Modeling naturalized decision problems in linear logic

The following is a model of a simple decision problem (namely, the 5 and 10 problem) in linear logic. Basic familiarity with linear logic is assumed (enough to know what it means to say linear logic is a resource logic), although knowing all the operators isn’t necessary.

The 5 and 10 problem is, simply, a choice between taking a 5 dollar bill and a 10 dollar bill, with the 10 dollar bill valued more highly.

While the problem itself is trivial, the main theoretical issue is in modeling counterfactuals. If you took the 10 dollar bill, what would have happened if you had taken the 5 dollar bill? If your source code is fixed, then there isn’t a logically coherent possible world where you took the 5 dollar bill.

I became interested in using linear logic to model decision problems due to noticing a structural similarity between linear logic and the real world, namely irreversibility. A vending machine may, in linear logic, be represented as a proposition “$1 → CandyBar”, encoding the fact that $1 may be exchanged for a candy bar, being consumed in the process. Since the $1 is consumed, the operation is irreversible. Additionally, there may be multiple options offered, e.g. “$1 → Gumball”, such that only one option may be taken. (Note that I am using “→” as notation for linear implication.)

This is a good fit for real-world decision problems, where e.g. taking the $10 bill precludes also taking the $5 bill. Modeling decision problems using linear logic may, then, yield insights regarding the sense in which counterfactuals do or don’t exist.

First try: just the decision problem

As a first try, let’s simply try to translate the logic of the 5 and 10 situation into linear logic. We assume logical atoms named “Start”, “End”, “$5”, and “$10”. Respectively, these represent: the state of being at the start of the problem, the state of being at the end of the problem, having $5, and having $10.

To represent that we have the option of taking either bill, we assume the following implications:

TakeFive : Start → End ⊗ $5

TakeTen : Start → End ⊗ $10

The “⊗” operator can be read as “and” in the sense of “I have a book and some cheese on the table”; it combines multiple resources into a single linear proposition.

So, the above implications state that it is possible, starting from the start state, to end up in the end state, yielding $5 if you took the five dollar bill, and $10 if you took the 10 dollar bill.

The agent’s goal is to prove “Start → End ⊗ $X”, for X as high as possible. Clearly, “TakeTen” is a solution for X = 10. Assuming the logic is consistent, no better proof is possible. By the Curry-Howard isomorphism, the proof represents a computational strategy for acting in the world, namely, taking the $10 bill.

Second try: source code determining action

The above analysis is utterly trivial. What makes the 5 and 10 problem nontrivial is naturalizing it, to the point where the agent is a causal entity similar to the environment. One way to model the agent being a causal entity is to assume that it has source code.

Let “M” be a Turing machine specification. Let “Ret(M, x)” represent the proposition that M returns x. Note that, if M never halts, then Ret(M, x) is not true for any x.

How do we model the fact that the agent’s action is produced by a computer program? What we would like to be able to assume is that the agent’s action is equal to the output of some machine M. To do this, we need to augment the TakeFive/TakeTen actions to yield additional data:

TakeFive : Start → End ⊗ $5 ⊗ ITookFive

TakeTen : Start → End ⊗ $10 ⊗ ITookTen

The ITookFive / ITookTen propositions are a kind of token assuring that the agent (“I”) took five or ten. (Both of these are interpreted as classical propositions, so they may be duplicated or deleted freely).

How do we relate these propositions to the source code, M? We will say that M must agree with whatever action the agent took:

MachineFive : ITookFive → Ret(M, “Five”)

MachineTen : ITookTen → Ret(M, “Ten”)

These operations yield, from the fact that “I” have taken five or ten, that the source code “M” eventually returns a string identical with this action. Thus, these encode the assumption that “my source code is M”, in the sense that my action always agrees with M’s.

Operationally speaking, after the agent has taken 5 or 10, the agent can be assured of the mathematical fact that M returns the same action. (This is relevant in more complex decision problems, such as twin prisoner’s dilemma, where the agent’s utility depends on mathematical facts about what values different machines return)

Importantly, the agent can’t use MachineFive/MachineTen to know what action M takes before actually taking the action. Otherwise, the agent could take the opposite of the action they know they will take, causing a logical inconsistency. The above construction would not work if the machine were only run for a finite number of steps before being forced to return an answer; that would lead to the agent being able to know what action it will take, by running M for that finite number of steps.

This model naturally handles cases where M never halts; if the agent never executes either TakeFive or TakeTen, then it can never execute either MachineFive or MachineTen, and so cannot be assured of Ret(M, x) for any x; indeed, if the agent never takes any action, then Ret(M, x) isn’t true for any x, as that would imply that the agent eventually takes action x.

Interpreting the counterfactuals

At this point, it’s worth discussing the sense in which counterfactuals do or do not exist. Let’s first discuss the simpler case, where there is no assumption about source code.

First, from the perspective of the logic itself, only one of TakeFive or TakeTen may be evaluated. There cannot be both a fact of the matter about what happens if the agent takes five, and a fact of the matter about what happens if the agent takes ten. This is because even defining both facts at once requires re-using the Start proposition.

So, from the perspective of the logic, there aren’t counterfactuals; only one operation is actually run, and what “would have happened” if the other operation were run is undefinable.

On the other hand, there is an important sense in which the proof system contains counterfactuals. In constructing a linear logic proof, different choices may be made. Given “Start” as an assumption, I may prove “End ⊗ $5” by executing TakeFive, or “End ⊗ $10” by executing TakeTen, but not both.

Proof systems are, in general, systems of rules for constructing proofs, which leave quite a lot of freedom in which proofs are constructed. By the Curry-Howard isomorphism, the freedom in how the proofs are constructed corresponds to freedom in how the agent behaves in the real world; using TakeFive in a proof has the effect, if executed, of actually (irreversibly) taking the $5 bill.

So, we can say, by reasoning about the proof system, that if TakeFive is run, then $5 will be yielded, and if TakeTen is run, then $10 will be yielded, and only one of these may be run.

The logic itself says there can’t be a fact of the matter about both what happens if 5 is taken and if 10 is taken. On the other hand, the proof system says that both proofs that get $5 by taking 5, and proofs that get $10 by taking 10, are possible.

How to interpret this difference? One way is by asserting that the logic is about the territory, while the proof system is about the map; so, counterfactuals are represented in the map, even though the map itself asserts that there is only a singular territory.

And, importantly, the map doesn’t represent the entire territory; it’s a proof system for reasoning about the territory, not the territory itself. The map may, thus, be “looser” than the territory, allowing more possibilities than could possibly be actually realized.

What prevents the map from drawing out logical implications to the point where it becomes clear that only one action may possibly be taken? Given the second-try setup, the agent simply cannot use the fact of their source code being M, until actually taking the action; thus, no amount of drawing implications can conclude anything about the relationship between M and the agent’s action. In addition to this, reasoning about M itself becomes harder the longer M runs, i.e. the longer the agent is waiting to make the decision; so, simply reasoning about the map, without taking actions, need not conclude anything about which action will be taken, leaving both possibilities live until one is selected.

Conclusion

This approach aligns significantly with the less-formal descriptions given of subjective implication decision theory and counterfactual nonrealism. Counterfactuals aren’t real in the sense that they are definable after having taken the relevant action; rather, an agent in a state of uncertainty about which action it will take may consider multiple possibilities as freely selectable, even if they are assured that their selection will be equal to the output of some computer program.

The linear logic formalization increases my confidence in this approach, by providing a very precise notion of the sense in which the counterfactuals do and don’t exist, which would be hard to make precise without similar formalism.

I am, at this point, less worried about the problems with counterfactual nonrealism (such as global accounting) than I was when I wrote the post, and more worried about the problems of policy-dependent source code (which requires the environment to be an ensemble of deterministic universes, rather than a single one), such that I have updated towards counterfactual nonrealism as a result of this analysis, although I am still not confident.

Overall, I find linear logic quite promising for modeling embedded decision problems from the perspective of an embedded agent, as it builds critical facts such as non-reversibility into the logic itself.

Appendix: spurious counterfactuals

The following describes the problem of spurious counterfactuals in relation to the model.

Assume the second-try setup. Suppose the agent becomes assured that Ret(M, “Five”); that is, that M returns the action “Five”. From this, it is provable that the agent may, given Start, attain the linear logic proposition 0, by taking action “Ten” and then running MachineTen to get Ret(M, “Ten”), which yields inconsistency with Ret(M, “Five”). From 0, anything follows, e.g. $1000000, by the principle of explosion.

If the agent is maximizing guaranteed utility, then they will take the $10 bill, to be assured of the highest utility possible. So, it cannot be the case that the agent can be correctly assured that they will take action five, as that would lead to them taking a different action.

If, on the other hand, the agent would have provably taken the $5 bill upon receiving the assurance (say, because they notice that taking the $10 bill could result in the worst possible utility), then there is a potential issue with this assurance being a self-fulfilling prophecy. But, if the agent is constructing proofs (plans for action) so as to maximize guaranteed utility, this will not occur.

This solution is essentially the same as the one given in the paper on UDT with a known search order.

Topological metaphysics: relating point-set topology and locale theory

The following is an informal exposition of some mathematical concepts from Topology via Logic, with special attention to philosophical implications. Those seeking more technical detail should simply read the book.

There are, roughly, two ways of doing topology:

  • Point-set topology: Start with a set of points. Consider a topology as a set of subsets of these points which are “open”, where open sets must satisfy some laws.
  • Locale theory: Start with a set of opens (similar to propositions), which are closed under some logical operators (especially and and or), and satisfy logical relations.

What laws are satisfied?

  • For point-set topology: The empty set and the full set must both be open; finite intersections and infinite unions of opens must be open.
  • For local theory: “True” and “false” must be opens; the opens must be closed under finite “and” and infinite “or”; and some logical equivalences must be satisfied, such that “and” and “or” work as expected.

Roughly, open sets and opens both correspond to verifiable propositions. If X and Y are both verifiable, then both “X or Y” and “X and Y” are verifiable; and, indeed, even countably infinite disjunctions of verifiable statements are verifiable, by exhibiting the particular statement in the disjunction that is verified as true.

What’s the philosophical interpretation of the difference between point-set topology and locale theory, then?

  • Point-set topology corresponds to the theory of possible worlds. There is a “real state of affairs”, which can be partially known about. Open sets are “events” that are potentially observable (verifiable). Ontology comes before epistemology. Possible worlds are associated with classical logic and classical probability/utility theory.
  • Local theory corresponds to the theory of situation semantics. There are facts that are true in a particular situation, which have logical relations with each other. The first three lines of Wittgenstein’s Tracatus Logico-Philosophicus are: “The world is everything that is the case. / The world is the totality of facts, not of things. / The world is determined by the facts, and by these being all the facts.” Epistemology comes before ontology. Situation semantics is associated with intuitionist logic and Jeffrey-Bolker utility theory (recently discussed by Abram Demski).

Thus, they correspond to fairly different metaphysics. Can these different metaphysics be converted to each other?

  • Converting from point-set topology to locale theory is easy. The opens are, simply, the open sets; their logical relations (and/or) are determined by set operations (intersection/union). They automatically satisfy the required laws.
  • To convert from locale theory to point-set topology, construct possible worlds as sets of opens (which must be logically coherent, e.g. the set of opens can’t include “A and B” without including “A”), which are interpreted as the set of opens that are true of that possible world. The open sets of the topology correspond with the opens, as sets of possible words which contain the open.

From assumptions about possible worlds and possible observations of it, it is possible to derive a logic of observations; from assumptions about the logical relations of different propositions, it is possible to consider a set of possible worlds and interpretations of the propositions as world-properties.

Metaphysically, we can consider point-set topology as ontology-first, and locale theory as epistemology-first. Point-set topology starts with possible worlds, corresponding to Kantian noumena; locale theory starts with verifiable propositions, corresponding to Kantian phenomena.

While the interpretation of a given point-set topology as a locale is trivial, the interpretation of a locale theory as a point-set topology is less so. What this construction yields is a way of getting from observations to possible worlds. From the set of things that can be known (and knowable logical relations between these knowables), it is possible to conjecture a consistent set of possible worlds and ways those knowables relate to the possible worlds.

Of course, the true possible worlds may be finer-grained than these consistent set; however, it cannot be coarser-grained, or else the same possible world would result in different observations. No finer potentially-observable (verifiable or falsifiable) distinctions may be made between possible worlds than the ones yielded by this transformation; making finer distinctions risks positing unreferenceable entities in a self-defeating manner.

How much extra ontological reach does this transformation yield? If the locale has a countable basis, then the point-set topology may have an uncountable point-set (specifically, of the same cardinality as the reals). The continuous can, then, be constructed from the discrete, as the underlying continuous state of affairs that could generate any given possibly-infinite set of discrete observations.

In particular, the reals may be constructed from a locale based on open intervals whose beginning/end are rational numbers. That is: a real r may be represented as a set of (a, b) pairs where a and b are rational, and a < r < b. The locale whose basis is rational-delimited open intervals (whose elements are countable unions of such open intervals, and which specifies logical relationships between them, e.g. conjunction) yields the point-set topology of the reals. (Note that, although including all countable unions of basis elements would make the locale uncountable, it is possible to weaken the notion of locale to only require unions of recursively enumerable sets, which preserves countability)

If metaphysics may be defined as the general framework bridging between ontology and epistemology, then the conversions discussed provide a metaphysics: a way of relating that-which-could-be to that-which-can-be-known.

I think this relationship is quite interesting and clarifying. I find it useful in my own present philosophical project, in terms of relating subject-centered epistemology to possible centered worlds. Ontology can reach further than epistemology, and topology provides mathematical frameworks for modeling this.

That this construction yields continuous from discrete is an added bonus, which should be quite helpful in clarifying the relation between the mental and physical. Mental phenomena must be at least partially discrete for logical epistemology to be applicable; meanwhile, physical theories including Newtonian mechanics and standard quantum theory posit that physical reality is continuous, consisting of particle positions or a wave function. Thus, relating discrete epistemology to continuous ontology is directly relevant to philosophy of science and theory of mind.