Deductive Algorithmic Knowledge

Abstract

The framework of algorithmic knowledge assumes that agents use algorithms to compute the facts they explicitly know. In many cases of interest, a deductive system, rather than a particular algorithm, captures the formal reasoning used by the agents to compute what they explicitly know. We introduce a logic for reasoning about both implicit and explicit knowledge with the latter defined with respect to a deductive system formalizing a logical theory for agents. The highly structured nature of deductive systems leads to very natural axiomatizations of the resulting logic when interpreted over any fixed deductive system. The decision problem for the logic, in the presence of a single agent, is NP-complete in general, no harder than propositional logic. It remains NP-complete when we fix a deductive system that is decidable in nondeterministic polynomial time. These results extend in a straightforward way to multiple agents.

Belief ascription under bounded resources

“…The number of inference rules applied at each step is unbounded, but the rules are not immediately applied again to the newly derived formulas at the same time step. So these models correspond to the step logic of Perlis et al rather than to for example Konolige's deduction model of belief (Konolige, 1986) or algorithmic knowledge (Pucella, 2006) where the set of beliefs is assumed to be deductively closed under inference, and the time taken to achieve this closure is not taken into account. It would be straightforward to introduce systems with deductively closed belief sets as well, but we do not believe that they constitute an interesting case for the study of resource-bounded belief ascription problem.…”

“…Yet another account of epistemic logic called algorithmic knowledge, which treats explicit knowledge as something which has to be computed by an agent, was introduced in (Halpern et al, 1994), and further developed in, e.g., (Fagin et al, 1995;Pucella, 2006). In the algorithmic knowledge approach, agents are assumed to possess a procedure which they use to produce knowledge.…”

“…In the algorithmic knowledge approach, agents are assumed to possess a procedure which they use to produce knowledge. In later work (Pucella, 2006), this procedure is assumed to be given as a set of rewrite rules which are applied to the agent's knowledge to produce a closed set, so, like Konolige's approach, algorithmic knowledge is concerned with the result rather than the process of producing knowledge.…”

Reasoning about Other Agents’ Beliefs under Bounded Resources

“…An account of epistemic logic called algorithmic knowledge, which treats explicit knowledge as something which has to be computed by an agent, was introduced in [9], and further developed in e.g. [1,10]. In the algorithmic knowledge approach, agents are assumed to possess a procedure which they use to produce knowledge.…”

“…In the algorithmic knowledge approach, agents are assumed to possess a procedure which they use to produce knowledge. In later work [10] this procedure is assumed to be given as a set of rewrite rules which are applied to the agent's knowledge to produce a closed set, so, like Konolige's approach, algorithmic knowledge is concerned with the result rather than the process of producing knowledge. In [11,12] Duc proposed logics for non-omniscient epistemic reasoners which will believe all consequences of their beliefs eventually, after some interval of time.…”