Knowledge and Ability

In this chapter we relate epistemic logics with logics for strategic ability developed and studied in computer science, artificial intelligence and multi-agent systems. We discuss approaches from philosophy and artificial intelligence to modelling the interaction of agents’ knowledge and abilities and then focus on concurrent game models and the alternating-time temporal logic ATL. We show how ATL enables reasoning about agents’ coalitional abilities to achieve qualitative objectives in concurrent game models, first assuming complete information and then under incomplete information and uncertainty about the structure of the game model. We then discuss epistemic extensions of ATL enabling explicit reasoning about the interaction of knowledge and strategic abilities on different epistemic levels, leading inter alia to the notion of constructive knowledge. Chapter 11 of the Handbook of Epistemic Logic, H. van Ditmarsch, J.Y. Halpern, W. van der Hoek and B. Kooi (eds), College Publications, 2015, pp. 543–589. 544 CHAPTER 11. KNOWLEDGE AND ABILITY Our aim in this chapter is to survey logics that attempt to capture the interplay between knowledge and ability. The term “ability”, in the sense we mean it in this chapter, corresponds fairly closely to its everyday usage. That is, ability means the capability to do things, and to bring about states of affairs. There are several reasons why ability is worth studying in a formal setting: • First, the concept of ability is surely worth studying from the perspective of philosophy, and in particular the philosophy of language. In this context, the point is to try to gain an understanding of what people mean when they make statements like “I can X”, and in particular to understand what such a claim implies about the mental state of the speaker. • Second, and more recently, researchers in computer science and artificial intelligence are interested in the notion of what machines can achieve. For example, imagine an artificial intelligence to whom we can issue instructions in natural English. Then, if we give the machine an instruction “X” (where X might be “make me a cup of tea” or, famously, “bring me a beer”), then the question of what this instruction means in relation to the abilities of the machine becomes important. An artificial intelligence that is presented with an instruction to X surely needs to understand whether or not it actually can in fact X before it proceeds; this implies some model or theory of ability, and the ability of the artificial intelligence to reason with this model or theory. The remainder of this chapter is structured as follows. We begin, in the following section, with a discussion on the way that philosophers have considered knowledge and ability. In section 11.2, we move on to discuss how the concept of ability has been considered and formalised in artificial intelligence; we focus in particular on the seminal and enormously influential work of Robert Moore on the relationship between knowledge and ability. We then go on to review more recent contributions to the logic of ability, arising from the computer aided verification community: specifically, Alternating-time Temporal Logic (ATL) and the issues that arise when attempting to integrate ATL with a theory of knowledge. 11.1 Philosophical Treatments of Knowledge and Ability We will begin by reviewing the way that the concept of ability has been considered in philosophy. We will start with the work of the philosopher 11.1. PHILOSOPHICAL TREATMENTS 545 Gilbert Ryle (1900–1976). Ryle was greatly interested in the concept of mind, and one of the questions to which he addressed himself was the distinction between knowing how and knowing that. Crudely, “knowing that” is the concept of knowledge with which this handbook is largely concerned: we think of “knowing that” as a kind of relation between agents and true propositions, and in this book we write Ki' to mean that agent i knows that ', where ' is some proposition. The concept of “knowing how” seems related, but is clearly different: it is concerned with the knowledge of how to achieve things. One question in particular that Ryle considered was whether know-how could be reduced to know-that: that is, whether know-how was in fact just a type of know-that. Ryle argued that such a reduction was not possible. His argument against such a reduction has since become quite celebrated (it is commonly known as “Ryle’s regress”, and has been applied in various other settings). Broadly, the argument goes as follows: If know-how were a species of know-that, then, to engage in any action, one would have to contemplate a proposition. But, the contemplation of a proposition is itself an action, which presumably would itself have to be accompanied by a distinct contemplation of a proposition. If the thesis that knowledgehow is a species of knowledge-that required each manifestation of knowledge-how to be accompanied by a distinct action of contemplating a proposition, which was itself a manifestation of knowledge-how, then no knowledge-how could ever be manifested. In other words, know-how does not reduce to know-that because if we had to know-that a proposition every time we exercised know-how, we would get an infinite regress of know-thats. There are other arguments against a reduction of know-how to knowthat. For example, consider the following sentence: Michael knows how to ski. (⇤) If know-how reduces to know-that, then there is some proposition, call it '⇤, such that knowing how to ski is equivalent to knowing '⇤. Presumably, telling somebody the proposition '⇤ would then be enough to convey the ability to ski, assuming of course the hearer of the message was able to process and understand it. We are trivialising the arguments at stake, of course, but nevertheless, anybody who has learned to ski will recognise that such a simplistic reduction of know-how to know-that is implausible. 546 CHAPTER 11. KNOWLEDGE AND ABILITY The exact relationship of know-how to know-that remains the subject of some debate, with arguments on either side; see, e.g., Stanley and Williamson for a contemporary discussion. Despite the philosophical objections of Ryle and others, researchers in artificial intelligence have largely adopted the view that know-how can be reduced to know-that. In the section that follows, we will see how researchers in artificial intelligence have attempted to formalise such reductions. Interestingly, with a sufficiently rich logical formalism, the two notions turn out distinct again, as we will argue in Section 11.5.3. There have been several other early philosophical approaches to developing logics of agency and ability, including works of Georg Henrik von Wright (1916–2003), Stig Kanger (1924–88), Brian Chellas, Mark Brown and Nuel Belnap and Michael Perloff. In particular, Brown proposes a modal logic with non-normal modalities formalising the idea that the modality for ability has a more complex, existential-universal meaning (the agent has some action or choice, such that every outcome from executing that action (making that choice) achieves the aim), underlying all further approaches to formalizing agents’ ability that will be presented here. At about the same time Belnap and Perloff developed the basics of their theory of seeing to it that, usually abbreviated to “STIT”. The STIT family of logics were formulated in an attempt to be able to give a formal semantics to the concept of agency, where an agent is loosely understood to be an entity that makes deliberate purposeful actions. Although accounts differ on details, the general aim is to be able to make sense of statements such as “the agent i brings it about that '”. We will denote that STIT operator by ` i '. The semantics of Belnap and Perloff’s operator (technically, known as the achievement stit operator), which makes the idea behind Brown’s ability operators explicit, was as follows. Formulae of the form ` i ' are interpreted in a branchingtime model containing histories of the world, each of which contains various “moments”. Branching in the model occurs as a result of choices made by agents. Intuitively, in making a particular choice, an agent constrains the possible future histories of the world. Roughly speaking, we then say that ` i ' is true in a world/moment pair if there was some earlier moment in the history at which point the agent i made a choice such that: • ' is true in all histories consistent with that choice; • at the point at which the choice was made, the status of ' was not settled, i.e., there were histories arising from the choice moment in which ' was false. Expressed differently, ` i ' means that the agent i made a choice such that ' was a necessary consequence of this choice, while ' would not necessarily 11.1. PHILOSOPHICAL TREATMENTS 547 have been true had the agent not made this choice. Although we have not presented the full formal semantics for Belnap and Perloff’s operator, it should be obvious that the operator is semantically rather complex, at least compared to conventional tense and dynamic operators (the semantics constitute a “formidable definition” in the words of Belnap, Perloff, and Xu). The main source of difficulty is that an expression ` i ' seems to refer to both the past (the moment at which the relevant choice was made) and the present/future (the moments at which ' is true). Several subsequent proposals have been made to refine and simplify the logic of STIT, with the aim of dealing with some of their counterintuitive properties, and contemporary formulations are much simpler and more intuitive. Here is a list of simple candidate axioms and deduction rules for a logic of agency, due to Troquard: (M) ` i(' ^ ) ! ( ` i ' ^ ` i ) (C) ( ` i ' ^ ` i ) ! ` i(' ^ )