Edited by: Harold Bekkering, University of Nijmegen, Netherlands
Reviewed by: James Kilner, Institute of Neurology, UK; Patrick Shafto, University of Louisville, USA
*Correspondence: Iris van Rooij, Centre for Cognition, Donders Institute for Brain, Cognition, and Behavior, Radboud University Nijmegen, Montessorilaan 3, 6525 HR Nijmegen, Netherlands. e-mail:
This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.
Human intentional communication is marked by its flexibility and context sensitivity. Hypothesized brain mechanisms can provide convincing and complete explanations of the human capacity for intentional communication only insofar as they can match the computational power required for displaying that capacity. It is thus of importance for cognitive neuroscience to know how computationally complex intentional communication actually is. Though the subject of considerable debate, the computational complexity of communication remains so far unknown. In this paper we defend the position that the computational complexity of communication is not a constant, as some views of communication seem to hold, but rather a function of situational factors. We present a methodology for studying and characterizing the computational complexity of communication under different situational constraints. We illustrate our methodology for a model of the problems solved by receivers and senders during a communicative exchange. This approach opens the way to a principled identification of putative model parameters that control cognitive processes supporting intentional communication.
This paper introduces a formal methodology for analyzing the computational complexity of intentional communicative actions, i.e., actions designed to modify the mental state of another agent. The need for such a methodology is evident from the obvious discrepancies between intuitions on the complexity of human communication. Some neuroscientists have argued that intentional communication is easy: we have neural mechanisms that can directly extract communicative intentions from detectable sensory events through the filter provided by our motor abilities (Iacoboni et al.,
Evidently, human communication falls in between these two extremes. Trivial as well as intractable views of human communication fail to adequately characterize the complexity of communication problems solved by humans in everyday situations. After all, humans are often capable of communicating with each other with little or no error, and even when errors occur communicators are often quick to adapt their behaviors so as to resolve any ambiguities. This behavioral success suggests that humans are somehow able to quickly take contextual factors into account and use them to estimate the likely meanings of communicative behaviors in context. Yet, context-sensitive computations are notorious in cognitive computational science for the astronomical demands that they make on computation time (Pylyshyn,
The framework we propose builds explicit models of communicative problems in order to assess the computational (in)tractability of those problems under different situational constraints. Situational constraints that render communication tractable under the given models are then candidate explanations of the speed of everyday human communications. Crucially, these candidate explanations can then be empirically tested, assessing whether the same relation between situational constraints and computational demands predicted by the model fit with the behavior and cerebral processes observed in participants in the lab while solving the same communicative problems. This type of model-driven approach offers several benefits for cognitive neuroscience. First, it identifies putative model parameters that control the cognitive processes supporting intentional communication. Second, it provides a rigorous ground for empirical tests of those models. Third, it offers the possibility to test the neurophysiological plausibility and cognitive relevance of the model by comparing the predicted dynamics of relevant model parameters with the observed dynamics of communicators’ internal states (i.e., cerebral signals measured with neuroimaging methods).
We will illustrate our proposed approach using one particular model of intentional communication as a case study. The model is an extension of the
Our model of communication builds on an existing model of how humans infer
According to the BIP model, observers assume that actors are “rational” in the sense that they tend to adopt those actions that best achieve their goals. Here “best” may, for instance, be defined in terms of (expected or believed) efficiency of a set of actions for achieving a given (combination of) goal(s). Say, a person has a single goal of tying his shoe laces. Then this person could make the necessary moves of the fingers, or he could start the finger movements, pause to scratch his chin, and then continue making the finger movements until his shoe laces are tied. If “rationality” is defined in terms of efficiency then the latter sequence of actions would be considered less rational for the goal of “tying one's laces” than the former sequence of actions. Be that as it may, the latter sequence of actions
Given the assumption of rationality, (probabilistic) knowledge of the world and how actions are affected by it, and a measure of relative rationality of action–goal pairs, one can compute the probability that an agent performs an action given its goals, denoted
or in shorter format:
An important insight of researchers such as Baker et al. (
Of all the possible combinations of goals that an observer can (or does) entertain, the goal combination that maximizes the probability in Eq.
A representation of the probabilistic dependencies between actions, goals, and states and how these dependencies change over time, and a sequence of observed actions and world states.
A combination of goals that best explains the sequence of actions and world states against the background of the probabilistic dependencies between actions, goals, and world states and how these dependencies change over time.
To be able to analyze the computational complexity of the G
Having formalized all the relevant notions in the G
A BIP-Bayesian network
The most probable joint value assignment
The computational complexity of this model has previously been analyzed by Blokpoel et al. (
If indeed – as the BIP-model postulates – observers of actions infer goals from actions by means of an inference to the best explanation, then senders can use this knowledge to predict how a receiver will interpret their actions ahead of time. For instance, they could engage an internal simulation of a receiver's inferences to the best explanation when considering the suitability of candidate actions (Noordzij et al., 2009). Such a simulation subroutine could be called multiple times during the planning of instrumental and communicative behaviors, allowing a sender to converge on an action sequence which is both efficient and is likely to lead a receiver (if properly simulated) to attribute the correct (i.e., intended) communicative goal to the sender. This conceptualization of the computational bases of sender signal creation can be summarized by the following informal input–output mapping
A representation of the probabilistic dependencies between actions, goals, and states and how these dependencies change over time, and one or more communicative and instrumental goals.
A sequence of actions that will lead to the achievement of the instrumental goals and will lead a receiver to attribute the correct communicative goals to the sender.
Note that our model allows for the possibility that senders can have simultaneous instrumental and communicative goals as seems required for ecological validity. After all, in everyday settings communicative behaviors are typically interlaced with several sequences of instrumental actions – think, for instance, of a car driver signaling to another driver at night that his lights are off, while at the same time trying to drive safely and stay on route.
Building on the formalisms used in the G
A BIP-Bayesian network
A joint value assignment
Here the receiver function R
For completeness and clarity, we state the R
A representation of the probabilistic dependencies between actions, goals, and states and how these dependencies change over time, and a sequence of observed actions and world states.
A combination of communicative goals that best explains the sequence of actions and world states against the background of the probabilistic dependencies between actions, goals, and world states and how these dependencies change over time.
A BIP-Bayesian network
The most probable joint value assignment
Having defined both the generic S
In this section we state all computational complexity results. Readers interested in full details on the mathematical proofs are referred to the Section
Our two main computational intractability results are:
The R
The S
These results establish that both R
Importantly, our analyses do not stop at the intractability results (Results 1 and 2). On the contrary, we view such results as merely the fruitful starting point of rigorous analyses of the sources of complexity in human communication. For these further analyses we adopt a method for identifying sources of intractability in cognitive models developed by van Rooij and Wareham (
First, one identifies a set of potentially relevant problem parameters
The R
Parameter | Description |
---|---|
| |
The size of the set of communicative goals |
| |
The size of the set of instrumental goals |
| |
The size of the set of action nodes |
| |
The size of the set of states |
The number of time slices | |
1 − |
Here |
– | |G_{I}| | |G_{C}| | |G_{I}|, |G_{C}| | |
---|---|---|---|---|
– | NP-hard | fp-intractable | fp-intractable | fp-tractable |
| |
fp-intractable | fp-intractable | fp-intractable | fp-tractable |
1 − |
fp-intractable | fp-tractable^{a} | fp-intractable | fp-tractable |
| |
fp-intractable | fp-tractable | fp-intractable^{a} | fp-tractable |
– | NP-hard | fp-intractable | fp-intractable | fp-intractable |
| |
fp-intractable | fp-intractable | fp-intractable | fp-tractable |
1 − |
fp-intractable | fp-intractable | fp-intractable | fp-intractable^{a} |
| |
fp-intractable | ? | fp-intractable^{a} | fp-tractable |
^{a}
We start by considering the fp-(in)tractability of the R
The R
The R
Note that, by definition, if a problem is fp-tractable for a parameter set
Result 3 establishes that it is possible to compute the input–output mapping defined by the R
These mathematical results lead to a clear prediction. A receiver is able to quickly attribute communicative intentions to a sender's actions if the number of instrumental goals that the receiver assumes that the sender is pursuing in parallel to her communicative goals is small
Importantly and perhaps counter intuitively, low ambiguity of the signal is by itself not sufficient for tractability, as we have the following intractability result.
The R
Similarly, a small number of communicative goals is by itself not sufficient for tractability, as we also have the following fp-intractability result.
The R
However, if senders were to focus solely on communicating (and forgetting for the moment about any other instrumental goals they may also want to achieve), then |
Having seen that neither large
As the parameter |
The S
In other words, the R
The S
Result 8 can be understood as a consequence of the ability of a sender to use the same fp-tractable algorithm that the receiver can use to tractably infer the sender's communicative goals to predict the receiver's interpretation of a given action sequence, and then search the space of action sequences (which is exponential only in |
Is the S
What we do know is that no other combination of parameters excluding {|
We analyzed the complexity of two models of senders and receivers in a communicative exchange. In Section
We presented two new models of the tasks engaging senders and receivers during a communicative exchange: see the R
It could be argued that, since human communication is evidently tractable for real humans in the real world, the present results are useless. However, the mismatch between model and reality is informative for improving formal models of human communication, and for rejecting claims that intentional communication is computationally trivial to explain (Rizzolatti and Craighero,
A major advantage of the current approach lies in its ability to indentify, from first principles, which situational constraints render the generic R
The fp-tractability results are
As long as |
As soon as two parameters in the set {|
If these predictions were to be confirmed, then this result would provide evidence for both cerebral use of fp-tractable computations, and for the R
We end this section by remarking that complexity results, such as we have derived here, also expand the ways in which computational-level models can be tested for input–output equivalence with human communicators. This form of testing involves presenting a participant with a situation that can be described as a particular input in the model's input domain, computing the output that according to the model corresponds to that input, and testing if the responses of the participant match the predicted outputs. This test is quite common in cognitive science, and was also the method used by Baker et al. (
For our case study, we opted for analyzing the R
The models that we defined make the strong commitment that there is no direct (probabilistic) dependency between states and goals, as all such dependencies are assumed to be mediated by actions (see Figures
Although complete observability may hold true in some situations (and in those situations the models may thus apply without problem), it is likely that in many real-world situations not all relevant actions and states are observable. Think, for instance, of the possibility that part of a visual scene is occluded by nearby objects or even an eye blink. Then some actions and state changes may be observed directly, while others need to be inferred by an observer. Humans are often able to do this, and models of receivers should be able to explain how they can do so. Because our fp-tractability results reported in Section
Another important property of our models is that they assume that the sender and receiver know everything that is relevant to the context of the communicative exchange, including which action and goal variables are relevant, and their probabilistic interdependencies. Clearly, this assumption sweeps a considerable amount of computational complexity under the rug; after all, computing the set of relevant information itself may well be computationally very resource demanding. Be that as it may, our analyses apply in those cases where humans do have complete knowledge of everything relevant to the communication task, and this is a condition that can certainly be met in laboratory settings used to investigate sources of complexity in communication as suggested in Section
Last, we reflect on our choice of formalism: we opted for Bayesian networks as a formal representation of communicators’ situational knowledge and we used the mathematical notion of posterior probability to define the intuitive notion of “inference to the best explanation.” This choice of probabilistic formalism aligns our models with the current trend in cognitive neuroscience, which assumes that the brain implements its cognitive functions by means of probabilistic computations (Wolpert and Ghahramani,
Recent accounts of human communication have focused on the motoric abilities of individual agents (Iacoboni et al.,
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
In this Appendix we explain general concepts, notation and terminology that are used in the main text. In particular, we introduce the mathematical notions of dynamic Bayesian Networks and a special case of such networks, called Bayesian Inverse Planning (BIP-)Bayesian Networks (Baker et al.,
Bayesian or probabilistic networks are tools for modeling uncertain knowledge (Jensen and Nielsen,
A probabilistic network
What is the most probable joint value assignment
A probabilistic network
What is the most probable joint value assignment
While Bayesian networks denote static knowledge, they can be made dynamic by incorporating a discrete notion of time. In
A BIP-Bayesian network (BIPBN) is a dynamic Bayesian network with specific connectivity, that models goal inference in the form of inverse planning (Baker et al.,
A BIPBN is based on the assumption that agents choose actions that maximize the probability that their goals are achieved. In particular, a BIPBN assumes that agents solve a Markov Decision Problem (Bellman,
Some proofs require the degradation of dependencies, i.e., when the dependencies are per definition required in the instance but should not have an effect on the Bayesian inference. The following lemma states how to achieve this:
In computational complexity analyses one studies the amount of computational resources required to compute (or solve) a problem
We are interested in the time complexity of models in terms of the size of the input. The input
To see why this definition has merit, compare the speed with which polynomial functions (say, 2
Receiver | – | |G_{I}| | |G_{C}| | |G_{I}|, |G_{C}| |
---|---|---|---|---|
– | ||||
Theorem A | Theorem C | |||
| |
||||
Corollary A | ||||
1 − |
||||
Corollary B | ||||
| |
||||
Theorem D |
Sender | – | |G_{I}| | |G_{C}| | |G_{I}|, |G_{C}| |
---|---|---|---|---|
– | ||||
Theorem B | ||||
| |
||||
Corollary C | Theorem E | |||
1 − |
||||
Theorem F | ||||
| |
2 |
2 |
|||
---|---|---|---|---|
2 | 4 | 4 | 8 | 4 |
5 | 10 | 25 | 125 | 32 |
10 | 20 | 100 | 1000 | 1024 |
20 | 40 | 400 | 8000 | 1048576 |
50 | 100 | 2500 | 125000 | >10^{15} |
100 | 200 | 10000 | 1000000 | >10^{30} |
200 | 400 | 40000 | 8000000 | >10^{60} |
NP-hard problems are problems that cannot
The reductions in the proofs in Sections
An undirected graph
Does there exist a subset
A tuple (
Does there exist a truth assignment to the variables in
Our analyses not only consider (classical) tractability as in Definition 1, but also fixed-parameter tractability. The latter type of complexity assessment is done using the tools and proof techniques from parameterized complexity theory. This mathematical theory is motivated by the observation that many NP-hard problems can be computed by algorithms whose running time is polynomial in the overall input size |i| and non-polynomial only in one or more small aspects of the input. These aspects are called
Proving fixed-parameter tractability is conceptually straightforward: It suffices to produce just one algorithm that computes the problem in fixed-parameter tractable time (see, e.g., Sloper and Telle,
W[1]-hard problems are problems, including a set of parameters, that cannot
The proofs in the Sections
A undirected graph
Does there exist a subset
Finally, the following Lemma is used to propagate fp-(in)tractability results derived for one parameter set
A Bayesian network
A value assignment a to A, such that
A Bayesian network
The most probable value assignment
Assume an arbitrary order on the vertices in
Assume the basic structure of
Set
These state variables effectively function as conjunctions which ensure that there is some assignment
All dependencies between
For 1 ≤
These action variables ensure that a joint value assignment
For
These action variables ensure that a joint value assignment
Make the prior probability distribution for each goal variable uniform.
Set
As the number of conditional probability tables that are constructed by the reduction is proportional to the total number of variables (which is
To prove that the construction above is a valid reduction, we must show that the answer to the given instance of C
If the answer to the given instance of Clique is “Yes”, there exists a subset
If the answer to the constructed instance of R
Theorem C. R
Corollary A. R
Consider a variant of the reduction in the proof of Theorem A. In addition to
For 1 ≤
These action variables ensure that a joint value assignment
For
These action variables ensure that a joint value assignment
If
Because of the inheritance from the P
Let
where α is a sufficiently small number to guarantee that all probabilities are in [0,1]. It plays no further role in the proof, so we fix α = ε^{2}.
Pr (
Pr (
Pr (
Dependencies between
Dependencies between
All dependencies between
Using this reduction from 3SAT instances to Sender we will prove that any joint value assignment
The following now holds:
Recall that Pr (
Where Pr_{SAT}(
Thus, if there exists a value assignment
This proves Sender fp-intractable for the parameter set {|
The authors would like to thank the reviewers for their valuable comments on an earlier version of this article. Johan Kwisthout was supported by the OCTOPUS project under the responsibility of the Embedded Systems Institute. Mark Blokpoel was supported by a DCC PhD grant awarded to Iris van Rooij and Ivan Toni. Todd Wareham was supported by NSERC Personal Discovery Grant 228104. Jakub Szymanik was supported by NWO (VICI grant #277-80-001). Ivan Toni was supported by NWO (VICI grant #453-08-002).
^{1}In other words, in the BIP model, goal inference is conceptualized as a form of probabilistic inference to the best explanation, a.k.a.
^{2}We remark that the computational-level S
^{3}We are aware of computer simulation studies in the cognitive science literature that seem to suggest that Bayesian models can be efficiently approximated (Vul et al.,
^{4}More formally, this would be a time on the order of
^{1}This is true, assuming that the class of problems solvable in polynomial time (P) is not equal to the class of problems whose solutions can be verified in polynomial time (NP). This “P ≠ NP” conjecture is believed by most living mathematicians, both on theoretical and empirical grounds (for more details see Garey and Johnson,
^{2}This is true, assuming that FPT ≠ W[1] (Downey and Fellows,
^{3}A polytree is a directed acyclic graph for which there are no undirected cycles when the arc direction is dropped.