Thinking differently about strategy: comparing paradigms

Our paper shows that mainstream strategic thinking and research already challenges the established Newtonian-Cartesian paradigm. Newtonian thought is the customary mode of western thinking, but is that about to change? Some papers from a complexity standpoint have appeared in the mainstream journals but its precise implications and merits have yet to be systematically spelled out and debated. We aim to facilitate this debate by comparing the established Newtonian and emergent complexity paradigms, clarifying the implications of this new perspective for strategy research. We suggest that the complexity paradigm is better attuned to current strategic realities than its Newtonian-Cartesian counterpart

LIPIcs

A regular language L of finite words is composite if there are regular languages L₁,L₂,…,L_t such that L = ⋂_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of ℕ, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than L’s, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs

LIPIcs

A deterministic finite automaton (DFA) is composite if its language L() can be decomposed into an intersection ⋂_{i = 1}^k L(_i) of languages of smaller DFAs. Otherwise, is prime. This notion of primality was introduced by Kupferman and Mosheiff in 2013, and while they proved that we can decide whether a DFA is composite, the precise complexity of this problem is still open, with a doubly-exponential gap between the upper and lower bounds. In this work, we focus on permutation DFAs, i.e., those for which the transition monoid is a group. We provide an NP algorithm to decide whether a permutation DFA is composite, and show that the difficulty of this problem comes from the number of non-accepting states of the instance: we give a fixed-parameter tractable algorithm with the number of rejecting states as the parameter. Moreover, we investigate the class of commutative permutation DFAs. Their structural properties allow us to decide compositionality in NL, and even in LOGSPACE if the alphabet size is fixed. Despite this low complexity, we show that complex behaviors still arise in this class: we provide a family of composite DFAs each requiring polynomially many factors with respect to its size. We also consider the variant of the problem that asks whether a DFA is k-factor composite, that is, decomposable into k smaller DFAs, for some given integer k ∈ ℕ. We show that, for commutative permutation DFAs, restricting the number of factors makes the decision computationally harder, and yields a problem with tight bounds: it is NP-complete. Finally, we show that in general, this problem is in PSPACE, and it is in LOGSPACE for DFAs with a singleton alphabet

Directional adposition use in English, Swedish and Finnish

Directional adpositions such as to the left of describe where a Figure is in relation to a Ground. English and Swedish directional adpositions refer to the location of a Figure in relation to a Ground, whether both are static or in motion. In contrast, the Finnish directional adpositions edellä (in front of) and jäljessä (behind) solely describe the location of a moving Figure in relation to a moving Ground (Nikanne, 2003). When using directional adpositions, a frame of reference must be assumed for interpreting the meaning of directional adpositions. For example, the meaning of to the left of in English can be based on a relative (speaker or listener based) reference frame or an intrinsic (object based) reference frame (Levinson, 1996). When a Figure and a Ground are both in motion, it is possible for a Figure to be described as being behind or in front of the Ground, even if neither have intrinsic features. As shown by Walker (in preparation), there are good reasons to assume that in the latter case a motion based reference frame is involved. This means that if Finnish speakers would use edellä (in front of) and jäljessä (behind) more frequently in situations where both the Figure and Ground are in motion, a difference in reference frame use between Finnish on one hand and English and Swedish on the other could be expected. We asked native English, Swedish and Finnish speakers’ to select adpositions from a language specific list to describe the location of a Figure relative to a Ground when both were shown to be moving on a computer screen. We were interested in any differences between Finnish, English and Swedish speakers. All languages showed a predominant use of directional spatial adpositions referring to the lexical concepts TO THE LEFT OF, TO THE RIGHT OF, ABOVE and BELOW. There were no differences between the languages in directional adpositions use or reference frame use, including reference frame use based on motion. We conclude that despite differences in the grammars of the languages involved, and potential differences in reference frame system use, the three languages investigated encode Figure location in relation to Ground location in a similar way when both are in motion. Levinson, S. C. (1996). Frames of reference and Molyneux’s question: Crosslingiuistic evidence. In P. Bloom, M.A. Peterson, L. Nadel & M.F. Garrett (Eds.) Language and Space (pp.109-170). Massachusetts: MIT Press. Nikanne, U. (2003). How Finnish postpositions see the axis system. In E. van der Zee & J. Slack (Eds.), Representing direction in language and space. Oxford, UK: Oxford University Press. Walker, C. (in preparation). Motion encoding in language, the use of spatial locatives in a motion context. Unpublished doctoral dissertation, University of Lincoln, Lincoln. United Kingdo

Factorizations of languages and commutativity conditions

Representations of languages as a product (catenation) of languages are investigated, where the factor languages are "prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions do not necessarily exist. If they exist, they are not necessarily unique - the number of factors can vary even exponentially. The paper investigates prime decompositions, as well as the commuting of the factors, especially for the case of finite languages. In particular, a technique about commuting is developed in Section 4, where the factorization of languages L1 and L2 is discussed under the assumption L1L2 = L2L1

Grounding LTLf specifications in images

A critical challenge in neurosymbolic approaches is to handle the symbol grounding problem without direct supervision. That is mapping high-dimensional raw data into an interpretation over a finite set of abstract concepts with a known meaning, without using labels. In this work, we ground symbols into sequences of images by exploiting symbolic logical knowledge in the form of Linear Temporal Logic over finite traces (LTLf) formulas, and sequence-level labels expressing if a sequence of images is compliant or not with the given formula. Our approach is based on translating the LTLf formula into an equivalent deterministic finite automaton (DFA) and interpreting the latter in fuzzy logic. Experiments show that our system outperforms recurrent neural networks in sequence classification and can reach high image classification accuracy without being trained with any single-image label