Phi Lab. Why Integrated Information Gives Feed-Forward Networks Zero

Integrated information theory says that consciousness is integrated information, and it gives a number for it, big phi. One of its sharpest predictions is about wiring. A network where signals only flow forward, with no loop, has a phi of zero, however complex its behaviour. This console lets you wire a small network yourself and see that prediction hold.

Each value here was computed by PyPhi, the reference software for the theory, run on our build server. The browser does not compute phi. It looks up the value for the wiring and the state on screen.

What the console computes

Three nodes, A, B and C, are logic gates: A is an OR, B is an AND and C is an XOR. Each node reads the nodes that connect to it and switches on or off at the next step. A node with no input is off. Six connections are possible between the three nodes, so there are 64 wirings.

For every wiring and every state it can be in, PyPhi computes big phi as integrated information theory 3.0 defines it. Phi asks how much the system’s causes and effects on itself are lost when the system is cut into two parts, for the cut that loses the least. The console also shows that cut, and the concepts, the parts of the system that have causes and effects of their own.

• Chapter 1, Lab. The network runs by its own rule, one step a second, and phi follows the state. Phi depends on the state as well as the wiring.
• Chapter 2, All 64. Each wiring at its highest phi over its states.
• Chapter 3, Four nodes. Four majority gates connected all to all, against the same four gates connected only forward.

What it finds

• Every feed-forward wiring scores zero. Of the 64 wirings, some have no loop, and the highest phi among them is zero.
• A loop is needed but is not enough. Of the other wirings, each with a loop, only some reach a phi above zero in some state. The highest is shown in the console.
• Four nodes, the same rule. All to all, the majority gates reach their highest value when all four agree, all off or all on, and less while they swing between two states. Forward only, they reach zero in every state, and their activity dies out within a few steps.

The reason is in the cut. In a network without a loop, the nodes can be put in an order where every connection points forward. Cut that order in two, and every connection runs from the first part to the second. Removing the connections the other way removes nothing, because there are none, so the system loses nothing and phi is zero.

What the console shows and does not show

• The theory’s verdict, not a test of it. Phi is zero for feed-forward networks by the theory’s own definition. The console shows what the definition does. It does not show that the theory is right.
• A known objection. A feed-forward network can be built to give the same outputs as a network with loops for any input. The theory gives the first zero and the second more. Doerig and colleagues called this the unfolding argument: if the two behave the same, no experiment on behaviour can tell which is right.
• Small networks. Phi grows very expensive with the number of nodes, which is why the lab has three and the last chapter four.

Why this belongs on this site

This project studies consciousness as an emergent property of physical systems and keeps integrated information among its measures. It computes phi with PyPhi on the states it records from its own agent. Its pre-registered test, that phi would rise and fall with the agent’s binding signal, failed and is closed. This console shows the measure itself, on networks small enough to see whole.

Sources

• Oizumi, M., Albantakis, L. and Tononi, G. (2014). From the phenomenology to the mechanisms of consciousness: integrated information theory 3.0. PLOS Computational Biology 10(5), e1003588. doi:10.1371/journal.pcbi.1003588
• Mayner, W. G. P., Marshall, W., Albantakis, L., Findlay, G., Marchman, R. and Tononi, G. (2018). PyPhi, a toolbox for integrated information theory. PLOS Computational Biology 14(7), e1006343. doi:10.1371/journal.pcbi.1006343. PyPhi 1.2.0, licence GPL-3.0, was run as a tool. None of its code is in this site.
• Doerig, A., Schurger, A., Hess, K. and Herzog, M. H. (2019). The unfolding argument. Why IIT and other causal structure theories cannot explain consciousness. Consciousness and Cognition 72, 49 to 59. doi:10.1016/j.concog.2019.04.002
• The project’s phi verdict: phi1_phaseBalt_2026_05_24