Knowledge and action in labeled stit logics

Transcription

1 Knowledge and action in labeled stit logics John Horty Eric Pacuit April 10, 2014 Abstract Stit semantics supplements branching time models with an operator representing the agency of an individual in seeing to it, or bringing it about, that some state of affairs holds. In this paper, we extend the basic stit semantics with action types, in addition to the action tokens which were already present. We motivate the introduction of action types by showing how they are necessary to represent an epistemic sense of ability. Once introduced, the action types enable us to define a new epistemic stit operator that interacts in interesting ways with the standard stit. 1 Introduction Stit logics originating with a series of papers by Belnap, Perloff, and Xu, culminating in their [2] grows out of a modal tradition in the logic of action going back to St Anselm, but with more recent contribution by, among others, Anderson, Chellas, Fitch, Kanger, Lindhal, Pörn, and von Kutschera. 1 It is characteristic of this tradition to focus only on an operator representing the agency of an individual in seeing to it, or bringing it about, that some 1 A brief history of the subject, with references to the works of these writers and others, can be found in Segerberg [16]. 1

2 state of affairs holds, rather than on the particular action that the individual performs in doing so. A recent survey of the literature goes so far as to claim that no author in the Anselm-Kanger-Chellas line up through Belnap... has countenanced the existence of actions in logic: action talk, yes; ontology of action, no. 2 In fact, and as that survey notes, one of the present authors, working somewhat later in the tradition of stit logic, does speak explicitly of actions in [10], defining a deontic operator in terms of a preference ordering on the actions available to an individual. The actions discussed in that book, however, were all action tokens particular, concrete actions. There were no general kinds of actions, or action types, such as the action type of opening a window, for example. There were particular, concrete openings of particular windows, but nothing to group them together as actions of the same kind. The purpose of this paper is to fill this gap, supplementing stit semantics with action types, in addition to the action tokens that were already present. But we do not do this simply for the sake of doing it, or because the neglect of action types seems like a defect from an external perspective. Instead, working from a perspective internal to stit semantics, we motivate the introduction of action types by showing how they are necessary for allowing us to represent an important concept: an epistemic sense of ability. Once introduced, the action types are not idle either. They enable us to define a new epistemic stit operator that interacts in interesting ways with the standard stit. The paper is organized as follows. The next section provides a quick summary of stit semantics, leading up to the definition of a simple stit operator. Section 3 then considers the standard treatment of ability within the theory, shows that this treatment fails to capture the epistemic sense of this concept, and argues that simply supplementing the basic theory 2 Lindström and Segerberg [14, p. 1199]. 2

3 with an epistemic operator will not work either, but that, instead, action types are necessary as well. Finally, Section 4 expands the stit framework with both action types and epistemic ideas, allowing us to define a new stit operator representing an epistemic sense of agency. An axiomatic characterization of the resulting logic is provided in the Appendix. 2 A review of stit semantics 2.1 Branching time Stit semantics is cast against the background of a theory of indeterministic time, first set out by Prior [15] and developed in more detail by Thomason [17], according to which moments are ordered into a treelike structure, with forward branching representing the indeterminacy of the future and the absence of backward branching representing the determinacy of the past. This picture leads to a notion of branching time frames as structures of the form Tree, <, in which Tree is a nonempty set of moments and < is a strict partial ordering without backward branching: for any m, m, and m from Tree, if m < m and m < m, then either m = m or m < m or m < m. A maximal set of linearly ordered moments from Tree is a history, representing some complete temporal evolution of the world. If m is a moment and h is a history, then the statement that m h can be taken to mean that m occurs at some point in the course of the history h, or that h passes through m. Of course, because of indeterminism, a single moment might be contained in several distinct histories. We let H m = {h : m h} represent the set of histories passing through m; and when h belongs to H m, we speak of a moment/history pair of the form m/h as an index. A branching time model can be defined as a structure that supplements a branching time frame with a valuation function v mapping each propositional constant from some 3

4 background language into the set of m/h indices at which, intuitively, it is thought of as true. If we suppose that formulas are formed from truth functional connectives as well as the usual temporal operators P and F, representing past and future, the satisfaction relation = between indices and formulas true at those indices can be defined as follows. Definition 2.1 (Evaluation rules: basic operators) Where m/h is an index and v the evaluation function from a branching time model M, M, m/h = A if and only if m/h v(a), for A a propositional constant, M, m/h = A B if and only if M, m/h = A and m/h = B, M, m/h = A if and only if M, m/h = A, M, m/h = PA if and only if there is an m h such that m < m and M, m /h = A, M, m/h = FA if and only if there is an m h such that m < m and M, m /h = A. A formula A is then defined as settled true at a moment m from a model M just in case M, m/h = A for each h in H m, and as settled false just in case M, m/h = A for each h in H m. In branching time, the set of possible worlds accessible at a moment m can be identified with the set H m of histories passing through that moment; those histories lying outside of H m are taken to represent worlds that are no longer accessible. The propositions at m can thus be identified with sets of accessible histories, subsets of H m. And the particular proposition expressed by a sentence A at a moment m in a model M can be identified with the set A m M = {h Hm : M, m/h = A} of histories from H m in which that sentence is true. 4

5 In addition to the usual temporal operators, the framework of branching time allows us to define the concept of historical necessity, along with its dual concept of historical possibility: the formula A is taken to mean that A is historically necessary, while A means that A is still open as a possibility. The intuitive idea is that A is true at some moment if A is true at that moment no matter how the future turns out, and that A is true if there is still some way in which the future might evolve that would lead to the truth of A. The evaluation rule for historical necessity is straightforward. Definition 2.2 (Evaluation rule: A) Where m/h is an index from a branching time model M, M, m/h = A if and only if M, m/h = A for each history h H m. And historical possibility can then be characterized in the usual way, with A defined as A. 2.2 Agents and choices In stit logic, the idea that an agent α sees to it that A is taken to mean that the truth of A is guaranteed by an action performed by α. In order to capture this idea, we must be able to speak of individual agents, and also of their actions; and so the basic framework of branching time is supplemented with two additional primitives. The first is simply a set Agent of agents, individuals thought of as acting in time. The second is a device for representing the possible constraints that a particular agent is able to exercise upon the course of events at a given moment, the actions or choices open to the agent at that moment. These constraints are encoded through a function Choice, mapping each agent α and moment m into a partition Choice m α of the set H m of histories through m. The idea is that, by acting at m, the agent α is able to determine a particular one of the 5

6 equivalence classes from Choice m α within which the history to be realized must then lie, but that this is the extent of the agent s influence. We will simplify in this paper by supposing that, at any given moment, only one agent can act that is, for a moment m, there is only one agent α for whom Choice m α is a nontrivial partition of H m. Because of this simplification, the actions of different agents are guaranteed to be independent, so there is no need for a separate independence requirement. We do, however, need to impose the requirement of no choice between undivided histories, according to which, for each agent α, any two histories that are undivided at m that is, any two histories sharing a moment later than m must belong to the same equivalence class from Choice m α. If K is a choice cell belonging to Choice m α, one of the equivalence classes specified by this partition, we speak of K as an action or more precisely, an action token available to the agent α at the moment m, and we say that α performs the action K at the index m/h just in case h is a history belonging to K. We let Choice m α (h) (defined only when h H m ) stand for the particular action from Choice m α that contains the history h; Choice m α (h) thus represents the particular action performed by the agent α at the index m/h. A stit frame can now be defined as a structure of the form Tree, <, Agent, Choice, supplementing a branching time frame with the additional components Agent and Choice, as specified above. A stit model is a model based on a stit frame. A stit statement has the form [α stit: A], carrying the intuitive meaning that α sees to it that A. Such a statement is defined as true at the index m/h just in case the action token performed by α at that index guarantees the truth of A. Formally, we can say that some action K available to α at the moment m guarantees the truth of A just in case A holds at m/h for each history h from K. Since the action performed by α at the index m/h is 6

7 Choice m α (h), our semantic analysis can be captured through the following evaluation rule. Definition 2.3 (Evaluation rule: [α stit: A]) Where α is an agent and m/h an index from a stit model M, M, m/h = [α stit: A] if and only if Choice m α (h) A m M. Those familiar with stit logics will recognize this particular operator as the Chellas stit, first defined by Horty and Belnap [11], but drawing on ideas from Chellas [7]. 3 Ability, knowledge, and action types 3.1 Ability There are several ways of motivating the introduction of action types, in addition to tokens, into stit logics, but we concentrate here on a motivation deriving from problems with the stit characterization of personal ability. This notion of personal ability must be distinguished, of course, from that of mere possibility: even though it is possible for it to rain tomorrow, no agent has the ability to see to it that it will rain tomorrow. Nevertheless, it has been suggested in the stit literature that what an agent is able to do can reasonably be identified with what it is possible that the agent does. In particular, Horty and Belnap [11] proposed that a formula of the form [α stit: A], carrying the literal meaning that it is possible for the agent α to see to it that A, can usefully be taken to expresses the claim that α has the ability to see to it that A. The basic idea behind this proposal is that α has the ability to see to it that A at a given moment m just in case there is some action in this case, an action token available to α at m that guarantees the truth of A. 7

8 Since formulas of the form A [α stit: A], [α stit: A B] ( [α stit: A] [α stit: B]) are invalid, this proposal escapes the well-known objections to modal analyses of ability due to Kenny, in [12] and [13]: Carol might actually hit the bull s eye on a dart board without having the ability to hit the bull s eye, and she might be able to hit the black region or the white region of the dart board, without being able to hit the black region, or to hit the white region. The proposal has clear relations to the logic of ability developed by Brown [6], and also to coalition logic; these later relations were first explored by Broersen, Herzig, and Troquard in a series of papers beginning with [5]. We still feel that this proposal captures one important sense of ability, which we refer to here as the causal sense. There is also, however, another important sense of ability that the proposal does not capture, which we will describe as the epistemic sense. To illustrate: suppose a friend places all the cards from a deck face down on a table, and asks you to turn over the Jack of Hearts. Is that something you are able to do? The answer is Yes, in the causal sense of ability. There are, we can suppose, 52 actions available to you turning over any of the cards. Each of these actions guarantees that some particular card is turned over, and the Jack of Hearts is among them; so one of the actions available to you guarantees that the Jack of Hearts is turned over. But in the epistemic sense, the answer is No. Even if one of the actions available to you guarantees that the Jack of Hearts is turned over, you do not, in the epistemic sense, have the ability to turn that card over unless you know which action that is. 8

9 3.2 Knowledge In order to understand the epistemic sense of ability, it is natural, as a first step, to introduce a knowledge operator into stit logic. This can be done in the standard way. 3 Begin by positing, for each agent α, an equivalence relation α among the indices from a stit frame, where m/h α m /h is taken to mean that nothing α knows distinguishes m/h from m /h, or that m/h and m /h are indistinguishable by α; for later convenience, we can take Know α (m/h) = {m /h : m /h α m/h} as the entire set of indices indistinguishable by α from m/h. An epistemic stit frame can be defined as a structure of the form Tree, <, Agent, Choice, { α } α Agent, like a stit frame but with the additional component { α } α Agent, a set containing indistinguishability relations for the various agents from Agent. An epistemic stit model can be defined in the usual way, by supplementing an epistemic stit frame with a valuation function; and evaluation rules for operators of the form K α, representing knowledge for the agent α, can then be stated as follows. Definition 3.1 (Evaluation rule: K α A) Where α is an agent and m/h an index from an epistemic stit model M, M, m/h = K α A if and only if M, m /h = A for all m /h such that m /h α m/h. Once a knowledge operator has been introduced into stit semantics, it is tempting to suppose that the epistemic sense of ability can be captured through some combination of 3 We say that this treatment of knowledge is standard, and it does follow the usual pattern of epistemic logic; but it was not until Herzig and Troquard [9] that anyone even thought to explore epistemic ideas in the context of stit semantics. A later, but somewhat more systematic, presentation was set out Broersen in [3] and [4], and the initial ideas have since been developed in a number of fruitful directions by these three authors, working in various combinations and with various colleagues. 9

10 knowledge, impersonal possibilty, and agency perhaps the idea that α has the ability to see to it that A, in this epistemic sense, should be represented as K α [α stit: A], or perhaps as K α [α stit: A]. The first of these formulas, taken literally, means that the agent α knows that it is possible that α sees to it that A; the second means that it is possible that α knows that α sees to it that A. An analysis in accord with this second formula has been advocated both by Herzig and Troquard [9] and by Broersen [3], and has much to recommend it. Still, we do not feel that either analysis is exactly right, for reasons that are best explained with an example. h 1 h 2 h 3 h 4 A A A A m 2 K 3 K 4 m 3 K 5 K 6 m 1 K 1 K 2 Figure 1: The first coin game Consider two simple games, depicted in Figures 1 and 2, in each of which, at an initial moment m 1, the agent β places a coin on the table either heads up, by selecting the action K 1, or tails up, by selecting K 2. Next, the agent α bets whether the coin was placed heads up or tails up. This action occurs at one of the later moments m 2 or at m 3, depending on the initial choice by β. If β has selected K 1, then α bets heads by selecting K 3 and tails by selecting K 4 ; if β has selected K 2, then α bets heads by selecting K 5 and tails selecting K 6. If α bets correctly, she wins. There are four histories: h 1 and h 4 are the histories in which α wins, betting that the coin is heads up when it is, and tails up when it is; h 2 and h 3 are the 10

11 histories in which α loses. If we let A represent the statement that α wins, then we have A true at m 2 /h 1 and m 3 /h 4, and false at m 2 /h 2 and m 3 /h 3. h 1 h 2 h 3 h 4 A A A A m 2 K 3 K 4 m 3 K 5 K 6 m 1 K 1 K 2 Figure 2: The second coin game The two games we consider share this much structure what we might refer to as their causal structure but differ in whether or not the agent α knows whether the coin has been placed heads up or tails up, and so must be represented by different epistemic stit models. 4 In the first game, α does know whether the coin has been placed heads up or tails up, and so knows, in effect, whether she is at m 2 or m 3. This game can be represented by supposing that the indistinguishability relation for α partitions the indices from m 2 and the indices from m 3 into separate, and so distinguishable, equivalence classes; the game can be represented, that is, by taking α = {{m 2 /h 1, m 2 /h 2 }, {m 3 /h 3, m 3 /h 4 }}. In the second game, α does not know whether the coin has been placed heads up or tails up, and so, in effect, does not know whether she is at m 2 or m 3. This game can be represented by supposing that 4 Our convention for representing indistinguishability relations in diagrams is this: a dotted line between histories h and h emanating from moments m and m indicates that the indices m/h and m /h belong to the same indistinguishability class; further, indistinguishability classes are closed under reflexivity and transitivity. 11

12 the indistinguishability relation for α groups all the indices from m 2 and m 3 together into a single equivalence class that is, by supposing that α = {{m 2 /h 1, m 2 /h 2, m 3 /h 3, m 3 /h 4 }}. Now of course, in both of these games, the agent α has the ability to win in the purely causal sense: the formula [α stit : A] is settled true at both m 2 and m 3, regardless of the agent s knowledge. But when we turn to the epistemic sense of ability, it seems that knowledge should make a difference. In the first game, where the agent knows whether the coin has been placed heads up or tails up, we would like to be say that she does have the ability to win, even in the epistemic sense. But in the second game, where the agent does not know whether the coin has been placed heads up or tails up, we would now like to say that, although the causal ability is still there, she no longer has the ability to win in the epistemic sense. As it turns out, neither of the two formulas under consideration that is, neither K α [α stit: A] nor K α [α stit: A] allow us to say both of these things, and so to isolate the epistemic sense of ability. It is easy to see that K α [α stit: A] is settled true, and that K α [α stit: A] is settled false, at both m 2 and m 3 in both games. What we want as a representation of the agent s ability to win, in the epistemic sense, is a formula that holds in the first game, but fails in the second; but each of these formulas either holds in both games, or else fails in both. 3.3 Action types How, then, can we represent the epistemic sense of ability? The basic idea underlying the causal sense, recall, is that an agent α has the ability to see to it that A just in case there is some action available to α that guarantees the truth of A. We propose to start with an epistemic modification of this basic idea. Rather than requiring only that some action available to α should in fact guarantee the truth of A, what we suggest is that α has the 12

13 A h 1 h 2 A h 3 h 4 m 2 K 5 K 6 m 3 K 7 K 8 m 1 K 1 K 2 K 3 K 4 Figure 3: The third coin game ability to see to it that A in the epistemic sense just in case there is some action available to α that is known by α to guarantee the truth of A. But now, consider a variant of our coin game. Suppose this time that β has two coins, a nickel and a dime, and begins at m 1 by placing either one on the table, either heads up or tails up: we take K 1 as the action of placing the nickel heads up, K 2 as the action of placing the dime heads up, K 3 as the action of placing the nickel tails up, and K 4 as the action of placing the dime tails up. As before, α then bets whether the coin whichever coin was placed on the table was placed heads up or tails up, and wins if she bets correctly. We consider here only that portion of the game, depicted in Figure 3, in which the coin was in fact placed heads up. In that case, if β has placed the nickel heads up on the table, selecting K 1, then α bets heads by selecting K 5 and tails by selecting K 6 ; if β has placed the dime heads up on the table, selecting K 2, then α bets heads by selecting K 7 and tails by selecting K 8. Again, we take A as the statement that α wins, so that A is true at m 2 /h 1 and m 3 /h 3, and false at m 2 /h 2 and m 3 /h 4. Finally, we assume that, although α does not know which coin was placed on the table, she knows that whichever coin is now on the table was placed heads up that is, we assume α = {{m 2 /h 1, m 2 /h 2, m 3 /h 3, m 3 /h 4 }}. 13

14 In this case, since α knows that the coin was placed heads up, we would like to say that she has the ability to win even in the epistemic sense: all she needs to do, it seems, is bet heads that is an action she knows will guarantee a win. But betting heads, in this sense, is not among the actions available to α. If α is at m 2, then the action token K 5 is among her available actions, and if she is at m 3, then the action token K 7 is available. These are, however, two different concrete actions, both tokens of what we might call the action type of betting heads and it is this action type, not some individual token of the type, that α knows will guarantee a win. In order to understand the epistemic sense of ability, then, we now turn to the task of supplementing stit semantics to include action types, as well as tokens. The introduction of action types will allow us to define a new epistemic sense of action, distinct from what we might describe as the causal sense captured by ordinary stit semantics, and according to which an agent sees to it that some proposition holds by performing an action token guaranteeing its truth. In this new epistemic sense, an agent sees to it that a proposition holds, not just by performing an action token that happens to guarantee its truth, but by executing an action type that the agent knows to guarantee its truth. Once this epistemic notion of action has been defined, we can then arrive at the epistemic notion of ability by adapting our previous recipe, combining this new notion of action with ordinary impersonal possibility. 4 Labeled stit logics 4.1 Basic concepts We begin by postulating a set T ype = {τ 1, τ 2,..., τ n } of action types general kinds of action, as opposed to the concrete action tokens already present in stit logics. The intuition 14

15 is that an agent performs a concrete action token, at a particular moment in space-time, by executing some action type at that moment. A robot, for example, might perform the action type of raising its left arm four inches twice during the day, once at the lab in the morning and once at home in the evening, resulting in two concrete action tokens of the same type; a gambler might perform the action type of betting heads in two different games, or, given branching time, at different possible points in the same same game. We assume for simplicity that the set of action types is finite, though there is no reason to rule out an infinite set, perhaps specified by a compositional action description language. An action type τ can be reified as a partial function mapping each agent α and moment m into the particular action token [τ] m α that results when τ is executed by α at m. Of course, this action token must be one of those available to α at m that is, we must have [τ] m α Choice m α. The functions representing action types are partial because, from an intuitive standpoint, it is best to assume that not every action type can actually be executed by every agent at every moment. Just as an action type τ maps an agent α and a moment m into a particular action token [τ] m α from Choice m α, we postulate a one-one function Label mapping each action token K from Choice m α into a particular action type Label(K) from T ype. The interaction between types and labels is governed by two type/label constraints: If K Choice m α, then [Label(K)] m α = K, If τ T ype and [τ] m α is defined, then Label([τ] m α ) = τ. The first of these requires that, if K is an token available to α at m, and K is an action of a particular type Label(K), then the execution of an action of that type by α at m is K itself; the second requires that, if τ is an action type whose execution by α at m is a particular action token [τ] m α, then the type of that token is τ itself. 15

16 Our previous definition of the actions available to an agent at a moment, as well our definition of the particular action performed, or executed, by an agent at an index, can both be lifted from tokens to types in the natural way. Since Choice m α is the set of action tokens available to α at m, we can take T ype m α = {Label(K) : K Choice m α } as the set of available action types; and since Choice m α (h) is the particular action token performed by α at the index m/h, we can take T ype m α (h) = Label(Choice m α (h)) as the action type executed by that agent at that index. Finally, since our goal is to arrive at an epistemic sense of action, it is natural to impose certain constraints on the indistinguishability relation. There are various plausible candidates of varying degrees of strength, but we will limit ourselves in this paper only to the very simple constraint of indistinguishability between moments: If m/h α m /h, then m/h α m /h for each h H m and h H m. What this says is that, if any indices from two moments are indistinguishable, then all indices from those moments must be indistinguishable, with the result that indistinguishability can be thought of as holding between the moments themselves. Putting these various ideas together, we can define a labeled stit frame as a structure of the form Tree, <, Agent, Choice, { α } α Agent, T ype, Label, like an epistemic stit frame, but with the new components T ype and Label as specified above, where the indistinguishability relations are subject to the indistinguishability between moments. A labeled stit model results when such a frame is supplemented with a valuation. 16

17 4.2 The kstit operator We are now, at last, in a position to define a new kstit operator, allowing for statements of the form [α kstit: A] to carry the intuitive meaning that α sees to it that A in the epistemic sense. A statement like this will be defined as true at an index m/h just in case the action type executed by α at that index is one that α knows to guarantee the truth of A just in case, that is, the execution of this action type guarantees the truth of A at every moment m that is indistinguishable from m. The action type executed by α at the index m/h is T ype m α (h), as we have seen, and so the execution of this action type by α at another moment m is [T ype m α (h)] m α. Therefore, the evaluation rule for our new operator is as follows. Definition 4.1 (Evaluation rule: [α kstit: A]) Where α is an agent and m/h an index from a labeled stit model M, M, m/h = [α kstit: A] if and only if [T ype m α (h)] m α A m M for all m /h such that m /h α m/h. This rule can be illustrated by returning to our previous coin game examples, making explicit the action types that were already implicit in our informal descriptions of these games. We will suppose, then, that T ype = {τ 1, τ 2 }, where, intuitively, τ 1 is the action type of betting heads and τ 2 is the action type of betting tails. In the first two coin games, depicted in Figures 1 and 2, the concrete actions K 3 and K 5 are tokens of the type betting heads, while K 4 and K 6 are tokens of the type betting tails. We therefore have [τ 1 ] m 2 α = K 3 and [τ 1 ] m 3 α = K 5, and [τ 2 ] m 2 α = K 4 and [τ 2 ] m 3 α = K 6. Let us focus on the index m 2 /h 1, where Choice m 2 α (h 1 ) is K 3 and T ype m 2 α (h 1 ) is τ 1 the agent is performing the action token K 3 by executing the action type τ 1 so that [T ype m 2 α (h 1 )] m 2 α is again K 3, by the type/label constraints. In the first game, from Figure 1, the agent can 17

18 distinguish m 2 from m 3. Since m 2 is indistinguishable only from itself, then, and because [T ype m 2 α (h 1 )] m 2 α A m 2, it follows that [α kstit: A] holds at the index m 2 /h 1 the agent α sees to it that A even in the epistemic sense. In the second game, from Figure 2, the agent cannot distinguish m 2 from m 3. There is therefore a moment, m 3, indistinguishable from m 2 at which the action type executed at m 2 /h 1 fails to guarantee A we do not, that is, have [T ype m 2 α (h 1 )] m 3 α A m 3. Because of this, [α kstit: A] does not hold at m 2 /h 1. Even though the action token performed by α at that index guarantees the truth of A the formula [α cstit: A] holds the action type executed by α is not one that she know to guarantee the truth of A, and so α fails to see to it that A in the epistemic sense. In the third coin game, from Figure 3, the concrete actions K 5 and K 7 are tokens of the type betting heads, while K 6 and K 8 are tokens of the type betting tails. We therefore have [τ 1 ] m 2 α = K 5 and [τ 1 ] m 3 α = K 7, and [τ 2 ] m 2 α = K 6 and [τ 2 ] m 3 α = K 8. Focusing once more on the index m 2 /h 1, we see again that T ype m 2 α (h 1 ) is τ 1. Because both [τ 1 ] m 2 α A m 2 and [τ 1 ] m 3 α A m 3, we have [T ype m 2 α (h 1 )] m α A m for each moment m that is indistinguishable from m 2 for α, so that [α kstit: A] holds at m 2 /h 1. Betting heads is an action type that α knows to guarantee a win, even though she does not know whether it is the nickel or the dime that has been placed heads up. Finally and returning to our motivating problem we can now take the formula [α kstit: A] to represent the idea that the agent α has the ability, even in the epistemic sense, to see to it that A. The reader can verify that this proposal yields the correct results in our previous examples: the formula is settled true at the moment m 2, for instance, in the games depicted in Figures 1 and 3, telling us in both cases that the agent has the ability to win, but settled false in the game from Figure 2, telling us that, although the agent can win in the causal 18

19 sense, she does not have the ability to win in the epistemic sense. 4.3 Discussion We leave it to the reader to verify that this new kstit operator, representing an epistemic sense of agency, is an S5 modal operator. It is, in addition, strictly stronger than the ordinary stit operator defined earlier, representing agency only in a causal sense, but strictly weaker than a combination of the knowledge operator with this causal stit: more exactly, both of the formulas K α [α stit: A] [α kstit: A], [α kstit: A] [α stit: A] are valid, and both converses fail. The kstit operator is, moreover, a conservative extension of the ordinary stit operator, collapsing into the ordinary stit whenever the agent s indistinguishability relation is limited to indices from a single moment: more exactly, the statement [α stit: A] [α kstit: A] will hold at any index m/h for which Know α (m/h) H m. The upshot is that there is no difference between the causal and epistemic senses of agency as long as the agent knows everything about the past, leading up to the present moment but that the two senses of agency can come apart if the agent has some uncertainty about the past, in which case the epistemic sense is stronger. There are various other interesting differences between the stit and kstit operators. For example, while A [α stit: A] is valid in stit models, the formula A [α kstit: A] is invalid in labeled stit models. The reason for this is that the truth of A is settled at a single moment, which is enough to guarantee [α cstit: A], while the truth [α kstit: A] may depend on other moments that are indistinguishable from the original. The corresponding 19

20 validity for kstit therefore requires a stronger antecedent: K α A [α kstit: A]. A full discussion of validities for classes of labeled stit frames satisfying different conditions will be left for a different paper. For now, we note only that the validities on the class of stit frames defined in this paper admit of a simple axiomatization. In addition to the usual S5 axioms and rules for the, stit, K α, and kstit operators, the following formulas suffice: A1. A [α stit: A] A2. [α kstit: A] [α stit: A] A3. [α kstit: [α stit: A]] [α kstit: A] A4. A [α stit: A] A5. K α A K α A A6. K α A [α kstit: A] A proof of completeness (for the single agent case) is sketched in the Appendix. 20

21 A Completeness To simplify the presentation, we assume that there is only one agent α. Let F α denote the class of labeled stit frames for a single agent α satisfying the indistinguishability between moments condition. The axioms include the axioms from Section 4.3 (A1-A6) together with the usual propositional tautology axiom (which we denote A0 stating that all propositional tautologies in our language are axioms), the rule Modus Ponens (denoted MP). Soundness of these axioms is straightforward and left to the reader. Proposition A.1 The axioms A0-A6 are valid on class F α of labeled stit frames. Proof. The proof that the axioms A0, A1, A2, A4 and A5 are valid is straightforward and left to the reader. To illustrate the arguments, we give the proof of the validity of axioms A3 and A6. Let M be a labeled stit model based on a labeled stit frame from F α. Axiom A3 is valid: Suppose that M, m/h = [α kstit: [α stit: A]]. Then, for all m /h if m/h α m /h, then [T ype m α (h)] m α [α stit: A] m = {m /h M, m /h = [α stit: A]} = {m /h Choice m α (h ) A m } A m. Suppose that m/h α m /h. Then, [T ype m α (h)] m α A m ; and so, M, m/h = [α kstit: A]. Axiom A6 is valid: Suppose that M, m/h = K α A. Then for all m /h, if m/h α m /h, then M, m /h = A. Suppose that m /h is an index such that m/h α m /h. Let h [T ype m α (h)] m α. Then, since m/h α m /h, h H m and h H m, by the indistinguishability between moments condition, m/h α m /h. Thus, M, m /h = A, and so, M, m/h = [α kstit: A]. qed The proof that these axioms are complete for the class F α of labeled stit frames adapts an analogous proof for stit models from [8] (see also [1]). The key idea is to define a suitable Kripke model that can be transformed into a labeled stit model. 21

22 Definition A.2 (Labeled stit Kripke model) A labeled stit Kripke model for an agent α is a structure M = W, R, α, C α, K α, V where 1. R, α, C α and K α are all equivalence relations. 2. C α R : for all w, v W, if w C α v, then w R v. 3. C α K α : for all w, v W, if w C α v, then w K α v 4. For all w, v, x W, if w K α v and v C α x, then w K α x. 5. For all w, v, x, y W, if w α v, w R x, and v R y, then x α y. 6. K α α : for all w, v W, if w K α v, then w α v. For any relations R on W and state w W, let R(w) = {v w R v}. Our language (restricted so that it only includes modalities for the single agent α) can be evaluated on these Kripke structures in the standard way: Definition A.3 (Evaluation rules for labeled stit Kripke models) For each state w in a labeled stit Kripke model M: M, w = p iff p V (w) M, w = A iff M, w = A M, w = A B iff M, w = A and M, w = B M, w = A iff for all v W, if w R v, then M, v = A M, w = K α A iff for all v W, if w α v, then M, v = A M, w = [α stit: A] iff for all v W, if w C α v, then M, v = A 22

23 M, w = [α kstit: A] iff for all v W, if w K α v, then M, v = A Note that in a labeled stit Kripke model, we have for all w W C α (w) R (w) K α (w). Given a labeled stit model, it is not difficult to define a labeled stit Kripke model that satisfies the same formulas (the construction is given in the proof Theorem A.5). It is not hard to see that a model generated from a labeled stit model will satisfy a stronger constraint on the stit, kstit and historic necessity relations: C α (w) = R (w) K α (w). In fact, this property is needed in order to show that every labeled stit Kripke model can be transformed into a labeled stit model. Fortunately, we can transform any labeled stit Kripke model into one satisfying the above the property for all w W, C α (w) = R (w) K α (w). (This is analogous to the construction used when proving completeness for epistemic logic with a distributed knowledge operator.) Lemma A.4 For any labeled stit Kripke model M, there is another labeled stit Kripke model M validating the same formulas satisfying the property for all states w, C α (w) = R (w) K α (w). Proof. Let M be a labeled stit Kripke models and suppose that there are w, v such that w R v and w K α v but w C α v. Now, since v R (w), there is a v R (w) such that v C α (v ). Define a new model M = W, R, α, C α, K α, V as follows. Add copies of the states in C α (v ) to the set W. That is, let W = W {(x, 1) x C α (v )}. Define the relations in M as follows: Let R = R on W and for all states of the form (x, 1), (y, 1), let (x, 1) R (y, 1). Note that there are no states w and (x, 1) in W such that w R (x, 1). Let α= α on W ; (x, 1) α (y, 1) iff x α y; and w α (y, 1) and (y, 1) α w iff w α y. 23

24 C α = C α on W and (x, 1) C α (y, 1) for all states of the form (x, 1), (y, 1) W. Define K α such that: for all x such that x R (w), C α(x) = C α (x); K α(v ) = C α (v ); and K α(w) = K α (w) C α (v ) C α (v ) {1}. An induction on formulas shows that the two models satisfy the same formulas. Repeat the above procedure on the newly generated models until there are no states w, v such that w R v and w K α v but w C α v. qed Theorem A.5 A formula A is satisfiable in a labeled stit model based on a frame from F α if and only if A is satisfiable in a labeled stit Kripke model. Proof. ( ): Let M = Tree, <, Agent, Choice, α, T ype, V be a model based on a frame from F α such that M, m/h = A for some index m/h. Define a Kripke model M = W, R, { α, C α, K α, V as follows: W = {{m } H m there is h H m such that m/h α m /h } R = {((m, h ), (m, h )) m = m } C α = {((m, h ), (m, h )) m = m and there is a K Choice m α (h ) such that h K. } K α = {((m, h ), (m, h )) m /h α m /h and T ype m α (h ) = T ype m α (h )} α = {((m, h ), (m, h )) m /h α m /h } (m, h) V (p) iff m/h V (p). It is immediate that R, C α, K α and α are all equivalence relations. It is also not hard to see that the remaining properties from Definition A.2 are all satisfied. We only verify 24

25 property (iv): Suppose that (m 1, h 1 ) K α (m 2, h 2 ) and (m 2, h 2 ) C α (m 3, h 3 ). Then by the definition of C α, m 3 = m 2 and there is some K Choice m 2 α (h 2 ) such that h 3 K. Then since Label(K) = T ype m 2 α (h 2 ), we have h 3 [T ype m 2 α (h 2 )] m 2 α. Since T ype m 1 α (h 1 ) = T ype m 2 α (h 2 ), we also have h 3 [T ype m 1 α (h 1 )] m 2 α. Furthermore, since m 1 /h 2 α m 2 /h 2 and h 3 H m 2, by the knowledge-moment condition we have m 1 /h 2 α m 2 /h 3. Hence, (m 1, h 1 ) K α (m 3, h 3 ). A simple proof by induction shows that for all formulas B, M, m/h = B iff M, (m, h) = B. Hence, M, (m, h) = A. ( ) Let M = W, R, α, C α, K α, V be a labeled stit Kripke model satisfying all the properties in Definition A.2. Given Lemma A.4, we can assume, without loss of generality, that for all w W, C α (w) = R (w) K α (w). Define a labeled stit model M = Tree, <, Agent, Choice, α, T ype, V, as follows: Agent = {α} The moments are M = {R (w) w W } W R (w) < v iff v R (w). Hence, H R (w) = {{R (w), v} v R (w)}. T ype = {K α (w) w W } For v R (w), let X v = {{R (w), v } v C α (v)}. Then, Choice R (w) α = {X v v R (w)}. R (w)/{r (w), w} α R (v)/{r (v), v} iff w α v. For X v Choice R (w) α, let Label(X v ) = K α (v). Then, T ypeα R (w) ({R (w), v}) = K α (v). V (p) = {R (w)/{r (w), v} v V (p)} We first show that if Label is defined, then it is 1-1: Suppose that X Choice R (w) α, then X = X v for some v R (w). Then Label(X) = K α (v). Suppose X Choice R (w) α 25

26 such that X X. Then, X = X v for some v R (w) with C α (v) C α (v ). Then, v C α v. Therefore, since C α (v) = R (v) K α (v) either v K α v or v R v. Since, X v, X v Choice R (w) α, we have v, v R (w). Thus, v R v. Thus, we have K α (v) K α (v ). We now prove that for all w W, M, w = B iff M, R (w)/{w} = B. The proof is by induction on the structure of B. The base case and Boolean connectives follow as usual. Suppose that M, w = K α B. Then for all v W, if w α v, then M, v = B. By the induction hypotheses, for all v W, if w α v, then M, R (v)/{r (v), v} = B. But, by construction if w α v, then R (w)/{r (w), w} α R (v)/{r (v), v}. Hence, M, R (w)/{r (w), w} = K α B. Suppose that M, R (w)/{r (w), w} = K α B. Then for all R (v)/{r (v), v}, if R (w)/{r (w), w} α R (v)/{r (v), v}, then M, R (v)/{r (v), v} = B. By the induction hypothesis, M, v = B for all v W such that R (w)/{r (w), w} α R (v)/{r (v), v}. By construction, we have R (w)/{r (w), w} α R (v)/{r (v), v} iff w α v. Hence, M, w = K α B. Suppose that M, w = [α stit: B ]. Then, for all v W, if w C α v, then M, v = B. That is, for all v C α (w), M, v = B. By the induction hypothesis, this means that for all v C α (w), M, R (v)/{r (v), v} = B. By construction, {{R (v), v} v C α (w)} = {{R (w), v} v C α (w)} B R (w) (note that if v C α (w), then v R (w) and so since R is an equivalence relation, R (v) = R (w)). Furthermore {{R (w), v} v C α (w)} = Choice R (w) α ({R (w), w}). Therefore, M, R (w), {R (w), w} = [α stit: A]. Suppose that M, R (w)/{r (w), w} = [α stit: B ]. Then Choice R (w) α ({R (w), w}) B R (w) 26

27 = {{R (w), v} M, R (w)/{r (w), v} = B } = {{R (v), v} M, R (w)/{r (w), v} = B } By the induction hypothesis, for all {R (w), v} Choice R (w) α ({R (w), w}), M, v = B. By construction, if w C α v, then {R (w), v} Choice R (w) α. Hence, M, w = [α stit: B ]. Suppose that M, w = [α kstit: B ]. Then, for all v W, if w K α v, then M, v = B. By the induction hypothesis, for all v W, if w K α v, then M, R (v)/{r (v), v} = B. Now, by construction, we have T ype R (w) α ({R (w), w}) = K α (w) and T ypeα R (v) ({R (v), v}) = K α (v). Since v K α (w) and K α is an equivalence relation, we have K α (v) = K α (w). Therefore, {R (v), {R (v), v}} [T ype R (w) α ({R (w), w})] R (v) α. Since w K α v, we must have w α v. Hence, {R (w), {R (w), w}} α {R (v), {R (v), v}}. Therefore, M, R (w)/{r (w), w} = [α kstit: B ]. Suppose that M, R (w)/{r (w), w} = [α kstit: B ]. Then, for all R (v)/{r (v), v }, if R (w)/{r (w), w} α R (v)/{r (v), v }, then [T ypeα R (w) ({R (w), w})] α R (v) B R (v). By the induction hypothesis, if {R (v), v } [T ypeα R (w) ({R (w), w})] R (v) α, then M, v = B. Furthermore, by construction, if K α (v ) = K α (w), then {R (v), v } [T ypeα R (w) ({R (w), w})] R (v) α. Suppose that v is a state such that w K α v. Then, {R (v ), v } [T ype R (w) α ({R (w), w})] R (v ) α. Hence, M, v = B. Thus, M, w = [α stit: B ]. qed Completeness for the class of labeled stit models follows from standard techniques. A.1 Completeness when more than one agent is acting In this appendix, we have simplified our models by assuming that there is only one agent. The next natural question is: can we adapt the above argument to a setting with more than one agent while still maintaining the assumption that only one agent is acting at any given moment? We conclude by highlighting some issues that arise in the multiagent case. 27

28 A complete analysis will be left for a different paper. Note that if [α kstit: ] is true at an index m/h, then T ype m α (h) is not defined (this follows, in part, form the fact that α is reflexive). Thus, we can take the truth of [α kstit: ] at an index m/h to mean that agent α is not acting at m. Under this interpretation, the following two formulas express the fact that there is only one agent acting at any given moment: A7. α Agent [α kstit: ] A8. [α kstit: ] β α [β kstit: ] There is another natural constraint that we can impose on labeled stit frames. In general, there may be an index m/h in a model M such that M, m/h = [α kstit: ] K α [α kstit: ]. This happens if α is acting at m, but there is an index m /h such that m/h α m /h and α is not acting at m. That is, the agent α does not know that she is acting. The following two axioms state that the agents know whether they are acting: A9. [α kstit: ] K α [α kstit: ] A10. [α kstit: ] K α [α kstit: ] The corresponding constraints on the labeled stit frames is (1) if α is acting at a moment m, then α is acting at all moments m that are indistinguishable from m for α, and (2) if α is not acting at m, then α is not acting at any moment m that is indistinguishable from m for α. All that remains is to adapt the definition of a labeled stit Kripke model (Definition A.2) to a setting with more than one agent. The only difficulty involves the definition of the K α relation. This relation is used to evaluate the kstit modalities and the equivalence classes are the types in the constructed labeled stit model. Thus, each K α must be a partial equivalence relation on W, i.e., for each agent α, K α is an equivalence relation on a subset W W. The details of this definition and other issues that arise in the multiagent situation will be discussed in a different paper. 28

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