We present a computable model of consciousness that is a modi cation of an existing model of universal computation. This modi cation is partially motivated by two existing, but non harnessable, models of consciousness. We say a model of consciousness is harnessable i the following statement holds: if the model predicts that a system v is more conscious than another system u, then we should, in general, nd v more useful than u in a wide range of tasks. While there are no domain-general de nitions of what makes a system harnessable, we give a preliminary proposal here and assess our model against this yardstick.1 1 A preliminary version of this research was presented at the SRI 2017 Technology and Consciousness workshop series. We are grateful for the comments received during the workshop. Support from AFOSR and ONR has enabled the development of formal systems that underlie some of the work presented here, particularly moral cognition that requires de se reasoning. 2 Models such as by Tononi (2012) do not appear to be readily harnessable, see Aaronson (2014)
We present a computable model of consciousness by slightly altering an existing universal model of computation by adding in non-wellfounded objects. The model of computation that we use is the Kolmogorov-Uspensky one (Kolmogorov & Uspensky 1958). Non-wellfounded objects are ones that can contain themselves; see (Moss 2018) for an introduction. While common foundational theories of mathematics, such as Zermelo{Fraenkel theory with the axiom of choice (ZFC) are based on axioms that rule out non-wellfounded objects, variations of set theory based on axioms such as Aczel's anti-foundation axiom allow for such objects.
We also assert that any model of consciousness should be harnessable; i.e., if the model predicts that a system v is more conscious than another system u, then we should, in general, nd v more harnessable than u. While there are no domain-general de nitions of what makes a system harnessable, we give a preliminary proposal here and assess our model against this yardstick.2
The plan for the paper is as follows. First, we lay down conditions for what it means for a system to have harnessable consciousness. We then brie y discuss prior models of consciousness based on non-wellfounded objects. In (x4), we present a harnessable formal system . We arrive at this model by altering an existing model of universal computation, the Kolmogorov & Uspensky model of computation. Finally, we sketch examples showing the model satisfying some of the harnessability conditions laid out before. We end by discussing future work and next steps. 2 For a model of consciousness to be harnessable, we require the following set of high-level conditions be satis ed by any system that the model asserts is conscious. (These conditions should be considered a working set of conditions rather than a nalized set of conditions.) Condition 1 Di erentiation of de se/de re/de dicto beliefs : Any model of consciousness that predicts that a system is conscious should also require the system to di erentiate between de se/de re/de dicto beliefs. We illustrate these three kinds of beliefs with the following example. Let us say that there is a room in which there are two agents and an object, a ower. The agent on the right is looking at the ower. There are three statements with varying levels of self-reference that the agent can make, as shown in Figure 6. See Bringsjord & Govindarajulu (2013) for more details.
de dicto e cn de re e lffrre e e s r e p e e D de se
The secpoenrsdotnal est pTehrseosnecseoensdatafllolewsetr
That person That person on the right
sees a flower mIyself
I myself see a flower Condition 2 Unbounded Iterated Self Statements : Any model of consciousness that predicts that a system is conscious should also require the system to believe an unbounded sequence of iterated self-belief statements without arti cially kludging them in. For instance, if the system is perceiving a ower, as shown in Figure 2, the model should predict, at least under some conditions, the system to believe in expressions that correspond to the following statements: There is a ower, Agent x sees a ower, PIarste5.eToawardos wCoemrp,utIabsileitey that I see a ower, I see that I see that I see a ower, : : :.
Challenge Condition 3 Non-trivial Temporal Unboundedness : Any model of consciousness that predicts that a system is conscious should predict that the system acts over a large interval of time T 0 in a non-trivial fashion (unlike simple computational systems that can operate for decades).
Condition 4 Theory of Minds : The nal condition requires that any model of consciousness that predicts that a system is conscious should also allow for a mechanism that enables the system to form beliefs about other agents. 3
Miranker & Zuckerman (2009) present a model of consciousness that is based on non-wellfounded sets. Particularly, they take as a primitive a consciousness operator. This operator is not de ned further in terms of computable mechanisms. Corazza (2014) presents a model of consciousness derived from Advaita, one of the main branches of Vedic philosophy. Brie y, Advaita states only pure consciousness exists and the physical universe arises due to consciousness interacting with itself. This monistic conception is not easy to model in standard set theory. So Corazza leverages a version of set theory with non-wellfounded objects to present a model that satis es several principles of Advaita. While both these prior studies are robust philosophically and mathematically, they do not give us an easily computable or harnessable model. 4
KU machines were introduced by Kolmogorov & Uspensky (1958) to capture a general de nition of algorithms that more closely resembles human cognition. They have been used in attempted proofs of the Church Turing thesis due to their high-level of presentation, as compared with other models of computation.3 We present below a KU machine formalism rooted in formal logic. We have a formal logic F hL; Ii composed of a set of expressions from a language L and a nite set of inference schemata I. KU machines operate on a set of states S. States are directed graphs with labelled edges. Each node has either: (1) an associated expression from L; (2) or an inference schemata from I. Note: A KU machine state denotes one or more proofs in C.a a A KU machine state can be considered analogous to a workspace in the Slate system (Bringsjord et al. 2008). For a formalized version of Slate, see (Govindarajulu 2013, chap. 3). The algorithm proceeds by replacing the active patch with (P ) if the active patch is equivalent to P . Associated with each pair (Pi; (Pi)) is a mapping i between the nodes in the boundary of the active patch Pi to certain nodes in (Pi). The mapping i ensures that the new active patch aligns with the rest of the dataspace. We modify the KU machine formalism to allow any node to contain an entire state. A node n in a KU machine state S 2 S can have associated one of the following entities: 1. Any expression from L 2. Any inference schemata from I 3. Any state from S, including S The following de nition connects beliefs of an agent modeled using a KU machine with the state of the machine.
Belief De nition If an expression is within the active patch of a KU machine state S at time t and if that state corresponds to an agent a's state of mind at time t, we can 3 See (Smith 2013, chap. 45) for one such attempt.
Shadow Prover 8 A The gure below shows an input proTblem toeShaPdowuProrvero.Thie preobldem coLmeestter h l n from a situation in Edgar Allan Poe's The Purloined Letter. Note that situation contains many levels of iterated beliefs. ShadowProver solves this problem in around 5~5ms on a machine with 2:9 GHz CPU and 16 GB of memory.