39 KiB
Examples of usage of dds
Introduction
This document shows some examples of usage of the modules in dds
with Org-mode. It relies on emacs-ob-racket.
The following section describes how Org-mode can interact with
Racket, and how this interaction can be used for a fluid workflow
with dds. In particular, the code block munch-sexp is defined
in this section.
The subsequent sections show off some the functionalities of the
submodules of dds.
Org-mode, Racket, and dds <<intro>>
Installing dds locally
To install dds locally, you can simply run the following command
in dds.
raco pkg install
After this installation, you can import dds modules by simply
doing the following:
#lang racket
(file dds/networks)
(file dds/utils)Importing a module from file old
To require the modules from the files of dds, you can use the
following code (I only reset the prelude here because I set at the
top of this file):
#lang racket
(require (file "~/Candies/prj/racket/dds/networks.rkt"))
(require (file "~/Candies/prj/racket/dds/utils.rkt"))
Note that this code will not work with :results value. I think it
is because in this case the code is not really evaluated at top
level.
These initialisation lines can be put into the prologue of every
code block in a subtree by setting :prologue via
:header-args:racket: in the properties drawer. Check out the
properties drawer of this section for an example.
Alternatively, this property can be set via a #+PROPERTY line at
the top the file. For example, this file has such a line. Whenever
this property line changes, refresh the setup of the file by hitting
C-c C-c on the property line. This will update the prologue for
all racket code blocks.
Finally, you can also set :prologue (and other properties with
long values) in the following way:
(make-state '((a . 1)))'#hash((a . 1))
Output formats for results of evaluation of code blocks
This section of the Org manual describes various different formats
for presenting the results of code blocks. I find the following
three particularly useful as of [2020-02-22 Sat]: output, list,
and table.
The output result format is the simplest and the most natural
one. It works as if the code block were inserted into a module
which would then be evaluated.
(println "This is the first line of output.")
(println (+ 1 2))
(println "This the third line of output.")"This is the first line of output." 3 "This the third line of output."
The list result format typesets the result of the last line in the
code block as a list:
'(1 "hello" (and x y))- 1
- "hello"
- (and x y)
Note how nested lists are not recursively shown as nested Org-mode lists.
For some reason, the list output format does not work with the
result drawer:
'(1 "hello" (and x y))- (1 "\"hello\"" (and x y))
Finally, the table result format typesets the output as a table:
'((a . #t) (b . #f))| a | #t |
| b | #f |
This is clearly very useful for printing states (and hash tables, more generally):
(make-state '((a . 1) (b . #f) (c . "hello")))| a | 1 |
| b | #f |
| c | "hello" |
A note about printing update function forms
Automatic table typesetting may go in the way of readability for hash tables whose values are lists, as the following example shows:
#hash((a . (and a b)) (b . (not b)))| a | and | a | b |
| b | not | b |
To tackle this issue, dds/utils provides
stringify-variable-mapping (with the shortcut sgfy) which
converts all the values of a given variable mapping to strings:
(stringify-variable-mapping #hash((a . (and a b)) (b . (not b))))| a | "(and a b)" |
| b | "(not b)" |
Passing values between code blocks
The :var header argument allows using the output of a code block
as an input of another one. For example:
'(4 2)'(4 2)
Here's how you use its output:
input"'(4 2)\n"
The parentheses when calling block-1 in the header of the second
code block are optional.
There are a two main problems with what we see in the second code block: there is a trailing newline and a leading quote. The trailing newline is not hard to drop, but the leading quote is trickier:
(unorg input)''(4 2)
I had much trouble understanding how to get rid of the second
quote, I even thought it was impossible. To understand the
solution, replace ' with quote: ''(4 2) becomes (quote
(quote (4 2))). Keeping in mind that (quote (4 2)) is the
same thing as '(4 2), this expression effectively defines a list
whose first element is quote and whose second element is (4
2). Therefore, to get '(4 2) out of ''(4 2), I need to simply
get the second element out of the list with double quote:
(cadr (unorg input))'(4 2)
There's a simpler way to avoid having to deal with the double quote
altogether: use value instead of output in :results.
'(4 2)(unorg input)'(4 2)
The only way to pass values between Org-babel code blocks is by writing them to strings. I have looked around for passing the values natively, but it doesn't seem possible, and it actually makes sense: code blocks may be written in different languages, and the most natural way to pass values between them is by serialising to strings.
A good option for passing around native values is by using Noweb references. It does require adapting both code blocks however.
Reading Org-mode tables<<tabread>>
Org-mode allows supplying tables as arguments for code blocks.
| a | "(and a b)" |
| b | (or b (not a)) |
tab((a (and a b)) (b (or b (not a))))
Unfortunately, the same trick does not work with Racket directly, because Racket interprets the first elements in the parentheses as function applications:
tabapplication: not a procedure; expected a procedure that can be applied to arguments given: "a" arguments…: "(and a b)" context…: "/tmp/babel-qkvrRR/org-babel-c4wuju.rkt": [running body] temp37_0 for-loop run-module-instance!125 perform-require!78
Fortunately, we can easily remedy this problem by creating a named parameterised Elisp source block which will explicitly convert the table to a string:
(prin1 sexp)(("a" "(and a b)") ("b" "(or b (not a))"))
We can now correctly receive this table in a Racket source code
block by threading it through munch-sexp:
(println tab)"((\"a\" \"(and a b)\") (\"b\" \"(or b (not a))\"))"
dds/utils has several functions for parsing such strings, and
notably read-org-variable-mapping, with the shortcut unorgv:
(unorgv tab)'#hash((a . (and a b)) (b . (or b (not a))))
Of course, we can use munch-sexp to prepare any other table than
test-table for use with Racket:
| a | (not a) |
| b | (and a c) |
| c | (and a (not b)) |
(unorgv tab)'#hash((a . (not a)) (b . (and a c)) (c . (and a (not b))))
Inline graph visualisation with Graphviz
Some functions in dds build graphs:
(build-syntactic-interaction-graph (unorgv bf))#<unweighted-graph>
The graph library allows building a Graphviz description of the
constructed graph. (Note that you have to install the graph
library by running raco pkg install graph and require it. The
long property line at the top of this file defining the prologue for
racket source code blocks takes care of requiring graph.)
(display (graphviz (build-syntactic-interaction-graph (unorgv bf))))digraph G { node0 [label="c"]; node1 [label="b"]; node2 [label="a"]; subgraph U { edge [dir=none]; node0 -> node1; node2 -> node2; } subgraph D { node2 -> node0; node2 -> node1; } }
You can have an inline drawing of this graph by calling the previous
code block (igraph) via a noweb reference in Graphviz/DOT source
block:
<<igraph()>>
Note that the graph library draws self-loops as undirected edges.
It also draws double-sided edges as undirected edges (e.g., in the
preceding graph, b depends on c and c depends on b).
dds/networks and dds/functions
The dds/networks is a module for working with different network
models. A network is a set of variables which are updated according
to their corresponding update functions. The variables to be
updated at each step are given by the mode. This model can
generalise Boolean networks, TBANs, multivalued networks, etc.
dds/functions is a module for working with the functions
underlying the network models. Similarly to dds/networks, it
provides primitives for tabulating functions, reconstructing
functions from tables, generating random functions, etc.
Boolean networks
Consider the following Boolean network:
| a | b |
| b | (and (not a) c) |
| c | (not c) |
Note that if you define the formula of a as 0, it will set a to 1,
because 0 is not #f. For example, (if 0 1 2) evaluates to 1, and
not to 2.
Here's the unsigned syntactic interaction graph of this network:
((compose
dotit
build-syntactic-interaction-graph
make-boolean-network-form
unorgv)
simple-bn)<<simple-bn-syig()>>
Note that, while this definition is an easy one to check structurally, this is not how interaction graphs are typically defined. An interaction graph is usually defined based on the dynamics of the network: an arrow from a variable x to a variable y means that varying x and only x may have an influence on the value of y. It is easy to imagine a situation in which the syntactic interaction graph does not in fact agree with this criterion, the simplest example being the network y = x ∧ ¬ x.
Here is the unsigned interaction graph of the same network, this time constructed according to the canonical definition:
((compose
dotit
build-interaction-graph/form
make-boolean-network-form
unorgv)
simple-bn)<<simple-bn-ig()>>
In this particular case, the syntactic interaction graph is the same as the interaction graph constructed according to the conventional definition. This however may not necessarily be the case for all networks.
The function build-interaction-graph/form builds the interaction
graph from the syntactic definition of the network. For an already
built network, you can use build-interaction-graph.
Here's the signed interaction graph of this network:
((compose
dotit
build-signed-interaction-graph/form
make-boolean-network-form
unorgv)
simple-bn)<<simple-bn-sig()>>
Here is the full state graph of this network under the asynchronous dynamics:
((compose
dotit
pretty-print-state-graph
build-full-state-graph
make-asyn-dynamics
forms->boolean-network
unorgv)
simple-bn)<<simple-bn-sg()>>
Alternatively, you may prefer a slighty more compact representation of Boolean values as 0 and 1:
((compose
dotit
pretty-print-boolean-state-graph
build-full-state-graph
make-asyn-dynamics
forms->boolean-network
unorgv)
simple-bn)<<simple-bn-sg-bool()>>
Consider the following state (appearing in the upper left corner of the state graph):
| a | 0 |
| c | 1 |
| b | 1 |
These are the states which can be reached from it in at most 2 steps:
(let* ([bn (forms->boolean-network (unorgv simple-bn))]
[bn-asyn (make-asyn-dynamics bn)]
[s0 (booleanize-state (unorgv some-state))])
(dotit (pretty-print-boolean-state-graph (dds-build-n-step-state-graph bn-asyn (set s0) 2))))<<simple-bn-some-state()>>
Here is the complete state graph with edges annotated with the modality leading to the update.
((compose
dotit
pretty-print-boolean-state-graph
build-full-state-graph-annotated
make-asyn-dynamics
forms->boolean-network
unorgv)
simple-bn)<<simple-bn-sg-bool-ann()>>
For some networks, a single transition between two states may be due to different modalities. Consider the following network:
| a | (not b) |
| b | b |
((compose
dotit
pretty-print-boolean-state-graph
build-full-state-graph-annotated
make-asyn-dynamics
forms->boolean-network
unorgv)
input-bn)<<bn2-sgr()>>
Tabulating functions and networks
Here's how you can tabulate a function. The domain of x is {1, 2}, and the domain of y is {0, 2, 4}. The first column in the output corresponds to x, the second to y, and the third corresponds to the value of the function.
(tabulate (λ (x y) (+ x y)) '((1 2) (0 2 4)))| 1 | 0 | 1 |
| 1 | 2 | 3 |
| 1 | 4 | 5 |
| 2 | 0 | 2 |
| 2 | 2 | 4 |
| 2 | 4 | 6 |
Here's how you tabulate a Boolean function:
(tabulate/boolean (λ (x y) (and x y)))| #f | #f | #f |
| #f | #t | #f |
| #t | #f | #f |
| #t | #t | #t |
You can tabulate multiple functions taking the same arguments over the same domains together.
(tabulate*/boolean `(,(λ (x y) (and x y)) ,(λ (x y) (or x y))))| #f | #f | #f | #f |
| #f | #t | #f | #t |
| #t | #f | #f | #t |
| #t | #t | #t | #t |
Here's how to tabulate the network simple-bn, defined at the top
of this section:
(tabulate-network (forms->boolean-network (unorgv in-bn)))| a | b | c | f-a | f-b | f-c |
| #f | #f | #f | #f | #f | #t |
| #f | #f | #t | #f | #t | #f |
| #f | #t | #f | #t | #f | #t |
| #f | #t | #t | #t | #t | #f |
| #t | #f | #f | #f | #f | #t |
| #t | #f | #t | #f | #f | #f |
| #t | #t | #f | #t | #f | #t |
| #t | #t | #t | #t | #f | #f |
Random functions and networks
To avoid having different results every time a code block in this section is run, every code block seeds the random number generator to 0.
dds/networks can generate random functions, given a domain for
each of its arguments and for the function itself. Consider the
following domains:
| a | (#f #t) |
| b | (1 2) |
| c | (cold hot) |
Here's a random function taking values in the codomain (4 5 6):
(random-seed 0)
(define rnd-func (random-function/state (unorgv simple-domains) '(4 5 6)))
(tabulate-state rnd-func (unorgv simple-domains))| a | b | c | f |
| #f | 1 | cold | 4 |
| #f | 1 | hot | 5 |
| #f | 2 | cold | 4 |
| #f | 2 | hot | 4 |
| #t | 1 | cold | 5 |
| #t | 1 | hot | 6 |
| #t | 2 | cold | 4 |
| #t | 2 | hot | 5 |
We can build an entire random network over these domains:
(random-seed 0)
(define n (random-network (unorgv simple-domains)))
(tabulate-network n)| a | b | c | f-a | f-b | f-c |
| #f | 1 | cold | #f | 2 | hot |
| #f | 1 | hot | #f | 2 | cold |
| #f | 2 | cold | #t | 1 | cold |
| #f | 2 | hot | #t | 2 | hot |
| #t | 1 | cold | #f | 2 | cold |
| #t | 1 | hot | #t | 1 | cold |
| #t | 2 | cold | #f | 2 | hot |
| #t | 2 | hot | #t | 1 | cold |
Let's snapshot this random network and give it a name.
| a | b | c | f-a | f-b | f-c |
| #f | 1 | cold | #f | 2 | hot |
| #f | 1 | hot | #f | 2 | cold |
| #f | 2 | cold | #t | 1 | cold |
| #f | 2 | hot | #t | 2 | hot |
| #t | 1 | cold | #f | 2 | cold |
| #t | 1 | hot | #t | 1 | cold |
| #t | 2 | cold | #f | 2 | hot |
| #t | 2 | hot | #t | 1 | cold |
Here's how we can read back this table as a network:
(string->any rnd-network)'(("a" "b" "c" "f-a" "f-b" "f-c") ("#f" 1 "cold" "#f" 2 "hot") ("#f" 1 "hot" "#f" 2 "cold") ("#f" 2 "cold" "#t" 1 "cold") ("#f" 2 "hot" "#t" 2 "hot") ("#t" 1 "cold" "#f" 2 "cold") ("#t" 1 "hot" "#t" 1 "cold") ("#t" 2 "cold" "#f" 2 "hot") ("#t" 2 "hot" "#t" 1 "cold"))
You can use table->network to convert a table such as rnd-network
to a network.
(table->network (unorg rnd-network))(network '#hash((a . #<procedure:…ds/functions.rkt:145:4>) (b . #<procedure:…ds/functions.rkt:145:4>) (c . #<procedure:…ds/functions.rkt:145:4>)) '#hash((a . (#f #t)) (b . (1 2)) (c . (cold hot))))
Here's the state graph of rnd-network.
((compose
dotit
pretty-print-state-graph
build-full-state-graph-annotated
make-asyn-dynamics
table->network
unorg)
rnd-network)<<rnd-network-sg()>>
Here's the signed interaction graph of rnd-network.
((compose
dotit
build-signed-interaction-graph
table->network
unorg)
rnd-network)<<rnd-network-ig()>>
Standalone threshold Boolean functions (TBF)
Note: Before using the objects described in this section, consider whether the objects from the section on TBN aren't a better fit.
dds includes some useful definitions for working with threshold
Boolean functions (TBF). A TBF is defined as a vector of weights
and a threshold. For example, the following defines a
TBF implementing the logical AND.
(tbf #(1 1) 1)(tbf '#(1 1) 1)
This TBF only returns 1 when both inputs are activated, which brings their weighted some above 1: 1 ⋅ 1 + 1 ⋅ 1 = 2 > 1.
(apply-tbf (tbf #(1 1) 1) #(1 1))1
Let's actually check out the truth table of this TBF.
(tbf-tabulate (tbf #(1 1) 1))| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
This truth table corresponds indeed to the logical AND.
dds allows reading TBFs from Org tables. In this case, the last
column in each row is treated as the threshold, while the first
values are taken to be the weights. Consider, for example, the
following table:
| 1 | 1 | 0 |
| 1 | 1 | 1 |
You can read the two TBFs defined in this table in the following way:
(read-org-tbfs simple-tbf)(list (tbf '#(1 1) 0) (tbf '#(1 1) 1))
The first TBF implements the logical OR of its inputs, while the second TBF implements the logical AND. Let's check it by tabulating both functions:
(tbf-tabulate* (read-org-tbfs simple-tbf))| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 |
The first two columns of this table give the values of the two inputs. The third column gives the values of the first TBF, and the fourth column gives the values of the second TBF.
dds also provides a couple shortcuts to deal with SBF—sign
Boolean functions. SBF are TBF with threshold equal to 0:
(sbf? (tbf #(1 1) 1))
(sbf? (tbf #(1 1) 0))#f #t
You can read SBFs from Org tables, like TBFs:
| 1 | -1 |
| 2 | 2 |
(read-org-sbfs some-sbfs)(list (tbf '#(1 -1) 0) (tbf '#(2 2) 0))
Threshold Boolean networks (TBN) <<tbn>>
dds includes a number of useful definitions for working with
threshold Boolean networks: networks of threshold Boolean
functions. Since, standalone TBF do give names to their inputs,
dds also defines TBF operating on states:
(make-tbf/state '((a . 1) (b . 1)) 1)(tbf/state '#hash((a . 1) (b . 1)) 1)
As the example standalone TBF, this TBF only returns 1 when both inputs are activated:
(apply-tbf/state (make-tbf/state '((a . 1) (b . 1)) 1)
(make-state '((a . 1) (b . 1))))1
Here's how you can read this TBF from an Org-mode table:
| a | b | θ |
| 1 | 1 | 1 |
(read-org-tbfs/state simple-tbf/state)(list (tbf/state '#hash((a . 1) (b . 1)) 1))
Note that the header of the rightmost column is discarded.
read-org-tbfs/state can also read multiple TBFs at once:
| a | b | θ |
| 1 | 1 | 1 |
| -2 | 1 | 0 |
(read-org-tbfs/state simple-tbfs/state)(list (tbf/state '#hash((a . 1) (b . 1)) 1) (tbf/state '#hash((a . -2) (b . 1)) 0))
You can print a list of TBFs in the following way:
(print-org-tbfs/state (read-org-tbfs/state simple-tbfs/state))| a | b | θ |
| 1 | 1 | 1 |
| -2 | 1 | 0 |
All TBFs given to print-org-tbfs/state mush have exactly the same
inputs. This function does not check this property.
Here's how you can tabulate both of these TBFs in the same table (e.g., to compare their truth tables):
(tbf/state-tabulate* (read-org-tbfs/state simple-tbfs/state))| a | b | f1 | f2 |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 0 |
dds also includes functions for dealing with SBF operating on
states. In particular, to create an SBF, you can do:
(make-sbf/state '((a . -1) (b . 1)))(tbf/state '#hash((a . -1) (b . 1)) 0)
Most of the functions operating on TBF can be directly applied to SBF and therefore have no specialization for this particular case. However, there are variants of reading and printing functions for SBF.
Consider the following table giving the weights of two SBF:
| a | b |
| 1 | 1 |
| -2 | 1 |
You can read these SBFs in the following way:
(read-org-sbfs/state simple-sbfs/state)(list (tbf/state '#hash((a . 1) (b . 1)) 0) (tbf/state '#hash((a . -2) (b . 1)) 0))
You can print them back to an Org-mode table as follows:
(print-org-sbfs/state (read-org-sbfs/state simple-sbfs/state))| a | b |
| 1 | 1 |
| -2 | 1 |
Finally, dds defines utilities for working with TBN (networks of
TBF). For example, here is how you can define and read a TBN from
a table:
| - | x | y | θ |
| y | -1 | 0 | -1 |
| x | 0 | -1 | -1 |
(read-org-tbn tbfs-nots)(hash 'x (tbf/state '#hash((x . 0) (y . -1)) -1) 'y (tbf/state '#hash((x . -1) (y . 0)) -1))
To take a look at the behaviour of this TBN, you can convert it to
a network using tbn->network and build its state graph:
(dotit (build-tbn-state-graph (read-org-tbn tbfs-nots)))<<tbfs-nots-sg()>>
For convenience, there is a similar function read-org-sbn which
allows reading an SBN.
| - | A | B | C |
| A | -1 | 1 | 2 |
| B | 2 | -2 | -2 |
| C | -1 | 2 | -1 |
(dotit (build-tbn-state-graph (read-org-sbn sbn-figure2)))<<sbn-figure2-sg()>>
You can print TBNs using print-org-tbn in the following way:
(print-org-tbn (read-org-sbn sbn-figure2))| - | A | B | C | θ |
| A | -1 | 1 | 2 | 0 |
| B | 2 | -2 | -2 | 0 |
| C | -1 | 2 | -1 | 0 |
For convenience, dds also includes print-org-sbn, which allows
you to chop off the threshold column, which only contains zeros
anyway for the case of SBN:
(print-org-sbn (read-org-sbn sbn-figure2))| - | A | B | C |
| A | -1 | 1 | 2 |
| B | 2 | -2 | -2 |
| C | -1 | 2 | -1 |
dds also defines functions to draw the interaction graphs
of TBNs:
(dotit (tbn-interaction-graph (read-org-tbn tbfs-nots)))<<tbfs-nots-ig()>>
tbn-interaction-graph can optionally omit edges with zero weight:
(dotit (tbn-interaction-graph (read-org-tbn tbfs-nots) #:zero-edges #f))<<tbfs-nots-ig-no0()>>
You can print the note labels in a slightly prettier way using
pretty-print-tbn-interaction-graph:
(dotit (pretty-print-tbn-interaction-graph (tbn-interaction-graph (read-org-tbn tbfs-nots))))<<tbfs-nots-ig-pp()>>
As usual, dds includes a specific function for constructing the
interaction graph of SBN. This function does not include the
thresholds of the SBF in the interaction graph.
(dotit (sbn-interaction-graph (read-org-sbn sbn-figure2)))<<sbn-figure2-ig()>>
Reaction systems
Consider the following reaction system:
| a | x t | y | z |
| b | x | q | z |
Here is how we read this reaction into Racket code:
(read-org-rs input-rs)(hash 'a (reaction (set 'x 't) (set 'y) (set 'z)) 'b (reaction (set 'x) (set 'q) (set 'z)))
Here is how we can put it back into an Org-mode table:
(rs->ht-str-triples (read-org-rs input-rs))| a | "t x" | "y" | "z" |
| b | "x" | "q" | "z" |
Here is how we can apply this reaction system to a state:
(let ([rs (read-org-rs input-rs)])
(apply-rs rs (set 'x 't)))(set 'z)
Let's see which reactions got applied:
(let ([rs (read-org-rs input-rs)])
(list-enabled rs (set 'x 't)))- a
- b
You can also give a name to a list and read it with munch-sexp:
- x y
- z
- t
(read-context-sequence input-ctx)(list (set 'x 'y) (set 'z) (set) (set 't))
Let's see what the evolution of rs1 looks like with the context
sequence ctx1.
(dotit (pretty-print-state-graph (build-interactive-process-graph (read-org-rs input-rs) (read-context-sequence input-ctx))))digraph G { node0 [label="C:{z}{}{t}\nD:{z}"]; node1 [label="C:{}{t}\nD:{}"]; node2 [label="C:{x y}{z}{}{t}\nD:{}"]; node3 [label="C:{t}\nD:{}"]; node4 [label="C:\nD:{}"]; subgraph U { edge [dir=none]; node4 -> node4 [label="{}"]; } subgraph D { node0 -> node1 [label="{}"]; node1 -> node3 [label="{}"]; node2 -> node0 [label="{b}"]; node3 -> node4 [label="{}"]; } }
<<rs1-sgr()>>
Note that this graph includes the full context sequence in the name
of each state, which is how the states are represented in the
dynamics of reaction systems. You can use
build-reduced-state-graph to construct a similar graph, but
without the context sequences.
(dotit (pretty-print-reduced-state-graph
(build-reduced-state-graph (read-org-rs input-rs)
(read-context-sequence input-ctx))))<<rs1-sgr-no-ctx()>>
The graphical presentation for interactive processes is arguably less readable than just listing the contexts and the results explicitly. Here is how you can do it.
(build-interactive-process (read-org-rs input-rs) (read-context-sequence input-ctx))| (y x) | nil |
| (z) | (z) |
| nil | nil |
| (t) | nil |
| nil | nil |
The first column of this table shows the current context. The second column shows the result of application of the reactions to the previous state. The interactive process contains one more step with respect to the context sequence. This is to show the effect of the last context.
Note that empty sets are printed as nil.