Combinatory Chemistry: Towards a Simple Model of Emergent Evolution Germn Kruszewski1∗ and Tomas Mikolov2 1Naver Labs Europe, Grenoble, France 2CIIRC CTU, Prague, Czech Republic [email protected] Abstract A r An explanatory model for the emergence of evolvable units AA r1 IA((S(SK)I)A) 2 A((S(SK)I)A must display emerging structures that (1) preserve themselves in time (2) self-reproduce and (3) tolerate a certain amount of S I variation when reproducing. To tackle this challenge, here we A introduce Combinatory Chemistry, an Algorithmic Artificial Chemistry based on a minimalistic computational paradigm r3 A(SKA(IA)) r4 A(SKAA) named Combinatory Logic. The dynamics of this system comprise very few rules, it is initialized with an elementary S I tabula rasa state, and features conservation laws replicating natural resource constraints. Our experiments show that a sin- A gle run of this dynamical system with no external intervention r AA discovers a wide range of emergent patterns. All these struc- r5 A(KA(AA)) 6 tures rely on acquiring basic constituents from the environ- S ment and decomposing them in a process that is remarkably K AA similar to biological metabolisms. These patterns include au- topoietic structures that maintain their organisation, recursive Figure 1: Metabolic cycle (showing one of the possible path- ones that grow in linear chains or binary-branching trees, and most notably, patterns able to reproduce themselves, dupli- ways) of a self-reproducing structure that emerges from the cating their number at each generation. dynamics of Combinatory Chemistry. Starting from (AA), where A = (SI(S(SK)I)), it acquires three copies of A from its environment and uses two to create a copy of itself, Introduction metabolising the third one to carry out the process. Finding the minimal set of conditions that lead to open- ended evolution in a complex system is a central question in Artificial Life and a fundamental question of science in Artificial Chemistries seek to mimic as closely as possi- general. One prominent hypothesis in this line of research ble the properties of the chemistry that gave rise to life on is that living systems emerge from the complex interaction Earth (Flamm et al., 2010; Hogerl,¨ 2010; Young and Ne- of simple components. Environments like Avida (Ofria and shatian, 2013), others abstract away from the particulari- Wilke, 2004) or Tierra (Ray, 1991) have been used to ex- ties of natural chemistries to focus only on their hypoth- plore this question by allowing self-reproducing programs esized core computational properties (Fontana and Buss, to mutate and evolve in time. Yet, the reproductive and arXiv:2003.07916v2 [nlin.AO] 19 Jun 2020 1994; di Fenizio and Banzhaf, 2000; Tominaga et al., 2007; mutation mechanisms, as well as the organisms’ capacity Buliga and Kauffman, 2014). In line with this latter line to tolerate such mutations were fixed by design. Instead, of work, in this paper we introduce an Algorithmic Artifi- Artificial Chemistries try to uncover how such evolvable cial Chemistry based on Combinatory Logic (Schonfinkel,¨ units emerge in the first place by simulating the proper- 1924; Curry et al., 1958) featuring a minimalistic design and ties of natural chemical systems at different levels of ab- three key properties. First, it is Turing-complete, enabling it straction (see Dittrich et al. (2001) for a thorough review). to express an arbitrary degree of complexity. Second, it is The driving hypothesis is that complex organizations emerge strongly constructive (Fontana et al., 1993), meaning that as thanks to self-organising attractors in chemical networks, the system evolves in time it can create new components that which preserve their structure in time (Walker and Ashby, can in turn modify its global dynamics. Third, it features 1966; Wuensche et al., 1992; Kauffman, 1993). While some intrinsic conservation laws so that the number of atomic el- ∗ Work done while the author was at Facebook AI. ements remains always constant. Previous work has relied on applying extrinsic conserva- this system, which they called level 0 organisations. Further- tion laws, such as for instance keeping a maximum num- more, when these expressions were explicitly prohibited, a ber of total elements in the system by randomly removing more complex organization emerged where every expression exceeding ones (Fontana and Buss, 1994; di Fenizio and in a set was computed by other expressions within the same Banzhaf, 2000). Instead, intrinsic conservation laws allow set (level 1 organisations). Finally, mixing level 1 organisa- us to bound the total number of elements without introduc- tions could lead to higher order interactions between them ing extraneous perturbations. Furthermore, limiting the total (level 2 organisations). Yet, this system had some limita- amount of basic elements can create selective pressures be- tions. First, each level of organisation was only reached af- tween emergent structures. ter external interventions. Also, programs must necessarily We simulate a Chemical Reaction System (Hordijk et al., reach a normal form, which happens when there are no more 2015; Dittrich et al., 2001) based on Combinatory Logic, λ-calculus rules than can be applied. Thus, recursive pro- which starting from a tabula rasa state consisting of only grams, which never reach a normal form, are banned from elementary components, it produces a diversity explosion the system. Furthermore, two processes where introduced as that develops into a state dominated by self-organized emer- analogues of food and waste, respectively. First, when ex- gent structures, including autopoietic (Varela and Maturana, pressions are combined, they are not removed from the sys- 1973), recursive and self-replicating ones. Notably, all these tem, allowing the system to temporarily grow in size. Sec- types of structures emerge at different points in time dur- ond, expressions which after being combined with existing ing a single run of the system without requiring any external expressions do not match any λ-calculus reduction rules are interventions. Furthermore, these structures preserve them- removed. Without these processes, complex organisations selves by absorbing compounds from their environment and fail to emerge. Yet, it is not clear under which circumstances decomposing them step-by-step, in a process that has a strik- these external interventions would not be needed anymore ing resemblance with the metabolism of biological organ- in order for the system to evolve autonomously. Finally, isms. Finally, we introduce a heuristic to emulate the effects bounding the total number of expressions by randomly re- of having larger systems without having to compute them moving excess ones creates perturbations to the system that explicitly. This makes considerably more efficient the search can arbitrarily affect the dynamics. Fontana and Buss (1996) for these complex structures. later proposed MC2, a chemistry based on Linear Logic that The paper is organized as follows. First, we describe ear- addressed some of these limitations (notably, conservation lier work in Artificial Chemistry that is most related to our of mass), but we are not aware of empirical work on it. approach. Then, we explain the basic workings of Combi- Here, we propose an AC based on Combinatory Logic. natory Logic and how we adapted it into an Artificial Chem- This formalism has been explored before in the context of istry. Third, following earlier work, we discuss how autoca- AC by di Fenizio and Banzhaf (2000). While this work talytic sets can be used to detect emerging phenomena in this shares with us the enforcement of conservation laws, it relies system, and propose a novel measure of emergent complex- for it on a normalisation process that introduces noise into ity, which is well-adapted to the introduced system. Finally, the system dynamics. Furthermore, as AlChemy, it reduces we describe our experiments that showcase the emergence expressions until they reach their normal forms, explicitly of complex structures in Combinatory Chemistry. forbidding recursive and other type of expressions that do not converge. Artificial Chemistries Finally, Chemlambda (Buliga and Kauffman, 2014) is a Artificial Chemistries (AC) are models inspired in natural Turing-complete graph rewriting AC that allows the encod- chemical systems that are usually defined by three differ- ing of λ-calculus and combinatory logic operators. As such, ent components: a set of possible molecules, a set of reac- it is complementary in many ways with the system proposed tions, and a reactor algorithm describing the reaction vessel here. Yet, we are not aware of conservation laws defined and how molecules interact with each other (Dittrich et al., within this formalism, nor of any reactor algorithm allowing 2001). In the following discussion we will focus on algo- explorations of emerging phenomena. rithmic chemistries that are the closest to the present work. AlChemy (Fontana and Buss, 1994) is an AC where Combinatory Logic molecules are given by λ-calculus expressions. λ-calculus Combinatory Logic (CL) is a minimalistic computa- is a mathematical formalism that, like Turing machines, can tional system that was independently invented by Moses describe any computable function. In AlChemy, pairs of ran- Schonfinkel,¨ John Von Neumann and Haskell Curry (Car- domly sampled expressions are joined through function ap- done and Hindley, 2006). Other than its relevance to com- plication, and the corresponding result is added back to the putability theory, it has also been applied in Cognitive Sci- population. To keep the population size bounded, expres- ence as a model for a Language of Thought (Piantadosi, sions are randomly discarded. Fontana and Buss showed that 2016). One of the main advantages of CL is its formal expressions that computed themselves quickly emerged in simplicity while capturing Turing-complete expressiveness. In contrast to other mathematical formalisms, such as λ- the K combinator discards a part of the expression (the ar- calculus, it dispenses with the notion of variables and all gument g), S duplicates its third argument x.