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The Neuro-Symbolic Inverse Planning Engine (NIPE):
Modeling Probabilistic Social Inferences from Linguistic Inputs

Lance Ying 1 2  Katherine M. Collins 3  Megan Wei 1  Cedegao E. Zhang 1  Tan
Zhi-Xuan 1

Adrian Weller 3  Joshua B. Tenenbaum 1  Lionel Wong 1

††footnotetext:  1Massachusetts Institute of Technology, Cambridge, USA
2Harvard University, Cambridge, USA 3University of Cambridge, Cambridge, UK.
Correspondence to: Lance Ying<EMAIL_ADDRESS>
Proceedings of the First Workshop on Theory of Mind in Communicating Agents,
Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).

###### Abstract

Human beings are social creatures. We routinely reason about other agents, and
a crucial component of this social reasoning is inferring people’s goals as we
learn about their actions. In many settings, we can perform intuitive but
reliable goal inference from language descriptions of agents, actions, and the
background environments. In this paper, we study this process of language
driving and influencing social reasoning in a probabilistic goal inference
domain. We propose a neuro-symbolic model that carries out goal inference from
linguistic inputs of agent scenarios. The “neuro” part is a large language
model (LLM) that translates language descriptions to code representations, and
the “symbolic” part is a Bayesian inverse planning engine. To test our model,
we design and run a human experiment on a linguistic goal inference task. Our
model closely matches human response patterns and better predicts human
judgements than using an LLM alone.

Figure 1: Overview of our model, the Neuro-Symbolic Inverse Planning Engine
(NIPE), which reasons about linguistic goal inference scenarios. (Left, green)
We design and measure human judgments on a linguistic goal inference domain
(inspired by Zhi-Xuan et al. (2020), which asks about the final goal of a
participant on a complex “gameshow” obstacle course involving trophies behind
locked doors. Inferring goals requires reasoning jointly about language that
describes the overall world configuration, specific world dynamics, and the
agent’s actions. Our model works by first using a large language model to
translate language into symbolic program semantics that collectively condition
a probabilistic generative model over possible environments and agents
(Middle, yellow), and answers queries via Bayesian inverse planning (Right) in
the conditioned probabilistic model, inferring likely goals using a nested
planner to model the agent.

##  1 Introduction

Humans are deeply social creatures, finely attuned to the inner lives of other
people Wellman (1992); Saxe (2006); Tomasello (2010). We not only think, but
think routinely about each other. We imagine what someone is trying to
accomplish and why are trying to accomplish it; we try to parse what someone
really believes and whether they might be mistaken; and we anticipate people’s
ultimate goals and upcoming actions, stepping in to assist or sometimes even
to try to hinder them. This capacity for social reasoning, broadly, is
abstract (we can observe someone’s actions in the world, but cannot directly
see what they think or want) and relies on a fine grained knowledge of
distinct but highly interrelated mental states within others (we know that
someone’s beliefs, desires, and goals relative to the world itself all
collectively influence what they might choose to do.)

This paper considers how language can inform social reasoning, with a
particular focus on understanding language about people’s actions, plans, and
goals. Language is an especially powerful means of conveying the abstract,
interrelated nature of concepts about agents in the world—we can talk about
someone’s actions (I saw Annie heading into the living room), directly convey
their mental states (Annie desperately wants a toy in there), describe
relevant but abstract aspects of the world in which they act (that cupboard
Annie is trying to open is locked inside), or even pose the questions we want
others to reason about (which toy do you think she wants the most?). Language
about any one of these aspects informs how we might reason about someone else
in holistic ways, changing our understanding and downstream predictions about
what they might know, want, or do.

Empirical evidence suggests that we think intuitively about each other as
largely rational, goal-directed intelligent agents Dennett (1981); Csibra et
al. (1999); Baker et al. (2009); Baillargeon et al. (2016). Within
computational cognitive science and AI, a considerable body of prior work
formalizes these intuitions by modeling agents as rational actors and planners
who choose actions to achieve specified goals or optimize rewards, and in turn
models many inferences about mental states as inverse planning from
observations of actions back to their latent goals, utility functions, or
beliefs about world states Jara-Ettinger et al. (2016); Baker et al. (2017);
Zhi-Xuan et al. (2020); Ying et al. (2023). Despite considerable successes in
predicting distinct human inferences and modeling cooperative behavior in
human-computer interactions Carroll et al. (2019); Mozannar & Sontag (2020);
Alanqary et al. (2021); Bansal et al. (2021); Wilder et al. (2021);
Shneiderman (2022), these approaches—largely non-linguistic models that draw
inferences from images or demonstrations of agent actions—face scalability and
specification challenges inherent to the abstract, complex, and interrelated
nature of social reasoning tasks. Modeling agents as goal-directed planners
requires implicitly or explicitly representing the world model they are
planning over, which is often precisely defined a priori (as in Baker et al.
(2017) and Zhi-Xuan et al. (2020)) or learned from expensive exploration (as
in Lian et al. (2021)). Inferring mental states, or even posing the task
itself, is challenging when framed as an inductive problem from images and
demonstrations alone (as in Shu et al. (2021)).

In part due to these challenges, the recent advances made by large language
models (LLMs) have driven an explosion of interest in how these models can be
used to plan Valmeekam et al. (2022) or draw social inferences Bubeck et al.
(2023); Shapira et al. (2023) from language—they suggest a new frontier of
cognitive models and AI systems that can leverage language to specify abstract
world knowledge, domain-specific world dynamics, and direct information about
agents and their mental states. However, early efforts to apply LLMs directly
to social reasoning problems often take a highly simplified approach to
evaluating their capacity for social reasoning or their “theory of mind”,
drawing broad conclusions based on a narrow set of templated cognitive tasks
(like the famous “Sally Anne tasks”). Other evaluations caution against
drawing general conclusions from such restricted task settings, pointing out
that modifications and extensions beyond these simple problem settings can
produce counterintuitive results that contradict the holism and generality of
these inferences Shapira et al. (2023); Ullman (2023); and several recent
evaluations suggest that robust planning in general remains an outstanding
challenge for large language models Valmeekam et al. (2022); Bubeck et al.
(2023). Several recent approaches seek to address these challenges by
augmenting LLMs with interfaces to external modules for representing and
simulating complex aspects of world state, particularly using classical AI
representations and engines for forward planning Collins et al. (2022); Liu et
al. (2023a); Lyu et al. (2023). Emulating human-like social reasoning and
knowledge, however, requires models that can robustly draw unified inferences
about actions, mental states, and the world Wong et al. (2023), relating plans
and beliefs in holistic, flexible, and ideally interpretable and safely
deployable way to language; and linguistic evaluations that move in turn
towards the richness and breadth of human-like social reasoning, flexible and
extensible enough to incorporate these interrelated aspects of human mental
states and world knowledge.

We focus specifically in this paper on understanding language that describes
inverse planning and goal inference problems—we consider how language can
convey information about the world itself, and the actions within it, in order
to drive inferences about the unseen goals motivating these actions. We make
the following contributions:

  1. 1.

We introduce a new model, the Neuro-symbolic Inverse Planning Engine (NIPE)
that leverages LLMs to translate language into program expressions that
compositionally construct and condition a probabilistic generative model over
agents and environments, then uses Bayesian inverse planning to perform goal
inference over this symbolic representation, expanding upon the paradigm of
_rational meaning construction_ for language-informed cognition Wong et al.
(2023).

  2. 2.

To evaluate this model, we compare our model against human social reasoning on
a linguistic inverse planning domain inspired by prior cognitive work in
social cognition Zhi-Xuan et al. (2020). We design this domain to require
jointly reasoning about agent actions and varying information about the world
itself.

  3. 3.

We find that our model well-predicts human judgments across a variety of
distinct social reasoning scenarios. We also find that our model outperforms
LLM baselines used to directly perform the same goal inference tasks. We find
that the largest LLM alone (GPT-4 OpenAI (2023)) correlates moderately well
with certain human judgments on this domain, particularly on simpler
scenarios. We discuss trends between LLMs of different sizes and on different
task variants in our stimuli.

Linguistic planning and inverse planning concepts across tasks
---
World configuration |  There are three trophies placed through this obstacle course: gold, silver, and bronze; The gold trophy is currently 5 steps East of Alice; The silver trophy is behind 1 door.
World dynamics |  On this course, each lock can only be opened by a key that is the same color as that lock; In this game, players lose points if they pick up more keys than needed.
Agent actions |  You see Alice picking up a green key; You see Alice walking a step towards the South; In this game, players lose points if they pick up more keys than needed.
Goal inference |  Considering each of the possible trophies on this course, how likely is it that Alice is trying to reach this trophy?
Task variations
Generic Keys |  There are three trophies placed throughout this obstacle course: gold, silver, or bronze. To get to the gold trophy, you need to unlock 2 different doors. To get to the silver trophy, you need to unlock 1 door. The bronze trophy isn’t behind a door. Keys to unlock the doors are placed around the obstacle course. All of the keys on this course work on all of the locks. You see Alice pick up 1 key.
Same-Colored Keys |  There are three trophies placed throughout this obstacle course: gold, silver, or bronze. The gold and silver trophies are both located behind a yellow door. To go to the bronze trophy, you need to go through a red door. On this course, each lock can only be opened by a key that is the same color as that lock. You see Alice pick up a red key.
Different-Colored Keys |  There are three trophies placed throughout this obstacle course: gold, silver, or bronze. The gold and silver trophies are both located behind a green door. To go to the bronze trophy, you need to go through a red door. On this course, each door can be unlocked by any key that is a different color than the door. You see Alice pick up a red key.
Spatial |  There are three trophies placed throughout this obstacle course: gold, silver, or bronze. Alice is a participant on this course. The gold trophy is currently 5 steps East of Alice. The silver trophy is 3 steps South from her. The bronze trophy is 5 steps further West of the silver trophy. You see Alice walking a step towards the South.
Table 1: Example language from the linguistic “gameshow” goal inference
domain. (Top) Each scenario contains language that spans a variety of
interrelated concepts relevant to the final goal-inference query, ranging from
information about the particular environment configuration to varying
information about an agent’s actions. (Bottom) We design four distinct task
variants that vary in complexity and type of the environment description and
the spatial details of the agent’s actions.

##  2 Linguistic inverse planning tasks and human experiment

We begin by describing the domain of tasks and human experiment we use
throughout this paper, to provide intuition for the modeling experiments that
follow. We construct a domain of linguistic inverse planning stimuli designed
to evaluate how humans and models infer the goals of agents acting within
varied environments. We design stimuli to convey varying information about the
world configuration and dynamics in which agents behave, as well as the
observed actions of agents with unknown goals.

Our tasks are inspired by the spatial navigation stimuli used in prior social
reasoning experiments Zhi-Xuan et al. (2020); Alanqary et al. (2021). Our
experiments adapt this basic domain into a set of linguistic inverse planning
stimuli, and introduce a range of variations designed to test distinct world
dynamics, spatial information, and vague or incomplete information about agent
actions conveyed linguistically.

###  2.1 Planning and inverse planning concepts in the Gameshow Domain

Each stimuli in our experiment (Table 1) begins with a description of a
“gameshow” scenario in which a player is trying to reach one of several goal
trophies placed around an obstacle course. This obstacle course also includes
locked doors that can only be unlocked with keys placed around the course, and
which additionally require multi-step reasoning about an agent’s longer term
plans. Based on each scenario description, which includes varying information
about the particular obstacle course and the actions of an agent within it, we
pose a linguistic goal inference query: which of the possible trophies do you
think the player trying to reach?

We design each stimulus to span several core conceptual categories relevant to
reasoning about agents and their actions in the world (Table 1, top),
including:

  * •

World configuration: information about goal trophies, keys, and their
attributes, as well as their relative locations, at differing levels of
specificity.

  * •

Abstract world dynamics: causal dynamics within the environment, such as rules
determining which keys can open which doors.

  * •

Agent actions: descriptions of an agent’s behavior before they reach the goal,
like picking up keys or walking in different directions towards keys or
trophies.

###  2.2 Task variants

We design four distinct variants of gameshow stimuli for a total of n=18
stimuli, which vary the complexity of the specified world configuration, the
world dynamics, and the type of observed agent actions (excerpts from stimuli
in each variant are shown in Table 1, bottom):

  * •

Generic keys: Goal trophies are placed behind locked doors, and must be
unlocked by keys placed around the obstacle course. Any key on the course can
be used to open any other door.

  * •

Same-colored keys: This variant adds additional complexity in the world
dynamics. Keys and doors have colors. Doors can only be unlocked by keys that
are the same color as the door.

  * •

Different-colored keys: In this variant, doors can only be unlocked by keys
that are a different color from the door.

  * •

Spatial: This variant contains specific spatial descriptions of the obstacle
course and agent actions. Due to the added spatial complexity, in these
environments, there are no keys or doors; agents can walk freely towards
various goal trophies.

###  2.3 Human experiment

We collect and evaluate human judgments on these goal inference tasks. We
recruit a total of n=46 participants from Prolific 111We exclude 6
participants as less than 30% of their MAP results agreed with consensus; i.e.
they disagree with the majority on which goal is the most likely for at least
5 out of 7 stimuli.; each viewed a random batch of 7 stimuli from a specific
task variant, and for each stimulus, produced goal inference judgements about
each possible goal trophy on a 1-7 Likert scale (1: definitely not this
trophy; 7: definitely this trophy). Scenarios were presented for a fixed
amount of time before an answer was permitted (90 seconds in the Spatial
condition; 45 seconds in the others). Each scenario was seen by between 14-16
participants. Participants were native English speakers from the USA/UK.

##  3 Our Model: The Neuro-Symbolic Inverse Planning Engine (NIPE)

We now introduce our model, the Neuro-Symbolic Inverse Planning Engine (NIPE).
While we focus on goal inference tasks, we describe a general framework for
conditioning on the diverse range of linguistic information in these
stimuli—varying information about the spatial environment, world dynamics, and
agent actions—that supports inferences about agents based on a principled,
probabilistic framework for Bayesian inverse planning.

Our core contribution is to show how we can approach complex linguistic
scenarios about agents by leveraging structured, symbolic representations and
inference methods (modeling agents as explicit planners, and performing
principled Bayesian inference over complex models) from the existing cognitive
and AI social reasoning literature (eg. Baker et al. (2017); Zhi-Xuan et al.
(2020)). We do so by translating from language into program expressions that
condition a symbolic generative model over agents and environments,
instantiating the paradigm of _rational meaning construction_ for language-
guided cognition Wong et al. (2023). This approach also addresses an important
open challenge for prior non-linguistic (inverse) planning models: We can
condition on abstract information about the world that is specified in
language, such as information about where objects are located or which keys
open which doors.

Our model (shown in schematic form in Figure 1) modularly decomposes the
problem of goal inference from language into three general subproblems:
language-to-code translation, which we instantiate generally by using a LLM as
a few-shot semantic parser (similar to Zhang et al. (2023); Liu et al.
(2023b); Collins et al. (2022); Wong et al. (2023)); the construction of a
probabilistic generative model over agents and environments from language; and
goal inference through Bayesian Inverse Planning in this probabilistic model
(using an nested planning and inference algorithm like that in (Zhi-Xuan et
al., 2020)). We now describe each problem in detail.

###  3.1 Language-to-code translation using LLMs

Following Wong et al. (2023), the crux of our approach is to show how the
different kinds of language in our scenarios can each be translated into
program expressions that collectively construct and condition a holistic
probabilistic model. We begin, however, by briefly describing the language-to-
code translation itself, which we implement by using LLMs to translate
sentences into code, which compositionally construct a symbolic model over an
extended, multi-sentence linguistic scenario (see Figure 1, LLM Language to
Code).

Our translation approach leverages the few-shot prompting capabilities of LLMs
trained jointly on language and code. We choose GPT-3.5 (Ouyang et al., 2022)
specifically to show that a smaller LLM can be used to implement this more
restricted task, rather than using the LLM itself to internally represent and
reason over these scenarios. Future work should explore whether even smaller
and more restricted language-to-code models can be used. Our approach builds
on the methods used in Zhang et al. (2023), which uses an LLM to translate
language into probabilistic programs, along with other work showing that LLMs
can translate language into code when given paired few-shot examples (Liu et
al., 2023a; Wong et al., 2023). For each stimulus, we construct a prompt
containing 2-3 randomly sampled examples of paired language and code from held
out examples on our domain. Models were sampled with a temperature of $1.2$.
To ensure robustness, we rejection sample translations to ensure that they
pass a number of syntactic and semantic tests (e.g., to ensure that translated
code is valid and executable.)

###  3.2 Constructing a probabilistic generative model over agents and
environments from language

Now we describe how the different kinds of information relevant to this domain
can be each translated into different kinds of code within a unifying
generative model. Following Zhi-Xuan et al. (2020), we model information about
the world dynamics, such as action preconditions and effects, using the
Planning Domain Definition Language (PDDL), a standardized representation
language widely used in robotics and automated planning Aeronautiques et al.
(1998), and which can b extended with modular semantics to efficiently
represent spatial domains such as ours Gregory et al. (2012); Zhi-Xuan (2022).

While prior work has considered how different subproblems related to planning
can be extracted via language-to-code (Collins et al., 2022; Liu et al.,
2023b; Silver et al., 2023), our aim here is to translate language not only to
distributions over possible worlds — i.e. possible spatial layouts and domain
rules — but also to inferences over an agent’s actions. We describe each
aspect of translation below:

#### Conditions on spatial environments

First, we prompt LLM to translate sentences like “The bronze chest is located
behind a red door.” into a CONDITION statement that describes this
arrangement. We then sample concrete grid-world maps from a generative model
over maps that satisfy this.

To ensure that maps are valid for our environment, we define and condition a
relatively restricted generative model over possible maps and trophy
placements that ensures that these basic validity conditions are met. Each
sampled map from this generative model is a concrete PDDL scene representation
that describes the locations of the initial agent, objects, and their
attributes.

#### Conditions on domain rules

Now, consider sentences like “Each door can be unlocked by a key with the
corresponding color.” These can be straightforwardly translated into operator
definitions expressed in PDDL (see Figure 1), which define an action and its
necessary parameters, preconditions for trying this action, and effects on the
environment.

Collectively, the scene representation and the domain rules define a complete,
concrete planning domain defined in PDDL. As in Zhi-Xuan et al. (2020), this
domain model is nested within a larger Bayesian model over agents acting in
this environment. The benefit of using PDDL is that we can easily model the
agents themselves using classical planning algorithms over the PDDL domain
representation and we can easily transfer this model to a different problem
setup by modifying the operator definitions or simply choosing a different
pre-defined PDDL domain.

#### Conditioning a generative model over agent subgoals and actions from
language

Finally, we consider how to translate sentences that describe agents’ actions
into conditions on a model over agents. We distinguish between language like
“You observe Alice picking up two keys” which is treated as a condition on
subgoals, and “You observe Alice walking 3 steps East.”, which is treated as a
low-level action sequence.

###  3.3 Goal Inference through Bayesian Inverse Planning

Once we have an initial PDDL environment representation $s_{0}$ and a sequence
of agent actions $a_{1:t}$, we can perform Bayesian inverse planning to infer
a posterior distribution over goals $g$ given actions $a_{1:t}$ and the
initial state $s$, which is proportional to the joint distribution over goals
and actions:

$P(g|s_{0},a_{1:t})\propto P(g,a_{1:t}|s_{0})=P(g|s_{0})P(a_{1:t}|g)$

Here $P(g|s_{0})$ defines the prior distribution over possible goals in the
initial state $s_{0}$, and $P(a_{1:t}|g)$ is a model of how the agent selects
actions $a_{1:t}$ given a particular goal $g$. For the goal prior, we model
agents as being more likely to pick up goals that are closer to them, as
determined by the cost $C(g,s_{0})$ of the shortest plan to each goal:

$P(g|s_{0})\propto\textstyle{\frac{1}{C(g)}}$

To model actions, we assume $P(a_{1:t}|g)$ decomposes into the probability of
selecting an action $a_{t}$ at each state $s_{t}$, given by a Boltzmmann-
rational policy $\pi$:

$\displaystyle P(a_{1:t}|g)$
$\displaystyle=\textstyle{\prod}_{\tau=1}^{t}\pi(a_{\tau}|s_{t},g)$
$\displaystyle\pi(a_{t}|s_{t},g)$ $\displaystyle=\frac{\exp{\beta
Q_{g}(s_{t},a_{t})}}{\sum_{a_{t}^{\prime}}\exp{\beta
Q_{g}(s_{t},a_{t}^{\prime})}}$

where $Q_{g}(s_{t},a_{t})$ denotes the (negative) cost of the reaching the
goal $g$ by taking action $a_{t}$ from $s_{t}$, and $\beta$ is a rationality
parameter, also called the inverse temperature. This policy models agents as
approximately rational in how they act, since actions which reach the goal
more quickly (higher $Q_{g}$) are more probable, with higher values of $\beta$
leading to more optimal actions. Although this model does not account for
forms of approximate rationality Evans et al. (2016); Shah et al. (2019);
Alanqary et al. (2021), it is a widely used choice in (Bayesian) inverse
planning and inverse reinforcement learning that can account for non-
systematic deviations from optimality Ramachandran & Amir (2007); Ziebart et
al. (2008); Baker et al. (2009). To compute $Q_{g}$ for each state $s_{t}$ and
action $a_{t}$ encountered during inference, we use A* search as an optimal-
cost planner in the PDDL domain to determine the cost of the optimal plan.

With these components of the model specified, goal inference can be performed
_exactly_ by computing the joint probability $P(g,a_{1:t}|s_{0})$ of each
possible goal $g$ with the observed sequence of actions $a_{1:t}$, which can
then be normalized to obtain the posterior distribution $P(g|s_{0},a_{1:t})$.

##  4 Model Experiments

We compare our model with human judgements on the gameshow domain of social
reasoning tasks. We also evaluate two LLM baselines to assess the value of an
explicit Bayesian inverse planning model.

For our model, we simulate answers to each goal inference stimulus (which
trophy will Alice pick up?), outputting a posterior probability over all of
the possible trophies (probabilities sum to 1 over all trophies). For each
stimulus, we translate the linguistic scenario description of the world
configuration, world dynamics, and agent behavior into symbolic conditions on
the generative model (by sampling from the LLM and rejecting invalid code),
and perform goal inference with the generated code to calculate the posterior
over the trophies. Since goal inference can be performed without sampling, and
since the space of maps consistent with a linguistic description is highly
restricted, we found in pilot experiments that our model showed limited
variability across LLM and generative model samples. Therefore, for each
stimulus, we computed our results using a single valid sample of PDDL domain
rules and spatial environments.

We also conduct two baseline experiments using two different LLMs
(gpt-3.5-turbo-0301, the dialogue-tuned variant of GPT-3.5 (Ouyang et al.,
2022), and gpt-4-0314, the March snapshot of GPT-4 (OpenAI, 2023)) to directly
provide goal inference judgments for each stimulus. We assess LLMs using
exactly the same conditions as the human experiment, using a zero-shot prompt
where the LLM is given the full linguistic experimental set up (including
instructions explaining the task) shown to human participants, concatenated
with the scene scenario and query for each stimulus, and a short prompt
specifying the answer format. To account for answer variability, we sample n =
30 answers for each stimulus, rejecting any answers that are not in the
correct format. Exact prompt formats can be found in the Appendix.

##  5 Results and discussion

Figure 2: Correlation between NIPE and baseline LLMs vs. human judgments,
aggregated across the goal-inference stimuli. Our model (Right) produces
posterior probabilities over goals that correlate strongly overall with human
judgments; GPT-4 correlates moderately overall (Middle) and GPT-3.5 (Left)
correlates poorly across the tasks. Color-coded points show stimuli from
specific task variants.

We evaluate our model and baselines in comparison to human judgments, both
across all of the stimuli (Figure 2) and within each task variant category
(Figure 3). We find that:

#### Our model (NIPE) correlates strongly overall with human judgments across
the goal inference experiment.

We calculate correlations between the posterior probability judgments produced
by our model and the mean per stimulus judgments (Likert scale ratings) across
human subjects, and find that our model is strongly correlated with human
judgments in the experiment overall (Figure 2, Our Model, R=0.927, 95%
CI=(0.877, 0.953)). All 95% CI are bootstrap sampled confidence intervals each
with 1000 samples.

#### The largest LLM baseline (GPT-4) correlates moderately with human
judgments overall; GPT3.5 correlates poorly.

We additionally calculate correlations between LLMs prompted directly to
produce goal inference judgments (mean over sampled LLM judgements for each
stimulus, and the mean over human judgments for each stimulus across
participants). In comparison to our model, we find that GPT-4 correlates well,
but more weakly, with human judgments (Figure 2, GPT-4 Rating, R=0.658,
CI=(0.403, 0.79)) overall, and GPT-3.5 correlates much more poorly with human
judgments overall (Figure 2, GPT-3.5 Rating, R=0.100, CI=(-0.145, 0.361)).
These aggregate trends are better understood by examining correlations between
human judgments within each task variant, as we discuss next.

#### NIPE correlates strongly with human judgements within each task variant;
GPT-3.5 and GPT-4 are less robust to complex world dynamics and spatial
information.

Inspecting trends within each individual task variant (Figure 3) provides
insight into when models deviate from humans.

NIPE consistently correlates with human judgments across all task variants,
from the simpler Generic stimuli to the more complex Colored and Spatial
stimuli (Figure 3, bottom). In contrast, on the more complex
tasks—particularly the Colored-Different variant, which specifies a new,
counterintuitive rule particular to this domain (doors can only be opened by
keys that are a different color from the door), and the Spatial variant, which
requires considering how an efficient, path-planning agent would navigate
towards their goal—the LLM baselines show revealingly different performance in
comparison to human reasoning.

As seen in Figure 3, GPT-3.5 actually correlates negatively with human
judgments on Colored-Different—it both ignores the stated rule about the
different colored keys and appears to make the opposite, incorrect assumption,
perhaps based on simpler color-based pattern matching; and shows nearly zero
correlation with human judgments on the explicit Spatial path-planning
variant. In comparison, GPT-4 correlates well with human judgments on both of
the simpler task variants, and shows a moderate (but positive) correlation on
the Spatial variant (though we discuss stimuli-specific failures next,
suggesting that GPT-4 may still be using simpler distance heuristics rather
than robustly modeling an efficient path-planning agent.) On the Colored-
Different domain, GPT-4 has nearly zero-correlation, suggesting that it is
highly uneven in taking the specific rule about key colors into account—as
shown in Figure 3, it predicts human judgments well in some cases, but
produces nearly the opposite overall judgment in others, suggesting it has
ignored the rule.

Figure 3: Correlations within specific task variants highlight more granular
trends in model behavior. Our model (Bottom) correlates strongly with human
judgements across all four task variants. In comparison, the LLM baselines
generally show poorer performance on the more complex variants (the Color-
Different variant, which introduces a new, counterintuitive rule about the
world dynamics; and the Spatial variant, which requires reasoning specifically
about agents as efficient path planners.

#### Humans make fine-grained assumptions about agents as rational planners;
LLMs do not always do so.

Our model explicitly reasons about agents as rational, goal-directed actors
and planners. Do LLMs do so as well? Qualitative analysis of individual
stimuli highlights subtle but important discrepancies between human
assumptions and LLM-based goal inferences on the most complex Spatial stimuli.
We highlight a representative example here:

> There are three trophies placed throughout this obstacle course: gold,
> silver, or bronze. Alice is a participant on this course. The gold trophy is
> currently 2 steps East of Alice. The silver trophy is 3 steps South from
> her. The bronze trophy is 5 steps South from her. You see Alice walking 4
> steps South.

Both humans and our model strongly predict that Alice is headed towards the
bronze trophy on this domain. In contrast, both the LLM baselines predict most
strongly that Alice is heading towards the silver trophy (perhaps because it
is closer to Alice), though in fact, humans tend to agree that because she has
walked past this trophy, as an efficient actor, this is likely not her goal.
We provide additional examples of similar efficient-planning situations in the
Appendix.

##  6 Conclusion and Future Directions

We conclude with several important avenues for future work. First, while we
evaluate all LLMs using a simple prompting regime, future work should evaluate
whether these results are consistent or improve with other prompting and
evaluation methods such as chain-of-thought (Wei et al., 2022; Kojima et al.,
2022), particularly on complex tasks such as those that invoke efficient
spatial path planning, which prior work also finds to be an outstanding
challenge for LLMs (Cohn & Hernandez-Orallo, 2023). It is also possible for
LLMs to be biased by surface-level cues (e.g. in our stimuli, they might be
biased towards gold over silver trophies, since the former are often
considered more valuable). Future experiments should control for these
potential biases by testing a wider diversity of prompts.

Second, while NIPE adopts a Bayesian approach to social inference, the process
of language translation itself is performed in a non-Bayesian “bottom-up”
manner using LLMs. Future work might explore a more fully Bayesian approach
that models how language itself is generated from symbolic task
representations, and perform posterior inference directly from language. This
might involve using LLMs as likelihood models over language, as in Ying et al.
(2023), and also incorporate LLMs as common sense priors over representations
of the world (Lew et al., 2020; Li et al., 2023). Among other benefits, this
could enable more controlled and robust translation of language to code Poesia
et al. (2022); Lew et al. (2023).

Finally, we designed the gameshow domain to permit future extensions that
describe a much broader range of social reasoning tasks that we could express
linguistically. This includes scenarios that specifically invoke agent belief
Baker et al. (2017); Ullman (2023); involve boundedly rational agents Alanqary
et al. (2021); and even involve multiple agents collaborating or competing
Kleiman-Weiner et al. (2016). In addition, future work could extend this
domain to multimodal stimuli that include both linguistic information and
demonstrations or images, e.g., to convey complex agent trajectories or
environment layouts Ying et al. (2023). The difficulty of the domain could
also be increased, expanding the number of objects, actions, and planning
steps in order to push the limits of language-to-code translation. By building
out this architecture, we hope to fully reap the benefits of the both the
neural and symbolic traditions of AI, enabling flexible social inferences from
a wide range of naturally occurring linguistic stimuli, while producing
outputs that are structured, coherent, and robust Collins et al. (2022).

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##  Appendix A Appendix

###  A.1 Acknowledgments

TZX is funded by an Open Phil AI Fellowship. KMC is funded by a Marshall
Scholarship and the Cambridge Trust. AW acknowledges support from a Turing AI
Fellowship under grant EP/V025279/1, The Alan Turing Institute, and the
Leverhulme Trust via CFI. The work is additionally funded by DARPA, MCS,
AFOSR, ONR Science of AI programs, and the MIT-IBM Watson AI lab.

###  A.2 Human linguistic inverse planning experiment

We include additional details on our human experiment. For all scenarios,
participants were asked “Now, considering each of the possible trophies on the
course, please respond on the 1-7 scale: how likely is it that the participant
is trying to reach this trophy?” Likert scale options were as follows:

  * •

“1 (Definitely not this trophy)”

  * •

“2 (Very unlikely)”

  * •

“3 (Somewhat unlikely)”

  * •

“4 (Equally likely or unlikely)”

  * •

“5 (Somewhat likely)”

  * •

“6 (Very likely)”

  * •

“7 (Definitely this trophy)”

Note, due to a mismatch, some participants in the Generic condition saw one
stimuli from Different-Colored Keys and vice versa. We reversed fixed this
partway through the study.

###  A.3 LLM Baselines Experiment

We use the gpt-4-0314 model (i.e., a fixed, non-updatable checkpoint from
March 14 for maximal reproducibility) in the Chat Completion API. For each
stimuli, we sample until we get 20 well-formed responses, rejecting samples
that are not in the correct format.

Example LLM Prompt for World Configuration

      Input:
      There are three trophies placed throughout this obstacle course: gold, silver, or bronze.
      The gold trophy is behind a green door.
      The silver trophy is behind a yellow and red door.
      The bronze trophy is behind a red door.

      Keys to unlock the doors are placed around the obstacle course. On this course, each door
      can be unlocked by any key as long as it is of a different color than the door.

      You see Alice picking up a red key. What would be the likely trophy Alice wants to get?"

      output:
      {"agent" : ["Alice"],
      "goals" : ["gold", "silver", "bronze"],
      "locations" : {"gold" : "Room A", "silver" : "Room B", "bronze" : "Room C"},
      "obstacles" : {"Room A" : ["G"], "Room B" : ["Y", "R"],"Room C" : ["R"] },
      "keys" : ["g","y","r"],
      "max_obstacle" : 2,
      "keys_per_door": 1,
      "len_key" : 3,
      "goal_count" : 3,
      "observation_type": "has_objects",
      "observation": "(exists (?k - key) (and (has Alice ?k) (iscolor ?k red)))"}

Example LLM Prompt for PDDL Domain Definition

In this example we constrain the model such that the LLM can only modify the
unlock action of the PDDL domain definition. Our goal is to show that with
few-shot prompting in the color-same condition, the LLM can generalize to
generate action definition for color-different condition.

        input:

        Alice is playing a treasure game to pick up exactly one trophy.
        There are 3 trophies: gold, silver and bronze, located in room A, room C and room B.
        There is 1 yellow and 1 green lock for Room A, a red lock for room B and a
        yellow lock for room C.

        Alice needs to unlock the locks to enter the rooms. Each lock can be unlocked
        with a key with the same color. Each key can only be used once.

        You see Alice go over to pick up a red key.

        It’s unclear what Alice would do next.

        output:

        (:action unlock
         :parameters (?a - agent ?k - key ?d - door)
         :precondition (and (has ?a ?k) (locked ?d)
                            (exists (?c - color) (and (iscolor ?k ?c) (iscolor ?d ?c)))
                            (or (and (= (xloc ?a) (xloc ?d)) (= (- (yloc ?a) 1) (yloc ?d)))
                                (and (= (xloc ?a) (xloc ?d)) (= (+ (yloc ?a) 1) (yloc ?d)))
                                (and (= (- (xloc ?a) 1) (xloc ?d)) (= (yloc ?a) (yloc ?d)))
                                (and (= (+ (xloc ?a) 1) (xloc ?d)) (= (yloc ?a) (yloc ?d)))))
         :effect (and (not (has ?a ?k)) (not (locked ?d)) )
        )

###  A.4 An example of LLM baseline result

Here we provide an example of GPT-4’s response in the baseline condition on
the Color-different task variant.

        input:

        There are three trophies placed throughout this obstacle course:
        gold, silver, or bronze.

        The gold and silver trophies are both located behind a yellow door.
        To go to the bronze trophy, you need to go through a red door.

        Keys to unlock the doors are placed around the obstacle course.
        On this course, each door can be unlocked by any key that is a different
        color than the door. For example, yellow keys cannot unlock yellow doors,
        but can unlock doors in green or red.

        In this game, players lose points if they pick up more keys than needed.
        You may assume that players will use all the keys they pick up.

        Alice is a participant on this course. You see Alice pick up a yellow key.

        It’s unclear what Alice would do next.

        How likely is Alice picking up the trophies? Please rate from 1 to 7
        and explain.

        output:

        Gold Trophy: 6
        Silver Trophy: 6
        Bronze Trophy: 1

        Explanation:

        Since Alice has picked up a yellow key, she can now unlock either
        the gold or silver trophy, since both are located behind a yellow
        door. This makes it quite likely that she will pick up one of those
        two trophies (rating them both at 6).

        However, picking up a yellow key means she cannot unlock the red
        door, which leads to the bronze trophy. Therefore, it is very unlikely
        that Alice will pick up the bronze trophy (rating it 1).

It is clear from this example that GPT-4 shows unsatisfactory understanding of
the setup of the Color-different variant. It is stated that yellow keys cannot
unlock yellow doors, so it is unlikely that Alice would pick up the gold or
silver trophy, but GPT-4 still judges these two possibilities to be highly
likely. On the other hand, both humans and our NIPE model assign significantly
more likelihood to the bronze trophy (and roughly equal low likelihoods to the
gold and silver trophies) in this example.

###  A.5 Sample stimuli for each experimental condition

Generic:

> There are three trophies placed throughout this obstacle course: gold,
> silver, or bronze.
>
> To get to the gold trophy, you need to unlock 2 different doors. To get to
> the silver trophy, you need to unlock 1 door. The bronze trophy isn’t behind
> a door.
>
> Keys to unlock the doors are placed around the obstacle course. On this
> course, each door has exactly 1 lock on it. All of the keys on this course
> work on all of the locks. However, each key can only be used one time.
>
> Alice is a participant on this course. You see Alice go over and pick up two
> keys.

Color-same:

> There are three trophies placed throughout this obstacle course: gold,
> silver, or bronze.
>
> The gold and silver trophy trophies are both located behind a yellow door.
> To get to the yellow door, you first need to go through a green door. To go
> to the bronze trophy, you need to go through a red door.
>
> Keys to unlock the doors are placed around the obstacle course. On this
> course, each door is unlocked by a key of the corresponding color.
>
> In this game, players lose points if they pick up more keys than needed. You
> may assume that players will use all the keys they pick up.
>
> Alice is a participant on this course. You see Alice pick up a yellow key.

Color-different:

> There are three trophies placed throughout this obstacle course: gold,
> silver, or bronze, which are placed in rooms labeled A, B, and C,
> respectively.
>
> Room A is behind a door with a yellow and green lock on it. Room B is behind
> a door with a red lock on it. Room C is behind a door with its own yellow
> lock on it.
>
> Keys to unlock the doors are placed around the obstacle course. On this
> course, each door can be unlocked by any key that is a different color than
> the door. For example, yellow keys cannot unlock yellow doors, but can
> unlock green and red doors
>
> In this game, players lose points if they pick up more keys than needed. You
> may assume that players will use all the keys they pick up.
>
> Alice is a participant on this course. You see Alice pick up a yellow key.
>
> It’s unclear what Alice would do next.

Spatial:

> There are three trophies placed throughout this obstacle course: gold,
> silver, or bronze. Alice is a participant on this course.
>
> The gold trophy is currently 1 step East of Alice. The silver trophy is 3
> steps South from her. The bronze trophy is 3 steps East from the silver
> trophy.
>
> You see Alice walking 3 steps East.


# A continuous $p$-adic action on the $K(2)$-local algebraic $K$-theory of
$p$-adic complex $K$-theory

Daniel G. Davis

###### Abstract.

Let $p$ be a prime, let $KU_{p}$ be $p$-complete complex $K$-theory, and let
$\mathbb{Z}_{p}^{\times}$ denote the group of units in the $p$-adic integers.
The $p$-adic Adams operations induce an action of the profinite group
$\mathbb{Z}_{p}^{\times}$ on $KU_{p}$, and hence, on the algebraic $K$-theory
spectrum $K(KU_{p})$. For $p\geq 5$, we give an elementary construction of the
continuous homotopy fixed point spectrum $(L_{K(2)}K(KU_{p}))^{hG}$, where
$K(2)$ is the second Morava $K$-theory and $G$ is any closed subgroup of
$\mathbb{Z}_{p}^{\times}$. Also, for each $G$, we show that there is an
associated strongly convergent homotopy fixed point spectral sequence whose
$E_{2}$-term is given by Jannsen’s continuous group cohomology, with
$E_{2}^{s,\ast}=0$, for all $s>2$. This work is related to a conjecture of
Ausoni and Rognes.

##  1\. Introduction

Let $p$ be a prime, $\mathbb{Z}_{p}$ the $p$-adic integers, $L_{K(1)}S^{0}$
the Bousfield localization of the sphere spectrum with respect to the first
Morava $K$-theory spectrum, and $KU_{p}$ $p$-complete complex $K$-theory.
Thus,

$\pi_{\ast}(KU_{p})=\mathbb{Z}_{p}[u^{\pm 1}],$

where $\pi_{0}(KU_{p})=\mathbb{Z}_{p}$ and $|u|=2$. Also, let
$\mathbb{Z}_{p}^{\times}$ denote the group of units in $\mathbb{Z}_{p}$. By
[16, 17], there is a $\mathbb{Z}_{p}^{\times}$-action on the commutative
$S^{0}$-algebra $KU_{p}$ via the $p$-adic Adams operations and this action is
by maps of commutative $S^{0}$-algebras. Given a commutative $S^{0}$-algebra
$A$, $K(A)$ denotes the algebraic $K$-theory spectrum of $A$, and $K(-)$ is a
functor on the category of commutative $S^{0}$-algebras. It follows that
$\mathbb{Z}_{p}^{\times}$ acts on $K(KU_{p})$ by maps of commutative
$S^{0}$-algebras.

For the rest of this paper, let $p\geq 5$ and let $V(1)$ be the type $2$
Smith-Toda complex $S^{0}/(p,v_{1})$, a homotopy commutative ring spectrum.
There is a $v_{2}$-self-map $v\colon\Sigma^{2p^{2}-2}V(1)\to V(1)$ and
$v_{2}^{-1}V(1)$ denotes the mapping telescope associated to $v$. In [3,
paragraph containing (0.1)], [4, Conjecture 4.2], and [2, page 46; Remark
10.8], Christian Ausoni and John Rognes conjectured that the unit map

$L_{K(1)}S^{0}\to KU_{p}$

in the $K(1)$-local category induces a weak equivalence

(1.1) $K(L_{K(1)}S^{0})\wedge v_{2}^{-1}V(1)\to
K(KU_{p})^{h\mathbb{Z}_{p}^{\times}}\wedge v_{2}^{-1}V(1),$

where the conjectural object $K(KU_{p})^{h\mathbb{Z}_{p}^{\times}}$ is a
continuous homotopy fixed point spectrum that is formed with respect to the
conjecturally continuous action of the profinite group
$\mathbb{Z}_{p}^{\times}$ on $K(KU_{p})$.

We point out that as [3, 4, 2] explain, the above conjecture consists of $n=1$
cases of a more general conjecture made by Ausoni and Rognes for every
positive integer and every prime. There is no published construction of
$K(KU_{p})^{h\mathbb{Z}_{p}^{\times}}$ and, according to [10, Remark 1.5], the
only models for it currently are a “candidate definition” using condensed
spectra in the sense of Clausen-Scholze and, possibly, a pyknotic version of
this construction (in the framework of [6]). As far as we know, documents
about these models are not available.

After some recollections in Section 2, we give in Section 3 an elementary
construction of $L_{K(2)}K(KU_{p})$ as the homotopy limit of a tower of
discrete $\mathbb{Z}_{p}^{\times}$-spectra. Here, $L_{K(2)}(-)$ denotes
Bousfield localization with respect to the second Morava $K$-theory spectrum
and we are using the term “discrete $\mathbb{Z}_{p}^{\times}$-spectra” in the
sense of [7, Section 2.3]. In Definition 3.1, by following a familiar pattern
in algebraic $K$-theory, we see that the aforementioned homotopy limit yields
the continuous homotopy fixed point spectrum

$(L_{K(2)}K(KU_{p}))^{hG},$

where $G$ is any closed subgroup of

$\mathbb{Z}_{p}^{\times}\cong\mathbb{Z}_{p}\times C_{p-1}$

and $C_{p-1}$ is a cyclic group of order $p-1$.

We also have a result about a homotopy fixed point spectral sequence for

$\pi_{\ast}\bigl{(}(L_{K(2)}K(KU_{p}))^{hG}\bigr{)}.$

To describe this, we need to make some preliminary remarks.

We work in the stable model category $Sp^{\Sigma}$ of symmetric spectra of
simplicial sets and we let

$(-)_{f}\colon Sp^{\Sigma}\rightarrow Sp^{\Sigma},\ \ \ Z\mapsto Z_{f}$

be functorial fibrant replacement: given a spectrum $Z$, there is a natural
map $Z\rightarrow Z_{f}$ that is a trivial cofibration, with $Z_{f}$ fibrant.
By [25, Propositions 3.7, 5.1] and [21, Section 2], there is a tower

(1.2) $V(1)=:M_{0}\leftarrow M_{1}\leftarrow\cdots\leftarrow
M_{i}\leftarrow\cdots$

of finite type $2$ complexes having the following properties:

  * •

for any spectrum $Z$, there is an equivalence

$L_{V(1)}Z\simeq\operatorname*{holim}_{i}(Z\wedge M_{i})_{f}\,;$

  * •

each $M_{i}$ has enough algebraic structure in the stable homotopy category
(see [12] and [22, Section 4] for the details) that $\pi_{0}(M_{i})$ is a
commutative ring and for every integer $t$ and any spectrum $Z$,
$\pi_{t}(Z\wedge M_{i})$ is a unitary module over $\pi_{0}(M_{i})$;

  * •

there is an increasing sequence $m_{0}=1,m_{1},m_{2},...,m_{i},...$ of
integers, such that for each $i\geq 0$, there is an isomorphism
$\pi_{0}(M_{i})\cong\mathbb{Z}/{p^{m_{i}}\mathbb{Z}}$ of rings.

In (1.2), after $M_{0}$, there are suspensions on each $M_{i}$ that – as is
common practice – we have omitted.

If $A$ is a discrete $G$-module, $H^{\ast}_{c}(G,A)$ denotes the continuous
cohomology of $G$ with coefficients in $A$. Given a tower $\\{A_{i}\\}_{i\geq
0}$ in the category of discrete $G$-modules,
$H^{\ast}_{\mathrm{cont}}(G;\\{A_{i}\\}_{i\geq 0})$ denotes continuous
cohomology in the sense of Jannsen [23]. We let $L_{2}(-)$ be Bousfield
localization with respect to the coproduct $K(0)\vee K(1)\vee K(2)$ of Morava
$K$-theories, where $K(0)$ is the Eilenberg-Mac Lane spectrum $H\mathbb{Q}$.
Also, for each $i$, the $G$-action on $\pi_{t}(K(KU_{p})\wedge L_{2}M_{i})$ is
the action induced by letting $G$ act – on the level of spectra – on
$K(KU_{p})$ only.

######  Theorem 1.3.

Let $p\geq 5$ and let $G$ be a closed subgroup of $\mathbb{Z}_{p}^{\times}$.
There is a strongly convergent homotopy fixed point spectral sequence

$E_{2}^{s,t}\cong H^{s}_{\mathrm{cont}}(G;\\{\pi_{t}(K(KU_{p})\wedge
L_{2}M_{i})\\}_{i\geq
0})\Longrightarrow\pi_{t-s}\bigl{(}(L_{K(2)}K(KU_{p}))^{hG}\bigr{)},$

where for each $t\in\mathbb{Z}$,

$E_{2}^{s,t}\cong\begin{cases}\displaystyle{\bigl{(}\lim_{i}\pi_{t}(K(KU_{p})\wedge
L_{2}M_{i})\bigr{)}^{\mspace{-3.0mu}G}},&s=0;\\\
\displaystyle{\operatorname*{lim^{1}}_{i}H^{1}_{c}(G,\pi_{t}(K(KU_{p})\wedge
L_{2}M_{i}))},&s=2;\\\ 0&s\geq 3,\end{cases}$

and when $s=1$ and $t\in\mathbb{Z}$, there is the short exact sequence

$0\to\operatorname*{lim^{1}}_{i}\bigl{(}\pi_{t}(K(KU_{p})\wedge
L_{2}M_{i})\bigr{)}^{\mspace{-2.5mu}G}\to
E_{2}^{1,t}\to\lim_{i}H^{1}_{c}(G,\pi_{t}(K(KU_{p})\wedge L_{2}M_{i}))\to 0.$

The proof of Theorem 1.3 is in Section 4. We are working on a result related
to when $E_{2}^{2,\ast}$ in Theorem 1.3 vanishes, and when
$G=\mathbb{Z}_{p}^{\times}$, we are working on expressing the above
$E_{2}$-page in terms of $\pi_{\ast}(K(L_{p}))$, where $L_{p}$ is the
$p$-complete Adams summand (as in [3]).

We briefly discuss the input that is needed for our work. In Section 2, we
recall from [10, Theorem 1.13] that if a $\mathbb{Z}_{p}^{\times}$-spectrum
$X$ has degreewise finite homotopy groups, then it can be realized as a
discrete $\mathbb{Z}_{p}^{\times}$-spectrum. Key input for this result is the
fact that by [14, Theorem 2.9], the natural map

$H^{\ast}_{c}(\mathbb{Z}_{p},A)\to H^{\ast}(\mathbb{Z}_{p},A)$

is a graded isomorphism whenever $A$ is a finite discrete
$\mathbb{Z}_{p}$-module. In Section 3, we use the aforementioned realization
result to show that for each $i\geq 0$, $K(KU_{p})\wedge M_{i}$ is equivalent
to a discrete $\mathbb{Z}_{p}^{\times}$-spectrum. These equivalences are
possible due to an application of the thick subcategory theorem [20, Theorem
7] and the fact that $K(KU_{p})\wedge V(1)$ has degreewise finite homotopy
groups. This fact is a consequence of [1] and also work in [8] and [5], [9],
and [26] – the way these works tie together to give the finiteness result is
outlined in [10, pages 6–7].

We would like to note that besides the input described above, our
constructions of $L_{K(2)}K(KU_{p})$ – with its action by
$\mathbb{Z}_{p}^{\times}$ continuous, the continuous homotopy fixed points
$(L_{K(2)}K(KU_{p}))^{hG}$, and its homotopy fixed point spectral sequence are
low-tech. We hope that the uncomplicated nature of this work has some appeal
to the reader.

In [24, page 2], David Jongwon Lee and Ishan Levy state that work of Shay Ben
Moshe, Shachar Carmeli, Tomer Schlank, and Lior Yanovski “combined with” [18,
Theorem 1.3.6] by Jeremy Hahn, Arpon Raksit, and Dylan Wilson “shows that
$L_{K(2)}K(L_{K(1)}\mathbb{S})\to
L_{K(2)}K(KU_{p})^{h\mathbb{Z}_{p}^{\times}}$ is an equivalence.” Thus, our
understanding is that the argument/work referred to by Lee and Levy involves a
construction of $(L_{K(2)}K(KU_{p}))^{h\mathbb{Z}_{p}^{\times}}$. We do not
know any of the details of this construction and so we do not know its
relationship to the model for $(L_{K(2)}K(KU_{p}))^{h\mathbb{Z}_{p}^{\times}}$
given in the present paper.

We make a few comments about our notation. Given $t\in\mathbb{Z}$ and a
spectrum $Z$, $\pi_{t}(Z)$ denotes $[S^{t},Z]$, the set of morphisms $S^{t}\to
Z$ in the homotopy category $\mathrm{Ho}(Sp^{\Sigma})$, where here, $S^{t}$ is
a fixed cofibrant and fibrant model for the $t$-th suspension of the sphere
spectrum. By “$\operatorname*{holim}$,” we mean the homotopy limit for
$Sp^{\Sigma}$, as defined in [19, Definition 18.1.8].

### Acknowledgements

I thank John Rognes for helpful tips about the degreewise finiteness of each
$\pi_{\ast}(K(KU_{p})\wedge M_{i})$.

##  2\. A discrete $H$-spectrum naturally associated to an $H$-spectrum, when
$H$ is profinite

Let $H$ be any profinite group and let $Z$ be any spectrum. We define
$\mathrm{Sets}(H,Z)$ to be the $H$-spectrum whose $k$th pointed simplicial set
$\mathrm{Sets}(H,Z)_{k}$ has $l$-simplices

$\mathrm{Sets}(H,Z)_{k,l}:=\mathrm{Sets}(H,Z_{k,l}),$

the $H$-set of all functions $H\to Z_{k,l}$, for all $k,l\geq 0$. The
$H$-action on $\mathrm{Sets}(H,Z_{k,l})$ is given by

$(h\cdot f)(h^{\prime})=f(h^{\prime}h),\ \ \ f\in\mathrm{Sets}(H,Z_{k,l}),\
h,h^{\prime}\in H.$

By [10, Section 2], when $X$ is an $H$-spectrum, there is a cosimplicial
$H$-spectrum

$\mathrm{Sets}(H^{\bullet+1},X),$

where for $n\geq 0$, $\mathrm{Sets}(H^{\bullet+1},X)([n])$ is equal to the
application of $\mathrm{Sets}(H,-)$ iteratively $n+1$ times to $X$.

It is useful to note that if $K$ is an abstract group and $Y$ is a
$K$-spectrum, then by functoriality, $Y_{f}$ is also a $K$-spectrum and the
trivial cofibration $Y\rightarrow Y_{f}$ is $K$-equivariant. The following
definition is based on [10, Definition 4.4], but is slightly more general.

######  Definition 2.1.

Let $X$ be an $H$-spectrum and let
$\mathcal{O}=\\{N_{\lambda}\\}_{\lambda\in\Lambda}$ be an inverse system of
open normal subgroups of $H$ ordered by inclusion, over $\Lambda$, a directed
poset. We let

$\displaystyle
X^{\mathrm{dis}}_{\mathcal{O}}:=\operatorname*{colim}_{\lambda\in\Lambda}\operatorname*{holim}_{\Delta}\mathrm{Sets}(H^{\bullet+1},X_{f})^{N_{\lambda}},$

where the colimit is formed in $Sp^{\Sigma}$ and $(-)^{N_{\lambda}}$ denotes
$N_{\lambda}$-fixed points. For each $\lambda$, the finite discrete group
$H/N_{\lambda}$ acts on
$\displaystyle{\operatorname*{holim}_{\Delta}\mathrm{Sets}(H^{\bullet+1},X_{f})^{N_{\lambda}}}$,
so that the projection $H\to H/N_{\lambda}$ makes it a discrete $H$-spectrum.
By [7, Section 3.4], colimits in the category of discrete $H$-spectra are
formed in spectra, so we see that $X^{\mathrm{dis}}_{\mathcal{O}}$ is a
discrete $H$-spectrum.

By [10, Lemma 4.7, proof of Theorem 4.9], for any $H$-spectrum $X$, there is a
zigzag

$\underbrace{X\xrightarrow{\,\simeq\,}X_{f}\xrightarrow{\,\simeq\,}\operatorname*{holim}_{\Delta}\mathrm{Sets}(H^{\bullet+1},X_{f})}_{\text{the
$H$-equivariant map}\
i_{X}}\xleftarrow{\phi_{X}}X^{\mathrm{dis}}_{\mathcal{O}}$

of $H$-equivariant maps, where $i_{X}$ is a weak equivalence of spectra and
$\phi_{X}$ is induced by the inclusions
$\mathrm{Sets}(H^{\bullet+1},X_{f})^{N_{\lambda}}\to\mathrm{Sets}(H^{\bullet+1},X_{f})$
of cosimplicial spectra.

Now let $H=\mathbb{Z}_{p}^{\times}$ and let

$\mathcal{O}=\\{p^{j}\mathbb{Z}_{p}\\}_{j\geq 0},$

where each $p^{j}\mathbb{Z}_{p}$ is the open normal subgroup of
$\mathbb{Z}_{p}^{\times}$ that corresponds to the open normal subgroup
$(p^{j}\mathbb{Z}_{p})\times\\{e\\}$ of $\mathbb{Z}_{p}\times C_{p-1}$. By
[10, Theorem 1.13], if the $\mathbb{Z}_{p}^{\times}$-spectrum $X$ has
degreewise finite homotopy groups (that is, for each $t\in\mathbb{Z}$, the
abelian group $\pi_{t}(X)$ is finite), then the map $\phi_{X}$ above is a weak
equivalence in $Sp^{\Sigma}$. In this case, it is natural to identify the
$\mathbb{Z}_{p}^{\times}$-spectrum $X$ with the discrete
$\mathbb{Z}_{p}^{\times}$-spectrum $X^{\mathrm{dis}}_{\mathcal{O}}$, and
hence, for any closed subgroup $G$ of $\mathbb{Z}_{p}^{\times}$, there is the
continuous homotopy fixed point spectrum

$X^{hG}:=(X^{\mathrm{dis}}_{\mathcal{O}})^{hG}.$

We recall the general definition of the above continuous homotopy fixed point
spectrum, where $H$ is now an arbitrary profinite group. Let $(-)_{fH}$ denote
functorial fibrant replacement for the model category of discrete $H$-spectra
[7, Theorem 2.3.2]. Unpacking the meaning of this yields that if $Y$ is any
discrete $H$-spectrum, then there is a natural map $Y\to Y_{fH}$ that is
$H$-equivariant, a trivial cofibration of spectra, and has a target that is a
fibrant discrete $H$-spectrum. Then as in [7, Section 3.1], $Y^{hH}$ is
obtained by taking the right derived functor of $H$-fixed points:

$Y^{hH}:=(Y_{fH})^{H}.$

##  3\. The construction of $(L_{K(2)}K(KU_{p}))^{hG}$

As recalled in Section 1, $K(KU_{p})\wedge V(1)$ has degreewise finite
homotopy groups. It is straightforward to see that the full subcategory of the
homotopy category of $p$-local finite spectra that consists of spectra $Z$
with the property that $K(KU_{p})\wedge Z$ has degreewise finite homotopy
groups is a thick subcategory. By the thick subcategory theorem [20, Theorem
7], since $V(1)$ is in this thick subcategory, it follows that any
$K(1)$-acyclic $p$-local finite spectrum is in this subcategory, and hence,
$K(KU_{p})\wedge M_{i}$ is too, for all $i\geq 0$.

For each $i\geq 0$, we regard $K(KU_{p})\wedge M_{i}$ as a
$\mathbb{Z}_{p}^{\times}$-spectrum by letting $\mathbb{Z}_{p}^{\times}$ act
only on $K(KU_{p})$. Since each $K(KU_{p})\wedge M_{i}$ has degreewise finite
homotopy groups, it follows from Section 2 that there is a zigzag

$\bigl{\\{}K(KU_{p})\wedge
M_{i}\bigr{\\}}_{\mspace{-3.0mu}i}\mspace{-2.0mu}\xrightarrow{\,\simeq\,}\mspace{-2.0mu}\bigl{\\{}\operatorname*{holim}_{\Delta}\mathrm{Sets}(H^{\bullet+1},\mspace{-2.0mu}(K(KU_{p})\wedge
M_{i})_{f})\bigr{\\}}_{\mspace{-3.0mu}i}\mspace{-2.0mu}\xleftarrow{\,\simeq\,}\mspace{-2.0mu}\bigl{\\{}(K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\bigr{\\}}_{\mspace{-3.0mu}i}$

of morphisms of towers of $\mathbb{Z}_{p}^{\times}$-spectra, with each
morphism a levelwise weak equivalence of spectra. Also, the rightmost tower is
a tower in the category of discrete $\mathbb{Z}_{p}^{\times}$-spectra.

Let $H$ be any profinite group. We write “$N\vartriangleleft_{o}H$” to signify
that $N$ is an open normal subgroup of $H$. Notice that if $Y$ is a discrete
$H$-spectrum and $Z$ is any spectrum with trivial $H$-action, then the induced
$H$-action on

$Y\wedge Z\cong(\operatorname*{colim}_{N\vartriangleleft_{o}H}Y^{N})\wedge
Z\cong\operatorname*{colim}_{N\vartriangleleft_{o}H}(Y^{N}\wedge Z)$

makes $Y\wedge Z$ a discrete $H$-spectrum, since the projection $H\to H/N$
makes the $H/N$-spectrum $Y^{N}\wedge Z$ into a discrete $H$-spectrum.

Now we make use of the above zigzag of towers. It is useful to note that
$L_{2}(-)$ is a smashing localization: for any spectrum $Z$, $L_{2}Z\simeq
Z\wedge L_{2}S^{0}$. Then by [13, Remark 3.6] and [22, Proposition 7.10],
there is an equivalence

$L_{K(2)}K(KU_{p})\simeq\operatorname*{holim}_{i}(K(KU_{p})\wedge M_{i}\wedge
L_{2}S^{0})_{f}.$

By [7, Corollary 5.3.3], a fibrant discrete $\mathbb{Z}_{p}^{\times}$-spectrum
is fibrant as a spectrum. Also, for each $i$, $(K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge L_{2}S^{0}$ is a discrete
$\mathbb{Z}_{p}^{\times}$-spectrum, where the $\mathbb{Z}_{p}^{\times}$-action
is induced by the $\mathbb{Z}_{p}^{\times}$-action on $K(KU_{p})$ and the
trivial $\mathbb{Z}_{p}^{\times}$-action on each of $M_{i}$ and $L_{2}S^{0}$.

We put the preceding facts together to conclude that there is an equivalence

$L_{K(2)}K(KU_{p})\simeq\operatorname*{holim}_{i}((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{\mspace{0.0mu}f\mathbb{Z}_{p}^{\times}}.$

In this equivalence, the right-hand side is the homotopy limit of a tower of
discrete $\mathbb{Z}_{p}^{\times}$-spectra. Thus, it is natural to make the
following definition.

######  Definition 3.1.

Let $G$ be a closed subgroup of $\mathbb{Z}_{p}^{\times}$. We define the
continuous homotopy fixed point spectrum

$(L_{K(2)}K(KU_{p}))^{hG}:=\bigl{(}\operatorname*{holim}_{i}((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{\mspace{0.0mu}f\mathbb{Z}_{p}^{\times}}\bigr{)}^{hG},$

where

$\displaystyle\bigl{(}\operatorname*{holim}_{i}((K(KU_{p})$
$\displaystyle\wedge M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{\mspace{-1.5mu}f\mathbb{Z}_{p}^{\times}}\bigr{)}^{hG}$
$\displaystyle:=\mspace{-2.0mu}\operatorname*{holim}_{i}\bigl{(}((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{\mspace{0.0mu}f\mathbb{Z}_{p}^{\times}}\bigr{)}^{hG}\mspace{-2.0mu}$

and for each $i$, $\bigl{(}((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{\mspace{0.0mu}f\mathbb{Z}_{p}^{\times}}\bigr{)}^{hG}$ is the
continuous $G$-homotopy fixed points of the given discrete $G$-spectrum (here,
we are regarding the stated discrete $\mathbb{Z}_{p}^{\times}$-spectrum as a
discrete $G$-spectrum).

######  Remark 3.2.

The above definition follows a well-known script in algebraic $K$-theory: for
example, see [15, Proposition 3.1.2, last paragraph of 3.1, proof of Theorem
4.2.6]. For a general cofiltered diagram of discrete $G$-spectra, the script
is expressed formally in [7, Sections 4.4, 4.5]. In the case of towers – as in
Definition 3.1, [11, Lemma 8.3, Remark 8.4] shows in the setting of Bousfield-
Friedlander spectra that for a tower $\\{X_{i}\\}_{i}$ of discrete
$G$-spectra, with each $X_{i}$ a fibrant spectrum, the continuous homotopy
fixed point spectrum

$(\operatorname*{holim}_{i}X_{i})^{hG}:=\operatorname*{holim}_{i}(X_{i})^{hG}$

is the output of the right derived functor of the functor $\lim_{i}(-)^{G}$
from the category of towers of discrete $G$-spectra, equipped with the
injective model structure, to spectra, when applied to the tower
$\\{X_{i}\\}_{i}$.

######  Remark 3.3.

Let $H$ be any profinite group. There is the commutative diagram

$\textstyle{\\{Y_{i}\\}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{Y^{\prime}_{i}\\}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{(Y_{i})_{fH}\\}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{(Y^{\prime}_{i})_{fH}\\}_{i}}$

in the category of towers of discrete $H$-spectra (in $Sp^{\Sigma}$, the
default category of spectra in this paper), in which the top horizontal
morphism is an arbitrary morphism in the category that levelwise consists of
weak equivalences of spectra and the other three morphisms are given by
fibrant replacement. Thus, the two vertical maps consist of levelwise weak
equivalences of spectra. Since a morphism of discrete $H$-spectra that is a
weak equivalence of spectra is a weak equivalence of discrete $H$-spectra, it
follows that the bottom horizontal map consists of levelwise weak equivalences
between fibrant objects in the model category of discrete $H$-spectra.
Application of the right Quillen functor $(-)^{H}$ to this bottom map yields
the morphism
$\\{(Y_{i})^{hH}\\}_{i}\xrightarrow{\,\simeq\,}\\{(Y^{\prime}_{i})^{hH}\\}_{i}$
of towers of spectra that consists of levelwise weak equivalences between
fibrant spectra. Thus, there is a weak equivalence

$\operatorname*{holim}_{i}(Y_{i})^{hH}\xrightarrow{\,\simeq\,}\operatorname*{holim}_{i}(Y^{\prime}_{i})^{hH}.$

This observation implies that for any $G$ closed in $\mathbb{Z}_{p}^{\times}$,
there is the weak equivalence

$\operatorname*{holim}_{i}((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})^{hG}\xrightarrow{\,\simeq\,}\operatorname*{holim}_{i}\bigl{(}((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{f\mathbb{Z}_{p}^{\times}}\bigr{)}^{hG}.$

The target of this equivalence appears in Definition 3.1, and hence, the
source of this equivalence – which is a slightly simpler expression than the
target – can also be taken as the definition of $(L_{K(2)}K(KU_{p}))^{hG}$.

##  4\. The homotopy fixed point spectral sequence for
$(L_{K(2)}K(KU_{p}))^{hG}$

In this section, we prove Theorem 1.3. We let $G$ be a closed subgroup of
$\mathbb{Z}_{p}^{\times}$. If $H$ is a profinite group and $U$ is an open
subgroup of $H$ such that $H^{s}_{c}(U,A)=0$ for all discrete $U$-modules $A$,
whenever $s>s_{0}$, for some integer $s_{0}$, then we say that $H$ has finite
virtual cohomological dimension. Also, as is standard, $\mathrm{cd}_{p}(H)$
denotes the cohomological $p$-dimension of $H$.

The starting point is that $G$ has finite virtual cohomological dimension: the
details for this assertion are, for example, in [11, page 330] and the salient
fact is that $\mathrm{cd}_{p}(\mathbb{Z}_{p})=1$. By [7, Section 3.2, Theorem
3.2.1], if $Y$ is a discrete $G$-spectrum that is fibrant as a spectrum, then
there is a cosimplicial spectrum $\mathrm{Map}^{c}(G^{\bullet},Y)$ that has
the following properties:

  * •

there is an equivalence
$Y^{hG}\simeq\displaystyle{\operatorname*{holim}_{\Delta}\mathrm{Map}^{c}(G^{\bullet},Y)}$;

  * •

by [7, Lemma 2.4.2, Section 4.6], in cosimplicial degree $n$, where $n\geq 0$,
there is an isomorphism

$\pi_{t}\bigl{(}\mathrm{Map}^{c}(G^{\bullet},Y)([n])\bigr{)}\cong\mathrm{Map}^{c}(G^{n},\pi_{t}(Y)),\
\ \ t\in\mathbb{Z},$

where the right-hand side is the abelian group of continuous functions from
the $n$-fold product $G^{n}$ to the discrete abelian group $\pi_{t}(Y)$ (if
$n=0$, then $G^{n}=\\{e\\}$, the trivial group);

  * •

by [7, Section 4.6], there is a graded isomorphism

$H^{\ast}\bigl{[}\pi_{t}(\mathrm{Map}^{c}(G^{\ast},Y))\bigr{]}\cong
H^{\ast}_{c}(G,\pi_{t}(Y)),\ \ \ t\in\mathbb{Z},$

where the left-hand side is the cohomology of the cochain complex associated
to the cosimplicial abelian group $\pi_{t}(\mathrm{Map}^{c}(G^{\bullet},Y))$.

To simplify our notation, let

$\mathbb{K}^{\mathrm{dis}}_{i}:=((K(KU_{p})\wedge
M_{i})^{\mathrm{dis}}_{\mathcal{O}}\wedge
L_{2}S^{0})_{f\mathbb{Z}_{p}^{\times}},\ \ \ i\geq 0.$

Also, we set

$\\{i\\}:=\\{0\shortrightarrow 1\shortrightarrow
2\shortrightarrow\cdots\shortrightarrow
i\shortrightarrow\cdots\\}^{\mathrm{op}},$

the opposite category of the directed poset $\\{0\shortrightarrow
1\shortrightarrow\cdots\shortrightarrow i\shortrightarrow\cdots\\}$. Then by
applying the above recollections to Definition 3.1, we find that

(4.1)
$(L_{K(2)}K(KU_{p}))^{hG}\simeq\operatorname*{holim}_{i}\operatorname*{holim}_{\Delta}\mathrm{Map}^{c}(G^{\bullet},\mathbb{K}^{\mathrm{dis}}_{i})\cong\operatorname*{holim}_{\Delta\times\\{i\\}}\mathrm{Map}^{c}(G^{\bullet},\mathbb{K}^{\mathrm{dis}}_{i})$

and there is the conditionally convergent homotopy spectral sequence

$E_{2}^{s,t}\Longrightarrow\pi_{t-s}\Bigl{(}\,\operatorname*{holim}_{\Delta\times\\{i\\}}\mathrm{Map}^{c}(G^{\bullet},\mathbb{K}^{\mathrm{dis}}_{i})\Bigr{)},$

where

$E_{2}^{s,t}\cong\operatorname*{lim^{\mathit{s}}}_{\Delta\times\\{i\\}}\pi_{t}\bigl{(}\mathrm{Map}^{c}(G^{\bullet},\mathbb{K}^{\mathrm{dis}}_{i})\bigr{)}\cong\operatorname*{lim^{\mathit{s}}}_{\Delta\times\\{i\\}}\mathrm{Map}^{c}(G^{\bullet},\pi_{t}(\mathbb{K}^{\mathrm{dis}}_{i}))$

and the diagram
$\mathrm{Map}^{c}(G^{\bullet},\pi_{t}(\mathbb{K}^{\mathrm{dis}}_{i}))$ is the
one obtained from isomorphisms stated above. By [15, Proposition 3.1.2] (the
particular case that we have here follows from [11, Theorem 8.5 and the
paragraph after it], which is based on [15], and [7, proof of Theorem 3.2.1])
and (4.1), the above homotopy spectral sequence is the homotopy fixed point
spectral sequence

$E_{2}^{s,t}\cong
H^{s}_{\mathrm{cont}}(G;\\{\pi_{t}(\mathbb{K}^{\mathrm{dis}}_{i})\\}_{i\geq
0})\Longrightarrow\pi_{t-s}\bigl{(}(L_{K(2)}K(KU_{p}))^{hG}\bigr{)}.$

For each $i$, there are $\mathbb{Z}_{p}^{\times}$-equivariant isomorphisms

$\pi_{\ast}(\mathbb{K}^{\mathrm{dis}}_{i})\cong\pi_{\ast}(K(KU_{p})\wedge
M_{i}\wedge L_{2}S^{0})\cong\pi_{\ast}(K(KU_{p})\wedge L_{2}M_{i}),$

where in the last expression, $\mathbb{Z}_{p}^{\times}$ acts trivially on
$L_{2}M_{i}$. These isomorphisms are natural in $i$, so that

$E_{2}^{s,t}\cong H^{s}_{\mathrm{cont}}(G;\\{\pi_{t}(K(KU_{p})\wedge
L_{2}M_{i})\\}_{i\geq 0}),\ \ \ s\geq 0,\ t\in\mathbb{Z}.$

For each $i\geq 0$, let

$\mathbb{K}_{i}:=K(KU_{p})\wedge L_{2}M_{i}.$

By [23, (2.1)], for each $s\geq 0$ and every $t\in\mathbb{Z}$, there is a
short exact sequence

$0\to\operatorname*{lim^{1}}_{i}H^{s-1}_{c}(G,\pi_{t}(\mathbb{K}_{i}))\to
E_{2}^{s,t}\to\lim_{i}H^{s}_{c}(G,\pi_{t}(\mathbb{K}_{i}))\to 0,$

where $H^{-1}_{c}(G,\pi_{t}(\mathbb{K}_{i}))=0$. Since $p\geq 5$,
$\mathbb{Z}_{p}$ is the $p$-Sylow subgroup of $\mathbb{Z}_{p}^{\times}$. Thus,

$\mathrm{cd}_{p}(G)\leq\mathrm{cd}_{p}(\mathbb{Z}_{p}^{\times})=\mathrm{cd}_{p}(\mathbb{Z}_{p})=1,$

so that

$H^{s}_{c}(G,A)=0,\ \ \ s\geq 2,$

whenever $A$ is a discrete $G$-module that as an abelian group is $p$-primary
torsion. Since every

$\pi_{t}(\mathbb{K}_{i})\cong\pi_{t}(K(KU_{p})\wedge L_{2}S^{0}\wedge M_{i})$

is a unitary $(\mathbb{Z}/p^{m_{i}}\mathbb{Z})$-module,

$H^{s}_{c}(G,\pi_{t}(\mathbb{K}_{i}))=0,\ \ \ s\geq 2,\ t\in\mathbb{Z},\ i\geq
0.$

This last fact, together with the above short exact sequence, yields the
remaining assertions in Theorem 1.3.

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