Apparent actions and apparent goal-directedness

Apparent actions We assume that a passive observer perceives a sequence S of sensor values (s1, ..., sT ) with T ∈ N. At each instance t of the sequence, the observer has a memory or knowledge state mt which gives us a second sequence M = (m1, ...,mT ). The memory state of the agent might contain models of the observations or not. In the course of a sequence S there is an apparent action for observer M if for r, t ∈ {1, ..., T} we have (sr,mr) = (st,mt) and (sr+1,mr+1) 6= (st+1,mt+1). The intuition is that, since the same observation sr = st at different times r, t is followed by different observations and the observer’s states mr = mt do not indicate / predict the difference, the observer suspects a hidden mechanism causing the difference. We assume that the observer interprets all signs of hidden mechanisms as (apparent) actions. Using the empirical distribution pS,M (s′, s,m) = 1 T ∑T t=0 δs′st+1δsstδmmt (δxy is Kronecker’s delta) we can quantify the extent of apparent actions along the sequences as the conditional entropy H(S′|S,M). Apparent goal-directedness Our observer attributes the complete sequence S to be the result of an agent’s strategy. Note that even if there is no apparent action along a subsequence this could be due to the agent trying to avoid detection. The idea is that any directedness reveals itself as some pattern within the sequence and any pattern in the sequence will increase the compressibility of the sequence. So we here define apparent goal-directedness of the observed sequence S as its compressibility. Using a common compression algorithm (e.g. gzip) Z we can estimate compressibility as l(S)−Z(S) where l(S) is the binary length of all the data in the observed sequence and Z(S) the binary length of the compressed data. Note that an adversary’s goal-directedness can remain undetected only if S is completely random.