s (ψ)ψ {s (ψ)ψ} Φ s (ψ)θ Φ. 1 First it is easy to see that s (ψ)ψ by AX and RN s of 3.4, so, by 3.6 (1), we have: (d) s (ψ)ψ Φ. On the other hand, it is easy to prove that by AX, 2.4 (5) and SN s of 3.4, sγ s (ψ)γ for all formula γ, so, by 3.6 (7), we have: (e) sγ Φ s (ψ)γ Φ for all formula γ. Below we will prove that (f) s (ψ)ψ Φ. Given any s (ψ)γ s (ψ)ψ, then γ Ψ, so sγ sψ, so sγ Φ by Assumption sψ Φ, hence s (ψ)γ Φ by (e), so (f) holds. By (f), (d) and (c), we have: s (ψ)θ Φ, namely s(ψ θ) = sϕ Φ. Weak: so Ψ = Ψ 1 Ψ 2 for some Ψ 1, Ψ 2 and Ψ 1 ϕ. By the induction hypothesis to Ψ 1 ϕ, we have: sψ 1 Φ sϕ Φ. Since sψ 1 s(ψ 1 Ψ 2 ), so s(ψ 1 Ψ 2 ) Φ sϕ Φ, namely sψ Φ sϕ Φ. Hence sϕ Φ by Assumption sψ Φ. 1 This is why the definition of pseudo-modality need contain formulas. 11

12 A regular dynamic logic HIREAO of having executed actions on objects Cut: so Ψ Ψ 1 and Ψ, Ψ 1 ϕ for some Ψ 1. By the induction hypothesis to Ψ Ψ 1 and Ψ, Ψ 1 ϕ, we have: (g) sψ Φ sψ Φ for all ψ Ψ 1, and (h) s(ψ Ψ 1 ) Φ sϕ Φ. By Assumption sψ Φ and (g), we have: sψ 1 Φ, so s(ψ Ψ 1 ) Φ by sψ Φ again, hence sϕ Φ by (h). By 2.4 (8) and 3.7 with (a) in the proof of 3.7, we have: Corollary 3.8 Let Φ be maximal consistent (saturated). Then Φ ϕ ϕ Φ. Lemma 3.9 (Lindenbaum lemma for HIREAO) Every consistent set can be extended to a maximal consistent set. Proof. By 3.7 (2), it suffices to prove that every consistent set can be extended to a saturated one. Let Θ be a consistent set, i.e. Θ. Let {ϕ n n N} be an enumeration of all formulas of L, {[α n ] n N} an enumeration of all actions of Action, and for all o OS, {θ n n N} an enumeration of all formulas of PO(o). We shall inductively define an increasing sequence {Φ n n N} of formula sets extending Θ, and show that Φ = {Φ n n N} is saturated. Let Φ 0 = Θ and Φ n {ϕ n }, if Φ n ϕ n ; Φ n { ϕ n }, if Φ n ϕ n and ϕ n is not of the form s[α * ]ψ or s Eαo; Φ n+1 = Φ n { ϕ n, s[α k ]ψ}, if Φ n ϕ n, where ϕ n is of the form s[α * ]ψ and k is the least natural number such that Φ n s[α k ]ψ; Φ n { ϕ n, s(<α>θ # k θ # k )}, if Φ n ϕ n, where ϕ n is of the form s Eαo and k is the least natural number such that Φ n s(<α>θ # k θ # k ). It is easy to see that the k in the last two cases always exists: for if not, then Φ n s[α k ]ψ for all k N, so we have: Φ n s[α * ]ψ by Inf *, SN s and 2.4 (4) (as L = {ϕ n n N}), contradicting Φ n ϕ n. Analogously, if not, then Φ n s(<α>θ # k θ # k ) for all θ k PO(o), so we have: Φ n s Eαo by Inf EAo, SN s and 2.4 (4) (as PO(o) = {θ n n N}), contradicting Φ n ϕ n. Hence the definition of Φ n is correct. We will prove that all Φ n are consistent: this is shown with induction over n, using the consistency of Θ for the base case. If the first case applies and Φ n+1 were inconsistent, then Φ n ϕ n by 3.1 (1) and Ded. As Φ n ϕ n, Φ n by 2.4 (6), contradicting Consistency of Φ n. If the second case applies and Φ n+1 were inconsistent, then Φ n, ϕ n by 3.1 (1), so Φ n ϕ n by 2.4 (9), contradicting Φ n ϕ n. If the third case applies and Φ n+1 were inconsistent, then, by 3.1 (1) and Ded, Φ n s[α * ]ψ s[α k ]ψ. So Φ n s[α * ]ψ s[α k ]ψ by 2.4 (6), and therefore Φ n s[α k ]ψ by 2.4 (10), RM s of 3.4 and 2.4 (6), contradicting the definition of k. If the last case applies and Φ n+1 were inconsistent, then, by 3.1 (1) and Ded, Φ n s Eαo s(<α>θ # k θ # k ). 12

13 So Φ n s Eαo s(<α>θ k # θ k # ) by 2.4 (6), and therefore Φ n s(<α>θ k # θ k # ) by Axiom EAo, RM s of 3.4 and 2.4 (6) (also cf. the proof of (13) of 2.4), contradicting the definition of k. Now we will prove that Φ is saturated. Maximality of Φ follows directly from the definition of the Φ n. For finite -closure we argue via contradiction: let Ψ Φ be finite with Ψ ϕ and assume that ϕ Φ. Then ϕ Φ by Maximality of Φ, so Ψ { ϕ} Φ = {Φ n n N}, hence Ψ { ϕ} Φ n for some n. We have: Ψ { ϕ} ϕ, by Ψ ϕ and Weak; Ψ { ϕ} ϕ, by AX, 2.4 (5) and Weak. It is easy to prove that Ψ { ϕ}, and thus Φ n by Ψ { ϕ} Φ n and Weak, contradicting Consistency of Φ n. We will prove that Φ is s-inf * -closed. By contraposition, we assume s[α * ]ϕ Φ = {Φ n n N}. By the first case in the definition of Φ n+1, we have: Φ n s[α * ]ϕ for all n N. By the third case in the definition of Φ n+1, s[α k ]ψ Φ n+1 Φ for some k N. By Maximality of Φ, s[α k ]ψ Φ for some k N. So s{[α n ]ϕ n N} Φ. Finally we will prove that Φ is s-inf EAo -closed. By contraposition, we assume s Eαo Φ = {Φ n n N}. By the first case in the definition of Φ n+1, we have: Φ n s Eαo for all n N. By the last case in the definition of Φ n+1, s(<α>θ k # θ k # ) Φ n+1 Φ for some k N. By Maximality of Φ, s(<α>θ k # θ k # ) Φ for some k N. So s{<α>θ # θ # θ PO(o)} Φ. Definition 3.10 (Canonical model) The canonical model M = (W, R, V) is defined by W = {w w is maximal consistent}, R a = {(u, w) W 2 [a] w u} for all a AA, V(θ) = {w W θ w} for all θ PO. Lemma 3.11 (Truth lemma) Let M = (W, R, V) be the canonical model. For all w W and all ϕ L, ϕ Φ w V(ϕ). Proof. Given any w W and ϕ L. Induction over ϕ. The cases ϕ = θ PO, ϕ = ψ and ϕ = ψ θ are proved as usual. Firstly we will prove the case ϕ = [α]ψ, by induction over α. 1. α = a AA. First we will prove that (1) [a]ψ w u W ([a] w u ψ u). : Assume that [a]ψ w and for all u W, [a] w u. It is easy to see that ψ u. : We argue via contraposition: assume that [a]ψ w, so [a]ψ w by the maximal consistency of w. It suffices to show that there exists a maximal consistent set u such that [a] w u and ψ u. By the Lindenbaum lemma for HIREAO, it suffices to show that [a] w { ψ} is consistent. Assume that it is not, i.e. [a] w { ψ}, then [a] w ψ by 2.4 (9). Thus, by SN α, we have: {[a]ϕ [a]ϕ w} [a]ψ. We have: w [a]ψ by Weak, and hence [a]ψ w by 3.8, contradicting the assumption [a]ψ w. 13

14 A regular dynamic logic HIREAO of having executed actions on objects By (1) and the defition of the canonical model, we have: [a]ψ w u W (ur a w ψ u). Using Truth definition 2.6 (3), we have: [a]ψ w w V([a]ψ). 2. α = β γ. We have: [β γ]ψ w [β]ψ w and [γ]ψ w by Axiom AX w V([β]ψ) and w V([γ]ψ) by the induction hypotheses R β (w) V(ψ) and R γ (w) V(ψ) by 2.6 (3) R β (w) R γ (w) V(ψ) R β γ (w) V(ψ) by 2.6 (ch) w V([β γ]ψ). by 2.6 (3) 3. α = β ; γ. We have: [β ; γ]ψ w [γ][β]ψ w by Axiom AX ; w V([γ][β]ψ) by the induction hypothesis R γ (w) V([β]ψ) by 2.6 (3) v W (vr γ w v V([β]ψ)) v, u W (vr γ w and ur β v u V(ψ)) u W ( v W (ur β v and vr γ w) u V(ψ))) u W (ur β ; γ w u V(ψ)) by 2.6 (co) R β ; γ (w) V(ψ) w V([β ; γ]ψ). by 2.6 (3) 4. α = [β * ]. Note that the proof as a whole has the form of an induction over a well-ordering of formulas, where [β n ]ϕ is considered to be a subformula of [β * ]ϕ. So we have: w V([β * ]ψ) w V([β n ]ψ) for all n N by the proof of 2.8 [β n ]ψ w for all n N by the induction hypotheses [β * ]ψ w. by 2.4 (12) and 3.8 Finally, we consider the case ϕ = Eαo. By 2.4 (13) and 3.8, we have: {<α>θ # θ # θ PO(o)} w Eαo w. So, by the maximal consistency of w, (2){<α>θ # θ # θ PO(o)} w Eαo w. On the other hand, by the results that we have proved (the induction hypotheses), it is easy to prove that for all θ PO(o), w V(<α>θ # θ # ) <α>θ # θ # w. Hence w V({<α>θ # θ # θ PO(o)}) {<α>θ # θ # θ PO(o)} w. So w V({<α>θ # θ # θ PO(o)}) {<α>θ # θ # θ PO(o)} w. Hence (3) θ PO(o), w V(<α>θ # θ #d ) {<α>θ # θ # θ PO(o)} w. 14

15 So, by (2), (3) and Truth definition 2.6 (4), we have: w V(Eαo) w Eαo w. Note that in the Truth lemma, we do not prove Dual property for all α Action as follows (Dual property) ur α w [α] w u for all α Action. It holds by definition for atomic actions a, of course. Theorem 3.12 (Strong completeness theorem) Φ ϕ Φ ϕ for all Φ L and ϕ L. Proof. By contraposition, suppose that Φ ϕ. Assume that Φ { ϕ} is not consistent. Then Φ, ϕ by 3.1 (1), so Φ ϕ by 2.4 (9), contradicting that Φ ϕ. So Φ { ϕ} is consistent. By Lindenbaum lemma for HIREAO (3.9), Φ { ϕ} is extended to a maximal consistent set w such that Φ w and ϕ w. Now by Truth lemma 3.11, for the canonical model M = (W, R, V), w W, w V(Φ) and w V(ϕ), so Φ ϕ. We conclude this paper by the following remarks: (1) We can extend the language defined in 2.1 by adding a countable set PV = {p 1,, p n, } of propositional variables as another kind of atomic formulas, accordingly V defined in 2.5 is now defined as a mapping from PO PV into (W). It is easy to see that all the above results still hold. Of course, we can also view propositional variables in PV as 0-ary predicate symbols. This view is simpler. (2) We can modify the language defined in 2.1 as a first-order epistemic language, and then establish HIREAO by the classical first-order logic. Such HIREAO should be more refined. For example, let IV = {x 1,, x n, } be a countable set of individual variables such that OS PS IV =, then PO(o) in 2.1 (2) is altered as follows: for all P PS, if f(p) = 1, then let Po PO(o); if f(p) > 1, then for all x 1,, x n 1, x n + 1,, x f(p) IV, let Px 1 x n 1 ox n + 1 x f(p) PO(o). Here Axiom Eao can be altered as <α>(px 1 x n 1 ox n + 1 x f(p) ) # (Px 1 x n 1 ox n + 1 x f(p) ) #d Eαo. So, by the rule of existential antecedent generalization in the classical first-order logic, we have x 1 x n 1 x n + 1 x f(p) (<α>(px 1 x n 1 ox n + 1 x f(p) ) # (Px 1 x n 1 ox n + 1 x f(p) ) #d ) Eαo. 1 And Rule Inf EAo can be altered as {<α>(px 1 x n 1 ox n + 1 x f(p) ) # (Px 1 x n 1 ox n + 1 x f(p) ) # Px 1 x n 1 ox n + 1 x f(p) PO(o)} Eαo. So, by the rule of universal instantiation in the classical first-order logic, 2 we have { x 1 x n 1 x n + 1 x f(p) (<α>(px 1 x n 1 ox n + 1 x f(p) ) # (Px 1 x n 1 ox n + 1 x f(p) ) # ) Px 1 x n 1 ox n + 1 x f(p) PO(o)} Eαo. These are just what (#) in Section 1 says. Of course, to prove the strong soundness theorem and the strong completeness theorem, we have to accordingly modify Definition 2.6 and so on. For simplicity, we can even accordingly alter PO(o), Px 1 x n 1 ox n + 1 x f(p) and Eαo in above expressions as PO(x), Px 1 x f(p) and Eαx for all x IV. 1 Note that o is an individual constant symbol. Cf (9) and (2) in [8]. 2 Cf (5) in [8]. 15

16 A regular dynamic logic HIREAO of having executed actions on objects References [1] Li Xiaowu. A regular dynamic logic of having executed actions on objects. [2] Li Xiaowu. A regular dynamic logic of having executed actions. [3] D. C. Dennett, Cognitive wheels: the frame problem of AI, in C. Hookway (ed.), Mind, Machines, and Evolution: Philosophical Studies (1984), pp Cambridge University Press. [4] D. Harel, D. Kozen, J. Tiuryn. Dynamic Logic. The MIT Press, [5] P. Blackburn, M. de Rijke and Y. Venema. Modal Logic, Cambridge University Press, [6] G. R. de Lavalette, B. Kooi, R. Verbrugge. Strong completeness and limited canonicity for PDL and similar modal logics. ai/onderzoek/prepublications/ prepubs2004. [7] J. P. Burgess. Basic tense logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume II, pages D. Reidel Publishing Company, Dordrecht (the Netherlands), [8] Li Xiaowu. A Course in Mathematical Logic. Sun Yat-sen University Press, China ( 中国. 中山大学出版社 ),