A better model of computation for digital physics?
aa r X i v : . [ c s . L O ] F e b DRAFT September 13, 2018
A better model of computation for digital physics?
Anton Salikhmetov [email protected]
This note is meant to invite the reader to consider interaction nets, a relatively recently discoveredmodel of computation, as a possible alternative for cellular automata which are often employed asthe basis for digital physics. Defined as graph-like structures (in contrast to the grids for cellularautomata), interaction nets possess a set of interesting properties, such as locality, linearity, andstrong confluence, which together result in so-called clockless computation in the sense that they donot require any global clock in order to operate. We believe that an attempt of using interaction netsas a replacement for cellular automata may lead to a new view in digital physics.
The idea of representing the world as a process of executing a digital computer program seems to havebeen circulating since about 1960s. In recent works [3] cellular automata appear to have been the mostwidely used (if not the only one) model of computation put forward as the basis for modeling such adigital computer. Traditionally, the grid of a cellular automaton is seen as representation of space, andthe sequence of its states as representation of time.In computer science, cellular automata is one member in quite a big array of models of computation.Models of computation are equivalent in the sense that any solvable problem can be solved within thosemodels. However, properties which models possess can be very different. In this note we would like toadvertise interaction nets which we believe might possibly lead to a new view in digital physics.Interaction nets are one of graphical models of computation based on the notion of “computation asinteraction”. They were devised by Yves Lafont [4] as a generalisation of the proof structures of linearlogic. This model of computation benefits from the following properties: • locality in the sense that only two adjacent nodes can interact at a time, and each such step ofcomputation is completely independent from the rest of a net; • linearity in the sense that each step of computation can be done in constant time, therefore totalcomputation time is linear on the number of steps; • strong confluence in the sense that the order in which the steps of computation are performed doesnot influence the process of computation.Let us make some remarks regarding representation of space and time with interaction nets. First, asarbitrary graph-like structures, interaction nets are inherently three-dimensional, unlike cellular automatawhere one has to assign dimensionality to the system, perhaps artificially. Second, while causality doestake place in interaction nets just like for cellular automata, the notion of a global state is rather irrelevantdue to strong confluence, so computing with interaction nets is essentially clockless.The next section first gives a short overview of interaction nets and then discusses their propertiesmore formally among some other aspects. For a more thorough introduction as well as examples of howexactly interaction nets can perform actual computations, we urge the reader to follow [1]. Abetter model ofcomputation for digital physics?Figure 1: Primitives (a) Agent α x x ... x n (b) Wiring x ··· x k ω (c) Tree t ··· ≡ α ··· t t n ··· ··· ··· or tx This section gives a rather informal brief introduction to interaction nets and their textual representationcalled the interaction calculus [2]. Here, we intentionally omit the notion of interface which is specificto some applications but is not crucial for this model of computation.Interaction nets are graph-like structures consisting of primitives shown in Figure 1.
Agents of type α can be graphically represented as shown in Figure 1a. Agents have arity ar ( α ) ≥
0. If ar ( α ) = n , theagent α has n auxiliary ports x , . . . , x n in addition to its principal port x . All agent types belong to aset Σ called signature . Any port must be connected to exactly one edge. Wiring ω on Figure 1b consistssolely of edges. Inductively defined trees on Figure 1c correspond to terms t :: = α ( t , . . . , t n ) | x in theinteraction calculus, where x is called a name .Any net N can be redrawn using the previously defined wiring and tree primitives as follows: N ≡ v w ω ... v n w n ...... ...... which in the interaction calculus corresponds to a configuration h v = w , . . . , v n = w n i which is anunordered multiset of equations v i = w i , where v i and w i are arbitrary terms. The wiring ω translates tonames, and each name has to occur exactly twice in a configuration.For configurations, so-called α -conversion is defined as follows: both occurrences of any name canbe replaced with any new name if the latter does not occur in a given configuration. Configurations areconsidered equal up to α -conversion..Salikhmetov 3When two agents are connected to each other with their principal ports, they form an active pair . Foractive pairs one can introduce interaction rules which describe how the active pair rewrites to anothernet. Graphically, any interaction rule can be represented as follows: α β x ... x m y ... y n → x y n ... N ... x m y ≡ v w n ... ω ... v m w ...... ...... where α , β ∈ Σ , and the net N is redrawn using primitives of wirings and trees in order to translate therule into the interaction calculus as α [ v , . . . , v m ] ⊲⊳ β [ w , . . . , w n ] using Lafont’s notation. A net with noactive pairs is said to be in normal form . A signature Σ (with mapping ar defined on it) along with a setof interaction rules defined for agents α ∈ Σ together constitute an interaction system .Now, let us consider an example for the notions introduced above in this section. Figure 2 shows twointeraction rules for commonly used agents ε and δ and a simple interaction net to which these interactionrules are applied. Using Lafont’s notation, the erasing rule from Figure 2a is written as ε ⊲⊳ α [ ε , . . . , ε ] ,while the duplication rule given in Figure 2b can be represented as follows: δ [ α ( x , . . . , x n ) , α ( y , . . . , y n )] ⊲⊳ α [ δ ( x , y ) , . . . , δ ( x n , y n )] . Figure 2c provides an example of a non-terminating net which reduces to itself. In terms of the interactioncalculus, one can write this net as a configuration h δ ( ε , x ) = γ ( x , ε ) i .The interaction calculus defines reduction on configurations in more details than seen from graphrewriting defined on interaction nets. Namely, if α [ v , . . . , v m ] ⊲⊳ β [ w , . . . , w n ] , the following reduction: h α ( t , . . . , t m ) = β ( u , . . . , u n ) , . . . i → h t = v , . . . , t m = v m , u = w , . . . , u n = w n , . . . i is called interaction . For equations of the form x = u indirection can be applied resulting in substitutiont [ x : = u ] defined as the result of replacing the other occurrence of the name x in term t with term u : h x = u , t = w , . . . i → h t [ x : = u ] = w , . . . i . Figure 2: Example (a) Erasing εα ... → ε ... ε (b) Duplication δα ... → α αδ ... δ ... ... (c) Non-termination γ γδ δεε ý ∗ Abetter model ofcomputation for digital physics?An equation t = x is called a deadlock if the name x has occurrence in the term t . Together, interactionand indirection define the reduction relation on configurations.Coming back to the example of a non-terminating net shown in Figure 2c, the infinite reductionsequence starting from the corresponding configuration in the interaction calculus is as follows: h δ ( ε , x ) = γ ( x , ε ) i →h ε = γ ( x , x ) , x = γ ( y , y ) , x = δ ( x , y ) , ε = δ ( x , y ) i → ∗ h x = ε , x = ε , x = γ ( y , y ) , x = δ ( x , y ) , x = ε , y = ε i → ∗ h δ ( ε , x ) = γ ( x , ε ) i → . . . Now, let us describe the properties of interaction nets more formally: • locality means that only active pairs can be rewritten; • linearity means that each interaction rule can be applied in constant time; • strong confluence also known as one-step diamond property means that if c → c and c → c , then c → c ′ and c → c ′ for some c ′ .Perhaps the simplest universal interaction system is that of interaction combinators [5]. It is definedby the signature Σ = { ε , δ , γ } with annihilation rules γ [ x , y ] ⊲⊳ γ [ y , x ] and δ [ x , y ] ⊲⊳ δ [ x , y ] in additionto the erasing and duplication rules shown above. The interaction system of interaction combinators isTuring-complete and can simulate any other interaction system or another model of computation.However, please note that signatures and sets of interaction rules are not required to be small or evenfinite. In fact, in many applications of interaction nets infinite sets of agent types and interaction rules areactually preferred as they often allow to represent desired structures in a much more natural and efficientway than using interaction combinators. In this note we briefly presented interaction nets, which we believe may be a worthwhile replacement forcellular automata to consider in digital physics. It would be interesting to see the consequences of usingclockless computation with graph-like structures as the basis for discrete, deterministic characterisationof the physical world, in particular with respect to the phenomenon of quantum entanglement.
References [1] Maribel Fern´andez (2009):
Interaction-Based Models of Computation . In: Models of Computation: AnIntroductiontoComputabilityTheory, chapter 7, Springer Science & Business Media, pp. 107–130.[2] Maribel Fern´andez & Ian Mackie (1999):
A calculus for interaction nets . In: Principles and Practice ofDeclarativeProgramming, Springer, pp. 170–187, doi:10.1007/10704567.[3] Gerard ’t Hooft (2016):
The Cellular Automaton Interpretation of Quantum Mechanics . Fundamental Theoriesof Physics, Springer International Publishing.[4] Yves Lafont (1990):
Interaction nets . In: Proceedingsof the 17th ACM SIGPLAN-SIGACT symposium onPrinciplesofprogramminglanguages, ACM, pp. 95–108, doi:10.1145/96709.96718.[5] Yves Lafont (1997):