Organize docs: front and back end; custom predicates.

book/src/SUMMARY.md

@@ -16,9 +16,9 @@
   - [Operations](./operations.md)
     - [Simple example](./simpleexample.md)
   - [Custom statements and custom operations](./custom.md)
-    - [Version 1](./custom1.md)
-    - [Version 1, written again](./custom1b.md)
+    - [Defining custom predicates](./custompred.md)
     - [Custom statement example](./customexample.md)
+    - [How to hash a custom statement](./customhash.md)
 - [POD types](./podtypes.md)
   - [SignedPOD](./signedpod.md)
   - [MainPOD](./mainpod.md)

book/src/backendtypes.md

@@ -1 +1,41 @@
 # Backend types
+
+On the backend, there is only a single type: `Value`.
+
+A `Value` is simply a tuple of field elements.  With the plonky2 backend, a `Value` is a tuple of 4 field elements.  In general, the backend will expose a constant `VALUE_SIZE`, and a `Value` will be a tuple of `VALUE_SIZE` field elements.
+
+## Integers and booleans
+
+The backend encoding stores integers in such a way that arithmetic operations (addition, multiplication, comparison) are inexpensive to verify in-circuit.
+
+In the case of the Plonky2 backend, an integer $x$ is decomposed as
+$$x = x_0 + x_1 \cdot 2^{32}$$
+with $0 \leq x_0, x_1 < 2^{32}$ and represented as
+$$\texttt{map}\ \iota\ [x_0, x_1, 0, 0],$$
+where $\iota:\mathbb{N}\cup\{0\}\rightarrow\texttt{GoldilocksField}$ is the canonical projection.
+
+On the backend, a boolean is stored as an integer, either 0 or 1; so logical operations on booleans are also inexpensive.
+
+## Strings
+
+The backend encoding stores strings as hashes, using a hash function that might not be zk-friendly.  For this reason, string operations (substrings, accessing individual characters) are hard to verify in-circuit.  The POD2 system does not provide methods for manipulating strings.
+
+In other words: As POD2 sees it, two strings are either equal or not equal.  There are no other relationships between strings.
+
+In the case of the Plonky2 backend, a string is converted to a sequence of bytes with the byte `0x01` appended as  padding, then the bytes are split into 7-byte chunks starting from the left, these chunks then being interpreted as integers in little-endian form, each of which is naturally an element of `GoldilocksField`, whence the resulting sequence may be hashed via the Poseidon hash function. Symbolically, given a string $s$, its hash is defined by
+
+$$\texttt{poseidon}(\texttt{map}\ (\iota\circ\jmath_\texttt{le-bytes->int})\ \texttt{chunks}_7(\jmath_\texttt{string->bytes}(s)\ \texttt{++}\ [\texttt{0x01}])),$$
+
+where `poseidon` is the Poseidon instance used by Plonky2, $\iota$ is as above, $\texttt{chunks}_{n}:[\texttt{u8}]\rightarrow [[\texttt{u8}]]$ is defined such that[^aux]
+
+$$\texttt{chunks}_n(v) = \textup{if}\ v = [\ ]\ \textup{then}\ [\ ]\ \textup{else}\ [\texttt{take}_n v]\ \texttt{++}\ \texttt{chunks}_n(\texttt{drop}_n v),$$
+
+the mapping $\jmath_\texttt{le-bytes->int}: [u8] \rightarrow{N}\cup\{0\}$ is given by
+
+$$[b_0,\dots,b_{N-1}]\mapsto \sum_{i=0}^{N-1} b_i \cdot 2^{8i},$$
+
+and $\jmath_\texttt{string->bytes}$ is the canonical mapping of a string to its UTF-8 representation.
+
+## Compound types
+
+The three front-end compound types (`Dictionary`, `Array`, `Set`) are all represented as Merkle roots on the backend.  The details of the representation are explained on a separate [Merkle tree](./merkletree.md) page.

book/src/custom.md

@@ -10,9 +10,33 @@ At the backend level, every definition of a predicate is either a conjunction or
 
 On the backend, custom predicates are defined in _groups_.  A group can contain one or more custom predicates and their associated sub-predicates.  Recursive definition is only possible within a group: the definition of a predicate in a group can only depend on previously existing predicates, itself, and other predicates in the same group.
 
-## Custom predicates and their IDs
+## Arguments of custom predicates
+
+The definition of a custom predicate might also be called an _operation_ or _deduction rule_.  It includes two (or, potentially, say, five) statement templates as conditions.  The arguments to the statement templates are decomposed as (origin, key) pairs: if statements are allowed to have arity at most 4, then the statement templates in a deduction rule will have at most 8 arguments (4 origins and 4 keys).  The same holds for the output statement.
+
+Each argument (origin or key) to an statement template is either a wildcard or a literal.  In the backend, the wildcard arguments will be identified as *1, *2, *3, ....
+
+## Examples
+
+See [examples](./customexample.md)
+
+## Hashing and predicate IDs
+
+Each custom predicate is defined as part of a _group_ of predicates. The definitions of all statements in the group are laid out consecutively (see [examples](./customexample.md)) and hashed.  For more details, see the pages on [hashing custom statements](./customhash.md) and [custom predicates](./custompred.md).
+
+## How to prove an application of an operation
+
+The POD proof format is inspired by "two-column proofs" (for an example, see [Wikipedia](https://en.wikipedia.org/wiki/Mathematical_proof)).  A POD contains a "tabular proof", in which each row includes a "statement" and an "operation".  The "operation" is the "reason" that justifies the statement: it is everything the circuit needs as a witness to verify the statement.
+
+For a custom statement, the "reason" includes the following witnesses and verifications:
+- the definition of the statement, serialized (see [examples](./customexample.md))
+  - if the statement is part of a group, the definition of the full group, serialized
+- verify that the hash of the definition is the statement ID
+- the definition will have some number of "wildcards" (*1, *2, ...) as arguments to statement templates; a value for each wildcard must be provided as a witness (each will be either an origin ID or key)
+- the circuit must substitute the claimed values for the wildcards, and the resulting statements (true statements with origins and keys) will appear as witnesses
+- the circuit must verify that all the input statement templates (with origins and keys) appear in the previous statements (in higher rows of the table)
+- the circuit also substitutes the claimed values for the wildcards in the output statement, and verifies that it matches the claimed output statement
 
-A custom predicate, like a built-in predicate, is identified by a _name_ on the front end and an _identifier_ on the back end.  The identifier is a cryptographic hash of the definition of the group.
 
 
 [^inherit]: What to call this?  One POD "inherits" from another?

book/src/customhash.md

@@ -0,0 +1,20 @@
+# How to hash a custom predicate
+
+Every predicate, native or custom, is identified on the backend by a predicate ID.
+
+The native predicates are numbered with small integers, sequentially.  The ID of a custom predicate is a hash of its definition; this guarantees that two different predicates cannot have the same ID (aside from the miniscule probability of a hash collision).
+
+This document explains in some detail how the definition of a custom predicate is serialized and hashed.
+
+Custom predicates are defined in _groups_ (also known as _batches_); see an [example](./customexample.md).  The definition of a custom predicate in a group involves other predicates, which may include:
+- native predicates
+- previously-defined custom predicates
+- other predicates in the same group.
+
+Predicate hashing is recursive: in order to hash a group of custom predicates, we need to know IDs for all the previously-defined custom predicates it depends on.
+
+The definition of the whole group of custom predicates is serialized (as explained below), and that serialization is hashed (using a zk-friendly hash -- in the case of the plonky2 backend, Poseidon) to give a _group ID_.  Each predicate in the group is then referenced by
+```
+predicate_ID = (group_ID, idx)
+```
+(here `idx` is simply the index of the predicate in the group).

book/src/custom1.md → book/src/custompred.md

@@ -1,4 +1,4 @@
-# Custom operations (or: how to define a custom predicate): VERSION 1
+# Defining custom predicates
 
 (Note: At the moment, we consider a "custom operation" to be exactly the same thing as the "definition of a custom predicate.")
 

book/src/front_and_back.md

@@ -1,4 +1,17 @@
 # Frontend and backend
 
-The frontend is what we want the user to see.
-The backend is what we want the circuit to see.
+The POD2 system consists of a frontend and a backend, connected by a middleware.  This page outlines some design principles for deciding which components go where.
+
+```
+user -- frontend -- middleware -- backend -- ZK circuit
+```
+
+The frontend is what we want the user to see; the backend is what we want the circuit to see.
+
+## Circuit and proving system
+
+The first implementation of POD2 uses Plonky2 as its proving system.  In principle, a future implementation could use some other proving system.  The frontend and middleware should not be aware of what proving system is in use: anything specific to the proving system belongs to the backend.
+
+## User-facing types versus in-circuit types
+
+The frontend type system exposes human-readable types to POD developers: strings, ints, bools, and so forth.  On the backend, all types are build out of field elements.  The middleware should handle the conversion.

book/src/values.md

@@ -1,7 +1,9 @@
 # POD value types
 From the frontend perspective, POD values may be one of the following[^type] types: two atomic types
 - `Integer`
+- `Bool`
 - `String`
+- `Raw`
 
 and three compound types
 - `Dictionary`
@@ -24,6 +26,8 @@ with $0 \leq x_0, x_1 < 2^{32}$ and representing it as
 $$\texttt{map}\ \iota\ [x_0, x_1, 0, 0],$$
 where $\iota:\mathbb{N}\cup\{0\}\rightarrow\texttt{GoldilocksField}$ is the canonical projection.
 
+## `Bool`
+In the frontend, this is a simple bool.  In the backend, it will have the same encoding as an `Integer` `0` (for `false`) or `1` (for `true`).
 
 ## `String`
 In the frontend, this type corresponds to the usual `String`. In the backend, the string will be mapped to a sequence of field elements and hashed with the hash function employed there, thus being represented by its hash.
@@ -42,7 +46,10 @@ $$[b_0,\dots,b_{N-1}]\mapsto \sum_{i=0}^{N-1} b_i \cdot 2^{8i},$$
 
 and $\jmath_\texttt{string->bytes}$ is the canonical mapping of a string to its UTF-8 representation.
 
+## `Raw`
+"Raw" is short for "raw value".  A `Raw` exposes a backend value on the frontend.
 
+With the plonky2 backend, a `Raw` is a tuple of 4 elements of the Goldilocks field.
 
 ## Dictionary, array, set