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Many researchers argue that logics and connectionist systems complement each other nicely. Logics are an expressive formalism for describing knowledge, they expose the common form across a class of content, they often come with pleasant meta-properties (e.g. soundness and completeness), and logic-based learning makes excellent use of knowledge. Connectionist systems are good for data driven learning and they’re fault tolerant, also some would argue that they’re a good candidate for tip-toe-towards-the-brain cognitive models. I thought I’d give d’Avila Garcez and Lamb (2006) a go [A Connectionist Computational Model for Epistemic and Temporal Reasoning, Neural Computation 18:7, 1711-1738].

I’m assuming you know a bit of propositional logic and set theory.

The modal logic bit

There are many modal logics which have properties in common, for instance provability logics, logics of tense, deontic logics. I’ll follow the exposition in the paper. The gist is: take all the usual propositional logic connectives and add the operators □ and ◊. As a first approximation, □P (“box P”) means “it’s necessary that P” and ◊P (“diamond P”) means “it’s possible that P”. Kripke models are used to characterise when a model logic sentence is true. A model, M, is a triple (Ω, R, v), where:

• Ω is a set of possible worlds.
• R is a binary relation on Ω, which can be thought of as describing connectivity between possible worlds, so if R(ω,ω’) then world ω’ is reachable from ω. Viewed temporally, the interpretation could be that ω’ comes after ω.
• v is a lookup table, so v(p), for an atom p, returns the set of worlds where p is true.

Let’s start with an easy rule:

(M, ω) ⊨ p iff ω ∈ v(p), for a propositional atom p

This says that to check whether p is true in ω, you just look it up. Now a recursive rule:

(M, ω) ⊨ A & B iff (M, ω) ⊨ A and (M, ω) ⊨ B

This lifts “&” up to our natural language (classical logic interpretation thereof) notion of “and”, and recurses on A and B. There are similar rules for disjunction and implication. The more interesting rules:

(M, ω) ⊨ □A iff for all ω’ ∈ Ω such that R(ω,ω’), (M, ω’) ⊨ A

(M, ω) ⊨ ◊A iff there is an ω’ ∈ Ω such that R(ω,ω’) and (M, ω’) ⊨ A

The first says that A is necessarily true in world ω if it’s true for all connected worlds. The second says that A is possibly true if there is at least one connected world for which it is true.

A sketch of logic programs and a connectionist implementation

Logic programs are sets of Horn clauses, A1 & A2 & … & An → B, where Ai is a propositional atom or the negation of an atom. Below is a picture of the network that represents the program {B & C & ~D → A, E & F → A, B}.

The thresholds are configured so that the units in the hidden layer, Ni, are only active when the antecedents are all true, e.g. N1 is only active when B, C, and ~D have the truth value true. The thresholds of the output layer’s units are only active when at least one of the hidden layer connections to them is active. Additionally, the output feeds back to the inputs. The networks do valuation calculations through the magic of backpropagation, but can’t infer new sentences as such, as far as I can tell. To do so would involve growing new nets and some mechanism outside the net interpreting what the new bits mean.

Aside on biological plausibility

Biological plausibility raises its head here. Do the units in this network model – in any way at all – individual neurons in the brain? My gut instinct says, “Absolutely no way”, but perhaps it would be better not even to think this as (a) the units in the model aren’t intended to characterise biological neurons and (b) we can’t test this particular hypothesis. Mike Page has written in favour of localists nets, of which this is an instance [Behavioral and Brain Sciences (2000), 23: 443-467]. Maybe more on that in another post.

Moving to modal logic programs and nets

Modal logic programs are like the vanilla kind, but the literals may have one of the modal operators. There is also a set of connections between the possible worlds, i.e. a specification of the relation, R. The central idea of the translation is to use one network to represent each possible world and then apply an algorithm to wire up the different networks correctly, giving one unified network. Take the following program: {ω1 : r → □q, ω1 : ◊s → r, ω2 : s, ω3 : q → ◊p, R(ω1,ω2), R(ω1,ω3)}. This wires up to:

Each input and output neuron can now represent □A, ◊A, A, □~A, ◊~A, or ~A. The individual networks are connected to maintain the properties of the modality operators, for instance □q in ω1 connects to q in ω2 and ω3 since R(ω1, ω2), R(ω1, ω3), so q must be true in these worlds.

The Connectionist Temporal Logic of Knowledge

Much the same as before, except we now have a set of agents, A = {1, …, n}, and a timeline, T, which is the set of naturals, each of which is a possible world but with a temporal interpretation. Take a model M = (T, R1, …, Rn, π). Ri specifies what bits of the timeline agent i has access to, and π(t) gives a set of propositions that are true at time t.

Recall the following definition from before

(M, ω) ⊨ p iff ω ∈ v(p), for a propositional letter p

Its analogue in the temporal logic is

(M, t) ⊨ p iff t ∈ π(p), for a propositional letter p

There are two extra model operators: O, which intuitively means “at the next time step” and K which is the same as □, except for agents. More formally:

(M, t) ⊨ OA iff (M, t+1) ⊨ A

(M, t) ⊨ KA iff for all u ∈ T such that Ri(t,u), (M, u) ⊨ A

Now in the translation we have network for each agent, and a collection of agent networks for each time step, all wired up appropriately.

Pages 1724-1727 give the algorithms for net construction. The proof of soundness of translation relies on d’Aliva Garcez, Broda, and Gabbay (2002), Neural-symbolic learning systems: Foundations and applications.

Some questions I haven’t got around to working out the answers to

• How can these nets be embedded in a static population coded network. Is there any advantage to doing so?
• Where is the learning? In a sense it’s the bit that does the computation, but it doesn’t correspond to the usual notion of “learning”.
• How can the construction of a network be related to what’s going on in the brain? Really I want a more concrete answer to how this could model the psychology. The authors don’t appear to care, in this paper anyway.
• How can networks shrink again?
• How can we infer new sentences from the networks?

Comments

I received the following helpful comments from one of the authors, Artur d’Avila Garcez (9 Aug 2006):

I am interested in the localist v distributed discussion and in the issue of biological plausibility; it’s not that we don’t care, but I guess you’re right to say that we don’t “in this paper anyway”. In this paper – and in our previous work – what we do is to say: take standard ANNs (typically the ones you can apply Backpropagation to). What logics can you represent in such ANNs? In this way, learning is a bonus as representation should precede learning.

The above answers you question re. learning. Learning is not the computation, that’s the reasoning part! Learning is the process of changing the connections (initially set by the logic) progressively, according to some set of examples (cases). For this you can apply Backprop to each network in the ensemble. The result is a different set of weights and therefore a different set of rules – after learning if you go back to the computation you should get different results.