Circumscriptive Ignorance

In formal systems that reason about knowledge, inferring that an agent actually does not know a particular fact can be problematic. Collins [l] has shown that there are many different modes of reasoning that a subject can use to show that he is ignorant of something; some of these, for example, involve the subject reasoning about the limitations of his own information-gathering and memory abilities. This paper will consider a single type of inference about ignorance, which we call circumscriptive ignorance. We present a partial formalization of circumscriptive ignorance and apply it to the Wise Man Puzzle.’ 1. Circumscriptive Ignorance The premise that there is a limited amount of information, resources, or strategies available for the solution of a problem is often an unstated but essential part of the problem abstraction. For example, in the Missionaries and Cannibals Puzzle, it is important that only a single boat is available to ferry people across the river; one cannot invoke a helicopter brigade from the Sudan to solve the puzzle. McCarthy [5] has investigated the first-order formalization of problem statements such as this, using a circumscription schema to capture unstated limitations on resources. In puzzles that involve reasoning about the knowledge agents possess, there are often unstated conditions on the initial information given an agent, as well as on the information he can acquire. In the Wise Man Puzzle (see Section 3 below for a full statement of this puzzle), it is common knowledge that each man can see his neighbors’ spots and knows from the king that there is at least one white spot. It is an unstated condition of the puzzle that this is the only knowledge that wise men have about the initial situation; in an adequate formalization of the puzzle it should be possible to prove that each wise man is ignorant of the color of his own spot in the initial situation. In effect, the knowledge that is available to the agents in ‘This short note describes current research collected in 141. The work presented here was supported by grant N0014-80-C-0296 from the Office of Naval Research. the puzzle is being circumscribed; informally one would say “The only facts that agent S knows about proposition p are F.” If from the facts F it is not possible for S to infer p ( or -p), then S does not know whether p is true. Proving ignorance based on a limitation on the knowledge available to an agent will be called reasoning by circumscriptive ignorance. Circumscriptive ignorance, especially for the Wise Man Puzzle, has been formalized in first-order axiomatizations of a possible-world semantics for knowledge (Goad [a], McCarthy [6], and Sato [7]). However, there has been no formalization in a modal logic of knowledge. The advantages of using a modal formalization are clarity of expression and the ability to use inference procedures, including decision procedures, that have been developed for modal logic. In the next few sections we outline a modal formalization of circumscriptive ignorance. 2. The Modal Logic K14 The modal logic we shall use is a propositional modal logic based on Sato’s K4 [7], which includes an axiomatization of common knowledge.2 K4 is a family of languages parameterized by the choice of propositional letters Pr and agents Sp. 0 E Sp is a reserved name for Fool, a fictitious agent whose knowledge is common knowledge. For a particular choice of Pr and Sp, the language K4 is the propositional calculus over Pr, together with a set of indexed unary modal operators [S], S E Sp. The intended meaning of [S](Y is that agent S knows a. The axiom schemata for K4 are (Al) All propositional tautologies (A4 [SIC-~ WV [Sla 3 PI w (1) (A4 [wm~wlaw~ (A5) Pl~xolcsl~ 7 where Q and ,f3 denote arbitrary sentences, and S denotes an arbitrary agent. Axioms Al-A4 give the system S4 2For simplici t y we use K4 rather than Sato’s more complicated KT4, which deals explicitly with time. 202 From: AAAI-82 Proceedings. Copyright ©1982, AAAI (www.aaai.org). All rights reserved. for each modality [S], while A5 is the common knowledge axiom: what any fool knows, any fool knows everyone knows. The two rules of inference are modus ponens and necessitation (from (Y, infer [Slay). In K4 and other modal logics of knowledge, an agent’s knowledge is described as a theory, that is, as a set of formulas that contain th .e axioms and are closed u nder the rules of inference. To see this, note that all instances of [S](r for which (Y is an axiom of K4 are provable, and that modus ponens is implemented by A4. We shall use the term agent’s theory to mean the set of formulas Q for which [S]cy can be proven in K4. There is a difference between the axiomatization of an agent’s theory and K4 itself. K4 is a Hilbert system in which no proper axioms are allowed; an agent’s theory allows proper axioms (i.e., we can assert formulas of the form [S]P).~ We define the a-theory of K4, for a fixed sentence cr, as the set of formulas p for which cr>/!? is a theorem of K4. We write ~+$9 if p is in the a-theory of iY4, i.e., cy>/?’ is a theorem of K4. h-4 bY itself 1s not sufficient to represent circumscriptive ignorance, since there is no way of limiting the proper axioms that could conceivably be used to derive knowledge in an agent’s theory. If we look at a particular th eory for an agent, where the only proper axiom is cr, it is impossible to derive proofs of certain formulas within that theory; but there is also no way to express this in K4 itself. bY To express circumscriptive ignorance, K4 is extended a family of new unary modal operators indexed by sentences of K4. These are called circumscriptive modalities, and are written as [cy], where Q is a sentence of K4. The extended language is called K14. In informal terms, [a] is intended to mean the a-theory of K4, that is, [cr]/3 holds just in case p is in the a-theory of K4. Thus the notion of provability is explicitly introduced into K14. The axiomatization of [cr] is problematic, since it involves formalizing not only which sentences of K4 are provable, but also which sentences are not. However, a sufficient set of axioms for KI4 can be obtained by making use of the intended interpretation of [CL] as provability in K4. The axioms of K14 are simply those of K4, together with the schemata