# We can build safe systems using mathematical proof

This post is partly excerpted from the preprint to:

Omohundro, Steve (forthcoming 2013) “Autonomous Technology and the Greater Human Good”, Journal of Experimental and Theoretical Artificial Intelligence (special volume “Impacts and Risks of Artificial General Intelligence”, ed. Vincent C. Müller).

A primary precept in medical ethics is “Primum Non Nocere” which is Latin for “First, Do No Harm”. Since autonomous systems are prone to taking unintended harmful actions, it is critical that we develop design methodologies that provide a high confidence of safety. The best current technique for guaranteeing system safety is to use mathematical proof. A number of different systems using “formal methods” to provide safety and security guarantees have been developed. They have been successfully used in a number of safety-critical applications.

This site provides links to current formal methods systems and research. Most systems are built by using first order predicate logic to encode one of the three main approaches to mathematical foundations: Zermelo-Frankel set theory, category theory, or higher order type theory. Each system then introduces a specialized syntax and ontology to simplify the specifications and proofs in their application domain.

To use formal methods to constrain autonomous systems, we need to first build formal models of the hardware and programming environment that the systems run on. Within those models, we can prove that the execution of a program will obey desired safety constraints. Over the longer term we would like to be able to prove such constraints on systems operating freely in the world. Initially, however, we will need to severely restrict the system’s operating environment. Examples of constraints that early systems should be able to provably impose are that the system run only on specified hardware, that it use only specified resources, that it reliably shut down in specified conditions, and that it limit self-improvement so as to maintain these constraints. These constraints would go a long way to counteract the negative effects of the rational drives by eliminating the ability to gain more resources. A general fallback strategy is to constrain systems to shut themselves down if any environmental parameters are found to be outside of tightly specified bounds.

## Avoiding Adversarial Constraints

In principle, we can impose this kind of constraint on any system without regard for its utility function. There is a danger, however, in creating situations where systems are motivated to violate their constraints. Theorems are only as good as the models they are based on. Systems motivated to break their constraints would seek to put themselves into states where the model inaccurately describes the physical reality and try to exploit the inaccuracy.

This problem is familiar to cryptographers who must watch for security holes due to inadequacies of their formal models. For example, this paper recently showed how a virtual machine can extract an ElGamal decryption key from an apparently separate virtual machine running on the same host by using side-channel information in the host’s instruction cache.

It is therefore important to choose system utility functions so that they “want” to obey their constraints in addition to formally proving that they hold. It is not sufficient, however, to simply choose a utility function that rewards obeying the constraint without an external proof. Even if a system “wants” to obey constraints, it may not be able to discover actions which do. And constraints defined via the system’s utility function are defined relative to the system’s own semantics. If the system’s model of the world deviates from ours, the meaning to it of these constraints may differ from what we intended. Proven “external” constraints, on the other hand, will hold relative to our own model of the system and can provide a higher confidence of compliance.

Ken Thompson was one of the creators of UNIX and in his Turing Award acceptance speech “Reflections on Trusting Trust” he described a method for subverting the C compiler used to compile UNIX so that it would both install a backdoor into UNIX and compile the original C compiler source into binaries that included his hack. The challenge of this Trojan horse was that it was not visible in any of the source code! There could be a mathematical proof that the source code was correct for both UNIX and the C compiler and the security hole could still be there. It will therefore be critical that formal methods be used to develop trust at all levels of a system. Fortunately, proof checkers are short and easy to write and can be implemented and checked directly by humans for any desired computational substrate. This provides a foundation for a hierarchy of trust which will allow us to trust the much more complex proofs about higher levels of system behavior.

## Constraining Physical Systems

Purely computational digital systems can be formally constrained precisely. Physical systems, however, can only be constrained probabilistically. For example, a cosmic ray might flip a memory bit. The best that we should hope to achieve is to place stringent bounds on the probability of undesirable outcomes. In a physical adversarial setting, systems will try to take actions that cause the system’s physical probability distributions to deviate from their non-adversarial form (e.g. by taking actions that push the system out of thermodynamic equilibrium).

There are a variety of techniques involving redundancy and error checking for reducing the probability of error in physical systems. von Neumann worked on the problem of building reliable machines from unreliable components in the 1950’s. Early vacuum tube computers were limited in their size by the rate at which vacuum tubes would fail. To counter this, the Univac I computer had two arithmetic units for redundantly performing every computation so that the results could be compared and errors flagged.

Today’s computer hardware technologies are probably capable of building purely computational systems that implement precise formal models reliably enough to have a high confidence of safety for purely computational systems. Achieving a high confidence of safety for systems that interact with the physical world will be more challenging. Future systems based on nanotechnology may actually be easier to constrain. Drexler describes “eutactic” systems in which each atom’s location and each bond is precisely specified. These systems compute and act in the world by breaking and creating precise atomic bonds. In this way they become much more like computer programs and therefore more amenable to formal modelling with precise error bounds. Defining effective safety constraints for uncontrolled settings will be a challenging task probably requiring the use of intelligent systems.

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