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An agent-based model of the emergence and evolution of a language system for boolean coordination

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Abstract

This paper presents an agent-based model of the emergence and evolution of a language system for Boolean coordination. The model assumes the agents have cognitive capacities for invention, adoption, abstraction, repair and adaptation, a common lexicon for basic concepts, and the ability to construct complex concepts using recursive combinations of basic concepts and logical operations such as negation, conjunction or disjunction. It also supposes the agents initially have neither a lexicon for logical operations nor the ability to express logical combinations of basic concepts through language. The results of the experiments we have performed show that a language system for Boolean coordination emerges as a result of a process of self-organisation of the agents’ linguistic interactions when these agents adapt their preferences for vocabulary, syntactic categories and word order to those they observe are used more often by other agents. Such a language system allows the unambiguous communication of higher-order logic terms representing logical combinations of basic properties with non-trivial recursive structure, and it can be reliably transmitted across generations according to the results of our experiments. Furthermore, the conceptual and linguistic systems, and simplification and repair operations of the agent-based model proposed are more general than those defined in previous works, because they not only allow the simulation of the emergence and evolution of a language system for the Boolean coordination of basic properties, but also for the Boolean coordination of higher-order logic terms of any Boolean type which can represent the meaning of nouns, sentences, verbs, adjectives, adverbs, prepositions, prepositional phrases and subexpressions not traditionally analysed as forming constituents, using linguistic devices such as syntactic categories, word order and function words.

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Notes

  1. \(\text{ Bool }\) is the type of Boolean values and \(\text{ Ind }\) the type of individuals. See Sect. 2.1.1 for a definition of symple types in the \(\lambda \)-Calculus.

  2. \(\text{ up }\) and \(\text{ le }\) are constants of type \(\text{ Ind }\rightarrow \text{ Bool }\) denoting the properties of being in an upper and left position respectively. If the agents used First Order Logic instead of Higher Order Logic to represent their meanings, they would approximate the meaning of this noun phrase by using formula \((\text{ up }(x) \vee \text{ le }(x)) \wedge \lnot (\text{ up }(x) \wedge \text{ le }(x))\).

  3. When we omit parentheses from functional types, we assume associativity is applied to the right, so that \(\text{ Ind }\rightarrow \text{ Ind }\rightarrow \text{ Bool }\) is equivalent to \(\text{ Ind }\rightarrow (\text{ Ind }\rightarrow \text{ Bool })\).

  4. Assuming square and upper have the standard natural language interpretations, \(square(o_1)\) will be true if \(o_1\) denotes an object that is a square, upper(square) will denote the property of being a square situated in an upper position, and \(upper(square(o_1))\) will be true if \(o_1\) is a square which is in an upper position.

  5. Treating move as a binary relation means interpreting \(move(x,r_1)\) as ‘\(r_1\) moves x’. Currying move implies that move(x) means ‘moves x’, and \(move(x)(r_1)\) means ‘\(r_1\) moves x’.

  6. Generalized quantifiers, e.g. \(\mathtt{every}_{\tau }\) or \(\mathtt{some}_{\tau }\), quantify over objects of type \(\tau \).

  7. See definition 6 on page 9 for formal definitions of lexicon and lexical entry.

  8. See Sect. 3.1 for a formal definition of the set of syntactic categories used in this paper.

  9. We will omit parentheses within categories, taking the forward slash to be left associative, the backward slash to be right associative, and the backward slash bind more tightly than the forward slash. Thus \((\text{ np }\backslash \text{ s })/ \text{ np }\) is equivalent to \(\text{ np }\backslash \text{ s }/ \text{ np }\).

  10. Lambek extended pure applicative categorial grammar according to a simple algebraic interpretation of the slashes. In the Lambek Calculus an expression is assigned to category \(A\slash B\) (respectively, to category \(B\backslash A\)) if and only if when it is followed (respectively, preceded) by an expression of category B, it produces an expression of category A. However, applicative categorial grammar only respects one half of the biconditional.

  11. Syntactic categories pr / pr or pr\(\backslash \)pr are used to place the word associated with not before or after the expression associated with its argument. Categories pr / pr / pr, pr\(\backslash \)pr / pr and pr\(\backslash \)pr\(\backslash \)pr place the words associated with and,or before, between or after the expressions associated with their arguments.

  12. Affixes are groups of letters attached to words.

  13. A conceptualisation of a subset of objects (i.e. a meaning) is a higher-order logic term that is true for all the objects in the subset and false for the rest of the objects in the speaker’s and hearer’s context.

  14. If  F is a constant of type \(\mathtt{\text{ Ind }\rightarrow \text{ Bool }}\), i.e. a basic property, invention is not necessary, because an entry for F already exists in the common lexicon.

  15. The reason for this is that the lexical entries for basic properties are part of the common lexicon of basic concepts initially shared by all the agents in the population.

  16. It should be noted that although the choice of words for expressing different Boolean operators is independent of each other, the choice of a lexical entry for expressing a particular operator in a given sentence is always the same for all the occurrences of such an operator in the higher-order term expressed by such a sentence. Likewise, the choice of a lexical entry for parsing a particular word in a given sentence is always consistent for all the occurrences of such word in the sentence.

  17. In particular, the Ciao Prolog System [10], available from www.clip.dia.fi.upm.es, has been used.

  18. The same arguments as those used in Sect. 4.1 can be used to justify the method used to compute the score of a particular meaning obtained by parsing a given expression.

  19. In the experiments, the syntactic category C of the expressions invented by the agents is pr.

  20. In the experiments, the agents do not try to simplify associations between expressions and more complex meanings, except in the case of learning the expressions used by other agents to refer to the grouping operator id, which we discuss in the next subsection in the context of repair operations.

  21. On the other hand, if a repair operation had been applied to a sentence that contained the logical operator ’or’ before any repair operation had been applied to a sentence that contained the logical operator ’and’, lexical entry 10 would have been replaced with lexical entry 14 in step 1, and lexical entry 13 would have been added to the agent’s lexicon in step 2. Later on, if the agent applies a repair operation to a sentence containing ’and’, it would replace lexical entry 9 with lexical entry 12 and step 2 would not be applied.

  22. We describe repair operations for arbitrary syntactic categories c, because they are applicable to lexical entries for coordinators of expressions of any Boolean type. However, in the experiments, c is always pr.

  23. They could have been triggered by the generation of ambiguous expressions for other types of meanings, e.g. \(\mathtt{and(\alpha )(not(\beta ))}\) or \(\mathtt{and(\alpha )(or(\beta )(\gamma ))}\). But in the present experiments this is not the case.

  24. We use again an arbitrary syntactic category c to describe the adoption of lexical entries constructed during the application of repair operations, because the mechanisms proposed are applicable to lexical entries for coordinators of expressions of any Boolean type. However, in the experiments, c is always pr.

  25. \(\mu \) is the initial score agents assign to the lexical entries they create during simplification or repair.

  26. Adaptation captures the speed at which an agent can adapt to a new situation, which determines the time it takes for a population to agree on a common language. Amplification relates to the extent to which an agent is able to escape from a behaviour which represents a suboptimal fixed point of its response function. This is achieved by amplifying small deviations from suboptimal equilibrium states, thus making them unstable.

  27. Basic properties are propositional constants representing properties such as being light, dark, in an upper or lower position, or on the right or left side of the whiteboard. [8, 20, 22] also use simulations with software agents and initialise the agents with a common vocabulary for basic concepts.

  28. The length of the sliding window used in the experiments described in [41, 42] was of ten games, and that used in the experiments described in [8] of fifty games.

  29. The results described in [41, 42] are the average of ten simulation runs, whereas those reported in [8] are the average of fifty simulation runs.

  30. Note that null lexical variability means that all the agents in the population prefer the same expression (both word and position) for naming the Boolean operators (and,or,not) and the grouping operator (id).

  31. In [18], the number of words invented per object in the naming game is estimated to be in O(n), where n is the population size, and the time for n / 2 words to spread in the population in \(O(n^2 \cdot \ln (n))\). The asymptotic notation O(g(n)) is defined as follows. For a given function g(n), we denote by O(g(n)) the set of functions \(O(g(n)) = \{f(n) \; : \; \text{ there } \text{ exist } \text{ positive } \text{ constants } \; c \; \text{ and } \; n_0 \; \text{ such } \text{ that } \; 0 \le f(n) \le c \cdot g(n) \; \text{ for } \text{ all } \; n \ge n_0\}\).

  32. The experiments described in [8, 56] also use a population of ten agents.

  33. The number of games required to reach full communicative success (or full coherence) reported in [41, 42] is the maximum number of games required in the ten simulation runs carried out in those experiments.

  34. The number of games during which the agents invent or adopt words reported in [41, 42] is the maximum for the ten simulation runs carried out in those experiments.

  35. The initial groups of elder, adult and young agents contain \(\frac{n}{3}, \frac{n}{3}+n\;{ mod}(3)\) and \(\frac{n}{3}\) agents respectively, where n is the population size, \(\frac{n}{3}\) is the integer division of n by 3 and \(n\;\text{ mod }(3)\) the remainder.

  36. Note that in Figs. 6 and 9, the length of the first turnover interval is different from 1000 games, but the length of the rest of the turnover intervals is of 1000 games.

  37. See appendix A of [42] for a discussion of the expressiveness of the language systems constructed in the experiments reported in that paper, and a proof of the ability of such language systems to unambiguously express every propositional logic formula, which are represented in [42] using Lisp-like notation [34, 35].

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Acknowledgements

I should like to express my gratitude to Manuel Alfonseca for reading several versions of this paper and providing valuable comments.

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  1. Department of Computer Science, Technical University of Catalonia, Barcelona, Catalonia, Spain

    Josefina Sierra-Santibáñez

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Correspondence to Josefina Sierra-Santibáñez.

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This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant GRAMM (TIN2017-86727-C2-1-R), grant APCOM (TIN2014-57226-P), and Generalitat de Catalunya (AGAUR) under projects ALBCOM (2017 SGR 786) and LARCA (2014 SGR 890).

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Sierra-Santibáñez, J. An agent-based model of the emergence and evolution of a language system for boolean coordination. Auton Agent Multi-Agent Syst 32, 417–458 (2018). https://doi.org/10.1007/s10458-018-9384-1

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