# Inpla: Interaction nets as a programming language ## What is Inpla Inpla is a multi-threaded parallel interpreter of interaction nets. Once you write programs for sequential execution, it works also in multi-threaded parallel execution. Each thread is managed on each CPU-core with POSIX-thread library. ![speedup-ratio](pic/benchmark_reuse.png) | | Haskell | SML | Python | Inpla1 | Inpla1_r | Inpla7 | Inpla7_r | | -------------- | :------: | :------: | :----: | :----: | :------: | :----: | :------: | | ack(3,11) | 2.31 | **0.41** | - | 4.76 | 4.36 | 0.98 | 0.88 | | fib 38 | 1.60 | **0.26** | 8.49 | 3.59 | 3.56 | 0.56 | 0.54 | | bsort 40000 | 34.81 | 11.17 | 76.72 | 22.85 | 18.40 | 5.53 | **2.94** | | isort 40000 | **0.02** | 2.97 | 36.63 | 10.59 | 8.74 | 2.59 | 1.43 | | qsort 800000 | **0.15** | 1.16 | 97.30 | 1.85 | 1.55 | 0.76 | 0.41 | | msort 800000 | **0.46** | 1.00 | 98.27 | 1.18 | 1.34 | 0.65 | 0.49 | - **Comparison in execution time** with other implementations: **Haskell** (GHC version 8.10.7), **Standard ML** v110.74 (interpreter mode) and **Python** 3.8.5 in execution time. - The above table contains execution time in second on average of ten times execution by using Linux PC (Core i7-9700 (8 threads, no Hyper-threading), 16GB memory). The fastest one is shown with bold style. There are scripts in the `comparison` directory. - Inpla*n* and Inpla*n*_**r** mean *n* threads without/with reuse-annotated execution, respectively. - "ack(3,11)" is computation of Ackermann function. There is a blank in Python due to stack size limitation error. - "fib 38" is computation to get the 38th Fibonacci number. - "bsort *n*", "isort *n*", "qsort *n*" and "msort *n*" are computation of bubble sort, insertion sort, quick sort and merge sort for random *n*-element lists, respectively. ### Feature of Version 0.4.2 - Line edit supports multi-line inputs. ### Feature of Version 0.4.1 - Integer numbers can be written the same as one of the first-class objects. - Interaction rules can re-allocate heaps of the rule agents to agents in nets. This re-allocation is specified by modifications such as (\*L), (\*R), called reuse annotation [1], to agents in nets. This re-allocation can improve execution performance in parallel. This is an experimental, thus in future version these should be decided automatically, so please do not care so much. - Weak reduction strategy is supported. It turns on by invoked with ```-w``` option, and then only connections that have living names are evaluated. - Nested guards in conditional rules are supporeted. ## Getting started - Requirement - gcc (>= 4.0), flex, bison - Build - Single-thread version: Use `make` command as follows (the symbol `$` means a shell prompt): ``` $ make ``` - Multi-thread version: Use `make` with `thread` option (it may also need `make clear` before that): ``` $ make thread ``` ## How to execute ### Interactive mode (single-thread version) - Inpla starts in the interactive mode by typing the following command (where the symbol `$` is a shell prompt): ``` $ ./inpla Inpla 0.4.1 : Interaction nets as a programming language [built: 21 Sept. 2021] >>> ``` - The symbol `>>>` is a prompt of Inpla. After the prompt you can write rules and nets. For instance, the following is a rule for incrementation `inc` and a net to bind the increment result of `10` to a name `r` (`//` is a comment): ``` >>> inc(ret) >< (int i) => ret~(i+1); // a rule for inc >< (int i) >>> inc(r)~10; // a net (1 interactions, 0.16 sec) >>> r; // show a connected net from the r 11 >>> ``` - To quit this system, use `exit` command: ``` >>> exit; ``` ### Interactive mode (multi-thread version) - There is an execution option `-t` in order to specify the number of threads in a thread pool. For instance, by invoking with `-t 4` Inpla starts with 4 threads in the pool: ``` $ ./inpla -t 4 ``` ### Batch mode and sample files - Inpla has also the batch mode in which a file is evaluated. This is available when invoked with an execution option `-f` *filename*. There are sample files in the `sample` folder. Here we introduce some of ones: #### Greatest common divisor - Sample file: `sample/gcd.in` ``` // Example program in Python // def gcd(a: Int, b: Int): Int = // if (b==0) a else gcd(b, a%b) // Rules gcd(ret) >< (int a, int b) | b==0 => ret ~ a | _ => gcd(ret) ~ (b, a%b); // Nets gcd(r) ~ (14,21); r; // it should be 7 ``` - Execution: ``` $ ./inpla -f sample/gcd.in Inpla 0.4.1 : Interaction nets as a programming language [built: 21 Sept. 2021] (4 interactions, 0.00 sec) 7 $ ``` #### Insertion sort ![insertion sort](pic/isort.png) - Sample file: `sample/isort.in` ``` // Rules insert(ret, int x) >< [] => ret~[x]; insert(ret, int x) >< (int y):ys | x<=y => ret~(x:y:ys) | _ => ret~(y:cnt), insert(cnt, x)~ys; isort(ret) >< [] => ret~[]; isort(ret) >< x:xs => insert(ret, x)~cnt, isort(cnt)~xs; // Nets isort(r)~[3,6,1,9,2]; r; ``` - Execution: ``` $ ./inpla -f sample/isort.in Inpla 0.4.1 : Interaction nets as a programming language [built: 21 Sept. 2021] (16 interactions, 0.00 sec) [1,2,3,6,9] $ ``` #### Quick sort ![quick sort](pic/qsort.png) - Sample file: `sample/qsort.in` ``` // Rules qsort(ret) >< [] => ret~[]; qsort(ret) >< (int x):xs => Append(ret, x:right)~left, part(smaller, larger, x)~xs, qsort(left)~smaller, qsort(right)~larger; // Note: `Append' is implemented as the following built-in agent: // Append(ret, b)~a --> ret ~ a++b part(smaller, larger, int x) >< [] => smaller~[], larger~[]; part(smaller, larger, int x) >< (int y):ys | y smaller~(y:cnt), part(cnt, larger, x)~ys | _ => larger~(y:cnt), part(smaller, cnt, x)~ys; // Nets isort(r)~[3,6,1,9,2]; r; ``` - Execution: ``` $ ./inpla -f sample/isort.in Inpla 0.4.1 : Interaction nets as a programming language [built: 21 Sept. 2021] (22 interactions, 0.00 sec) [1,2,3,6,9] $ ``` #### Other samples - Evaluation of a lambda term `245II` in [YALE encoding](http://dl.acm.org/citation.cfm?id=289434), where `2 4 5` mean church numbers of lambda terms respectively and `I` is a lambda term $\lambda x.x$: $ ./inpla -f sample/245II.in - Samples of linear systemT encoding (see [our paper](http://link.springer.com/chapter/10.1007%2F978-3-319-29604-3_6) presented at [FLOPS 2016](http://www.info.kochi-tech.ac.jp/FLOPS2016/)). $ ./inpla -f sample/linear-systemT.in ### Execution Options - When invoking Inpla, you can specify the following options: ``` $ ./inpla -h Usage: inpla [options] Options: -f Set input file name (Defalut is STDIN) -c Set the size of term cells (Defalut is 100000) -x Set the size of the AP stack (Default is 10000) -t Set the number of threads (Default is 1) -w Enable Weak reduction strategy (Default: false) -h Print this help message ``` (The option ```-t``` is available for the multi-thread version that is compiled by ```make thread```.) # Introduction to programming in Inpla Inpla evaluates *nets*, which are built by *connections between terms*. First, we learn about terms and connections. ## Terms **Terms** are built on *names* and *agents* as follows: ``` ::= | ::= ::= | ['(' ',' ... ',' ')'] ``` - **Name**: It works a buffer between terms and **the same name must occur at most twice in order to ensure one-to-one connections between terms**. The `````` is defined as strings start with a small letter, e.g. ```x``` and ```y```. To show connected terms from names, type the names. For instance, type just `x` to show a term connected from the 'x': ``` >>> x; >>> ``` Nothing is connected from the `x` yet, of course. - **Agent**: It works constructors and de-constructors (defined functions). Generally agents have one *principal port* and *n*-fixed *auxiliary ports*. The fixed number of auxiliary ports is called *arity*, and it is determined according to each agents. In graphical representation an agent term `A(x1,...,xn)`, whose arity is *n*, is drawn as the following picture, where its auxiliary ports and the principal port correspond to the occurrences of the `x1`,...`xn`, and `A(x1,...,xn)`, respectively: ![agent](pic/agent.png) The `````` is defined as strings start with a capital letter, e.g. ```A``` and ```Succ```, and also `````` followed by a open curry bracket```(```. So, ```foo(x)``` is recognised as an agent. ## Connections A **connection** is a relation between two terms, and it is expressed with the symbol `~`. For instance, a connection between a name `x` and an agent `A` (whose arity is 0) is denoted as `x~A`. There is no order in the left-hand and the right-hand side terms, thus `x~A` and `A~x` are identified as the same one. - **Connections between a name and an agent** are evaluated that the agent is connected from the name. For instance, `x~A` is evaluated that the `A` is connected from the `x`. Here, as an example, type `x~A` with the termination symbol `;` as follows: ``` >>> x~A; (0 interactions, 0.00 sec) >>> ``` To show the connected terms from the name x, type just `x`: ``` >>> x; A >>> ``` To dispose the name `x` and anything connceted from the `x`, use `free` command: ``` >>> free x; >>> x; >>> ``` - **Connections between names** are evaluated that these names are connected mutually in interaction nets. However, in Inpla, these are evaluated that the left-hand side name connects to the right-hand side name, thus only one way. For instance, `x~y` is evaluated that the `y` is connected from the `x`: ``` >>> x~y; >>> x; y >>> y; >>> ``` - **Connections between agents** are evaluated according to *interaction rules* explained later. One more connections are also evaluated. Connections whose the left-hand side is a name, such as `x~t`, are disposed and the occurrence `x` in other connections is replaced with the `t`. For instance, `x~A, x~y` are evaluated as `A~y` by replacing `x` with `A`, or possibly `y~A` by replacing the `x` with `y` when the second connection `x~y` is used. **We note** that the `x` is disposed because it is consumed by the substitution. Thus, **every connection is one-to-one via names, cannot be one-to-many**. ``` >>> x~A, x~y; (0 interactions, 0.00 sec) >>> y; A >>> x; // x has been consumed by the re-connection of x~A and x~y. >>> ``` Just in case for other examples, let the `y` disposed: ``` >>> free y; >>> y; >>> ``` ## Interaction rules Connections between agents are re-written according to **interaction rules**: ``` ::= '><' '=>' ';' ::= | '(' ',' ... ',' ')' ``` where - each name of the `````` must be **distinct**, and must **occur once** in the ``````. Something complicated? No problem! Let's us learn how to define the rules with some example! ### Example: Operations on unary natural numbers. Unary natural numbers are built by Z and S. For instance, 0, 1, 2, 3 are expressed as Z, S(Z), S(S(Z)), S(S(S(Z))). Here, let's think about an increment operation `inc` such that inc(n) = S(n). This is written as rules for Z and S(x) as follows: ``` inc(ret) >< Z => ret~S(Z); inc(ret) >< S(x) => ret~S(S(x)); ``` In the first rule, the name `result` occurs in its `` part once, so it satisfies the rule proviso. The second rule also satisfies because the `result` and `x` are distinct names and occur once in its `` part. When agents works can be separated into constructors and de-constructors, it could be good to use strings of all small letters for de-constructors such as `inc` , and for constructors ones start from a capital letter such as `Z` and `S`. Let's have the result of the increment operation for `S(S(Z))`: ``` >>> inc(r)~S(S(Z)); (1 interactions, 0.01 sec) >>> r; S(S(S(Z)) >>> ``` Good! We get the result `S(S(S(Z)))` of incrementation of `S(S(Z))` . To show the result as a natural number, use `prnat` command: ``` >>> prnat result; 3 >>> ``` Let's clean the result in case it could be used anywhere: ``` >>> free r; >>> ``` - **Exercise**: Addition on unary natural numbers. It is defined recursively in term rewriting systems as follows: - add(x, Z) = x, - add(x, S(y)) = add(S(x), y). This is written in interaction nets and Inpla as follows: ![add1](pic/add1.png) ``` add(ret, x) >< Z => ret~x; add(ret, x) >< S(y) => add(ret, S(x))~y; ``` The following is an execution example: ``` >>> add(r,S(Z))~S(S(Z)); (3 interactions, 0.00 sec) >>> r; S(S(S(Z))) >>> prnat result; 3 >>> free r; >>> ``` - **Exercise**: Another version of the addition. There is another version defined as follows in term rewriting system: - add(x,Z) = x, - add(x, S(y)) = S(add(x,y)). This is written in interaction nets and Inpla as follows: ![add2](pic/add2.png) ``` add(ret,x) >< Z => ret~x; add(ret,x) >< S(y) => ret~S(cnt), add(cnt, x)~y; ``` ## Built-in agents Inpla has built-in agents: ### Tuples - `Tuple0`, `Tuple2(x1,x2)`, `Tuple3(x1,x2,x2)`, `Tuple4(x1,x2,x3,x4)`, `Tuple5(x1,x2,x3,x4,x5)`, are written as `()`, `(x1,x2)`, `(x1,x2,x3)`, `(x1,x2,x3,x4)`, `(x1,x2,x3,x4,x5)`. - There is no Tuple1, and so `(x)` is evaluated as just `x`. ### Lists - `Nil`, `Cons(x,xs)` are written as `[]` and `x:xs`, respectively. - A nested `Cons` that terminated at `Nil` is written as a list notation using brackets `[` and `]`. For instance, `x1 : x2: x3 : Nil` is written as `[x1,x2,x3]` . ### Built-in rules for tuples and lists There are built-in rules for pairs of the same built-in agents that match and connect each element such that: ``` (x1,x2)><(y1,y2) => x1~y1, x2~y2 // This is already defined as a built-in rule. ``` We also have a built-in agent `Append` to append two lists as shown in the following pseudo code: ``` Append(r, listB) ~ listA --> r ~ (listA ++ listB) // pseudo code ``` - The following is an example of built-in agents: ``` >>> (x1,x2)~(Z, S(Z)); (1 interactions, 0.00 sec) >>> x1 x2; Z S(Z) >>> ``` ``` >>> [y1, y2, y3]~[Z, S(Z), S(S(Z))]; (1 interactions, 0.00 sec) >>> y1 y2 y3; Z S(Z) S(S(Z)) >>> ``` ``` >>> Append(r, [Z, S(Z)]) ~ [A,B,C]; (4 interactions, 0.00 sec) >>> r; [A,B,C,Z,S(Z)] >>> free x1 x2 y1 y2 y3 r; >>> ``` ## Attributes (integers) Agents can have integers as their ports. These integers are called *attributes*. - For instance, `A(100)` is evaluated as an agent `A` that holds an attribute of an integer value 100. ``` >>> x~A(100); (0 interactions, 0.01 sec) >>> x; A(100); >>> free x; >>> ``` It is possible to use integers the same as agents, but these are recognised as attributes of an anonymous built-in agent in Inpla: ``` >>> x~100; (0 interactions, 0.00 sec) >>> x; 100 >>> free x; >>> ``` ### Arithmetic expressions on attributes Attributes can be given as the results of arithmetic operation using `where` statement after connections: ``` ::= | 'where' * ';' ::= '=' ``` The symbol of addition, subtraction, multiplication, division and modulo are `+`, `-`, `*`, `/` and `%`, respectively. - For instance, the following is an expression using `where`: ``` >>> x~A(a) where a=3+5; (0 interactions, 0.00 sec) >>> x; A(8) >>> free x; >>> ``` - Arithmetic expressions can be written in arguments directly without using `where`: ``` >>> x~A(3+5); (0 interactions, 0.00 sec) >>> x; A(8) >>> free x; >>> ``` - Arithmetic expressions are also available for the anonymous agent: ``` >>> x~(3+5); // this is also written without brackets as x~3+5; (0 interactions, 0.00 sec) >>> x; 8 >>> free x; >>> ``` ### Interaction rules with expressions on attributes In interaction rules, attributes can be recognised by the modifier `int` in order to apply arithmetic expressions. We can use the same variable with the modifier `int` many times in the connection parts of interaction rules. - Example: Incrementor on an attribute: ``` >>> inc(result) >< (int a) => result~(a+1); >>> inc(r)~10; (1 interactions, 0.00 sec) >>> r; 11 >>> free r; >>> ``` - Example: Duplicator of integer lists: ``` >>> dup(a1,a2) >< (int i):xs => a1~(i:xs1), a2~(i:xs2), dup(xs1,xs2)~xs; >>> dup(a1,a2) >< [] => a1~[], a2~[]; >>> dup(a,b) ~ [1,2,3]; (4 interactions, 0.00 sec) >>> a b; [1,2,3] [1,2,3] >>> free a b; >>> ``` **We have to be careful for operations of two attributes**. For instance, we take the following rule of an `add` agent: ``` >>> add(result, int b) >< (int a) => result~(a+b); ``` Of course, it works as an addition operation on two attributes: ``` >>> add(r, 3) ~ 5; (1 interactions, 0.00 sec) >>> r; 8 >>> free r; >>> ``` However, it can cause core dump error in computation for results of something because the argument `b` of `add` must be an attribute when the rule is invoked. For instance, we take the following computation: ``` >>> add(r, b)~3, add(b, 10)~20; ``` If the `add(r, b)~3` is operated first, it causes core dump error because the `b` is not connected to an attribute, though it is OK if the `add(b, 10)~20` is operated first. To prevent this fragile situation, **we have to have extra rules to ensure that every port with the modifier `int` connects to an attribute**. For the rule of the `add` agent, the following is a **solution** to have the extra rule where `addn` is introduced as an extra agent: ``` >>> add(result, b) >< (int a) => addn(result, a) ~ b; >>> addn(result, int a) >< (int b) => r~(a+b); ``` ### Built-in rules of attributes in the anonymous agents There is a built-in rule for attributes on the two anonymous agents. This could be required to calculate attributes obtained as evaluation results of two nets. In this version, we have the following rules as built-in: ``` Add(result, y)><(int x) => Addn(result, x)~y; Addn(result, int x)><(int y) => result~(x+y); ``` - Example: ``` >>> Add(r, 3)~5; // Add is already defined as a bulit-in >>> r; 8 >>> ``` ### Interaction rules with conditions on attributes In interaction rules, conditional rewritings on attributes are available. The following is a general form: ``` ::= '><' '|' '=>' '|' '=>' ... '|' '_' '=>' ';' is an expression on attributes specified in the two . ``` The sequence of ` ` must be finished with the otherwise case `_`. - Example: The following shows rules to obtain a list that contains only even numbers: ``` // Rules (because it is long definition, so copy and paste is recommended) evenList(result) >< [] => r~[]; evenList(result) >< (int x):xs | x%2==0 => result~(x:r1), evenList(r1)~xs | _ => evenList(result)~xs; ``` ``` >>> evenList(r)~[1,3,7,5,3,4,9,10]; >>> r; [4,10] >>> free r; >>> ``` - Example: Fibonacci number: ``` fib(result) >< (int n) | n == 0 => result~0 | n == 1 => result~1 | _ => fib(r1)~(n-1), fib(r2)~(n-1), Add(result, r2)~r1; // * We cannot write result~(r1+r2) // because r1 and r2 are not recognised as attributes. // Therefore, we have to apply those to the built-in rule Add(r, int x)><(int y). ``` ``` >>> fib(r)~39; >>> r; 63245986 >>> free r; >>> ``` ## Commands - Inpla has the following commands: - `free` *name1* ... *name_n* `;` The *name1* ... *name_n* and connected terms from these are disposed. To dispose every living name and term connected from those, type `free ifce;`, where the `ifce` is an abbreviation of *interface* that is called for the set of living names. - *name1* ... *name_n* `;` Put terms connected from the *name1* ... *name_n*. To put every term connected from the interface, type `ifce;`. - `prnat` *name*`;` Put a term connected from the *name* as a natural number. - `use` `"`filename`";` Read the file named as "filename". - `exit;` Quit the system. - Inpla has the following macro: - `const` *NAME*`=` *i* `;` The *NAME* is replaced with the integer value *i* in nets and interaction rules. ## Extensions in Version 0.4 ### Reuse annotations In interaction rule definitions, we can specify which agent is reused in the nets again by annotations `(*L)` and `(*R)`, which means the left-hand side and the right-hand side agents in the rule. This annotations promote in-place computing, and as the result performance in parallel execution can be improved well. - For instance, in the rule `gcd(ret) >< (int a, int b)`, we can reuse the `gcd` and `Tuple2` in nets as follows: ``` gcd(ret) >< (int a, int b) | b==0 => ret ~ a | _ => (*L)gcd(ret) ~ (*R)(b, a%b); ``` ### Weak reduction strategy In this reduction strategy, only connections that have living names are evaluated. This is taken for non-terminate computation such as fixed point combinator and process networks. - Example: We have a sample net in `sample/processnet1.in` that keep producing natural numbers from 1 and output these to the port `r`: ``` // Rules dup(a1,a2) >< (int i):xs => a1~(i:xs1), a2~(i:xs2), dup(xs1,xs2)~xs; dup(a1,a2) >< [] => a1~[], a2~[]; inc(r) >< (int i):xs => r~(i+1):w, inc(w)~xs; inc(r) >< [] => r~[]; // Nets dup(r,w)~r1, inc(r1) ~ 0:w; // +-----+ +-----+ +---+ // r ----| | r1 | | | | // | dup |--->---| inc |---><---| 0 |----+ // +---| | | | | | | // | +-----+ +-----+ +---+ | // | | // +-----------------------------------------+ // w ``` This is executable in this reduction strategy as follows: ``` $ ./inpla -w Inpla 0.4.1 : Interaction nets as a programming language [built: 21 Sept. 2021] >>> use "sample/processnet1.in"; (2 interactions, 0.00 sec) >>> ifce; r Connections: r ->[1,... // this means a list of 1 and something. >>> a:b ~ r; (2 interactions, 0.00 sec) >>> ifce; a b Connections: a ->1 b ->[2,... >>> ``` # Publications - [1] Ian Mackie, Shinya Sato, [*In-place Graph Rewriting with Interaction Nets*](https://arxiv.org/abs/1609.03641), TERMGRAPH 2016, EPTCS 225, pp.15-24, 2016. - [2] Shinya Sato, [*Design and implementation of a low-level language for interaction nets*](http://sro.sussex.ac.uk/54469/), PhD Thesis, University of Sussex, September 2014. - [3] Abubakar Hassan, Ian Mackie and Shinya Sato, [*An implementation model for interaction nets*](http://arxiv.org/abs/1505.07164), Proceedings 8th International Workshop on Computing with Terms and Graphs, TERMGRAPH 2014, EPTCS 183, May 2015. - [4] Ian Mackie and Shinya Sato, [*Parallel Evaluation of Interaction Nets: Case Studies and Experiments*](http://journal.ub.tu-berlin.de/eceasst/article/view/1034), Electronic Communications of the EASST, Volume 73: Graph Computation Models - Selected Revised Papers from GCM 2015, March 2016. # Related works - [HINet: Interaction Nets in Haskell](http://www.cas.mcmaster.ca/~kahl/Haskell/HINet/) # License Copyright (c) 2021 [Shinya SATO](http://satolab.com/) Released under the MIT license http://opensource.org/licenses/mit-license.php