Documentation/rv: Add deterministic automaton documentation

Add documentation about deterministic automaton and its possible
representations (formal, graphic, .dot and C).

Link: https://lkml.kernel.org/r/[e-mail]

Cc: Wim Van Sebroeck <[e-mail]>
Cc: Guenter Roeck <[e-mail]>
Cc: Jonathan Corbet <[e-mail]>
Cc: Ingo Molnar <[e-mail]>
Cc: Thomas Gleixner <[e-mail]>
Cc: Peter Zijlstra <[e-mail]>
Cc: Will Deacon <[e-mail]>
Cc: Catalin Marinas <[e-mail]>
Cc: Marco Elver <[e-mail]>
Cc: Dmitry Vyukov <[e-mail]>
Cc: "Paul E. McKenney" <[e-mail]>
Cc: Shuah Khan <[e-mail]>
Cc: Gabriele Paoloni <[e-mail]>
Cc: Juri Lelli <[e-mail]>
Cc: Clark Williams <[e-mail]>
Cc: Tao Zhou <[e-mail]>
Cc: Randy Dunlap <[e-mail]>
Cc: [e-mail]
Cc: [e-mail]
Cc: [e-mail]
Signed-off-by: Daniel Bristot de Oliveira <[e-mail]>
Signed-off-by: Steven Rostedt (Google) <[e-mail]>

Documentation/trace/rv/deterministic_automata.rst

···
+Deterministic Automata
+======================
+
+Formally, a deterministic automaton, denoted by G, is defined as a quintuple:
+
+        *G* = { *X*, *E*, *f*, x\ :subscript:`0`, X\ :subscript:`m` }
+
+where:
+
+- *X* is the set of states;
+- *E* is the finite set of events;
+- x\ :subscript:`0` is the initial state;
+- X\ :subscript:`m` (subset of *X*) is the set of marked (or final) states.
+- *f* : *X* x *E* -> *X* $ is the transition function. It defines the state
+  transition in the occurrence of an event from *E* in the state *X*. In the
+  special case of deterministic automata, the occurrence of the event in *E*
+  in a state in *X* has a deterministic next state from *X*.
+
+For example, a given automaton named 'wip' (wakeup in preemptive) can
+be defined as:
+
+- *X* = { ``preemptive``, ``non_preemptive``}
+- *E* = { ``preempt_enable``, ``preempt_disable``, ``sched_waking``}
+- x\ :subscript:`0` = ``preemptive``
+- X\ :subscript:`m` = {``preemptive``}
+- *f* =
+   - *f*\ (``preemptive``, ``preempt_disable``) = ``non_preemptive``
+   - *f*\ (``non_preemptive``, ``sched_waking``) = ``non_preemptive``
+   - *f*\ (``non_preemptive``, ``preempt_enable``) = ``preemptive``
+
+One of the benefits of this formal definition is that it can be presented
+in multiple formats. For example, using a *graphical representation*, using
+vertices (nodes) and edges, which is very intuitive for *operating system*
+practitioners, without any loss.
+
+The previous 'wip' automaton can also be represented as::
+
+                       preempt_enable
+          +---------------------------------+
+          v                                 |
+        #============#  preempt_disable   +------------------+
+    --> H preemptive H -----------------> |  non_preemptive  |
+        #============#                    +------------------+
+                                            ^              |
+                                            | sched_waking |
+                                            +--------------+
+
+Deterministic Automaton in C
+----------------------------
+
+In the paper "Efficient formal verification for the Linux kernel",
+the authors present a simple way to represent an automaton in C that can
+be used as regular code in the Linux kernel.
+
+For example, the 'wip' automata can be presented as (augmented with comments)::
+
+  /* enum representation of X (set of states) to be used as index */
+  enum states {
+	preemptive = 0,
+	non_preemptive,
+	state_max
+  };
+
+  #define INVALID_STATE state_max
+
+  /* enum representation of E (set of events) to be used as index */
+  enum events {
+	preempt_disable = 0,
+	preempt_enable,
+	sched_waking,
+	event_max
+  };
+
+  struct automaton {
+	char *state_names[state_max];                   // X: the set of states
+	char *event_names[event_max];                   // E: the finite set of events
+	unsigned char function[state_max][event_max];   // f: transition function
+	unsigned char initial_state;                    // x_0: the initial state
+	bool final_states[state_max];                   // X_m: the set of marked states
+  };
+
+  struct automaton aut = {
+	.state_names = {
+		"preemptive",
+		"non_preemptive"
+	},
+	.event_names = {
+		"preempt_disable",
+		"preempt_enable",
+		"sched_waking"
+	},
+	.function = {
+		{ non_preemptive,  INVALID_STATE,  INVALID_STATE },
+		{  INVALID_STATE,     preemptive, non_preemptive },
+	},
+	.initial_state = preemptive,
+	.final_states = { 1, 0 },
+  };
+
+The *transition function* is represented as a matrix of states (lines) and
+events (columns), and so the function *f* : *X* x *E* -> *X* can be solved
+in O(1). For example::
+
+  next_state = automaton_wip.function[curr_state][event];
+
+Graphviz .dot format
+--------------------
+
+The Graphviz open-source tool can produce the graphical representation
+of an automaton using the (textual) DOT language as the source code.
+The DOT format is widely used and can be converted to many other formats.
+
+For example, this is the 'wip' model in DOT::
+
+  digraph state_automaton {
+        {node [shape = circle] "non_preemptive"};
+        {node [shape = plaintext, style=invis, label=""] "__init_preemptive"};
+        {node [shape = doublecircle] "preemptive"};
+        {node [shape = circle] "preemptive"};
+        "__init_preemptive" -> "preemptive";
+        "non_preemptive" [label = "non_preemptive"];
+        "non_preemptive" -> "non_preemptive" [ label = "sched_waking" ];
+        "non_preemptive" -> "preemptive" [ label = "preempt_enable" ];
+        "preemptive" [label = "preemptive"];
+        "preemptive" -> "non_preemptive" [ label = "preempt_disable" ];
+        { rank = min ;
+                "__init_preemptive";
+                "preemptive";
+        }
+  }
+
+This DOT format can be transformed into a bitmap or vectorial image
+using the dot utility, or into an ASCII art using graph-easy. For
+instance::
+
+  $ dot -Tsvg -o wip.svg wip.dot
+  $ graph-easy wip.dot > wip.txt
+
+dot2c
+-----
+
+dot2c is a utility that can parse a .dot file containing an automaton as
+in the example above and automatically convert it to the C representation
+presented in [3].
+
+For example, having the previous 'wip' model into a file named 'wip.dot',
+the following command will transform the .dot file into the C
+representation (previously shown) in the 'wip.h' file::
+
+  $ dot2c wip.dot > wip.h
+
+The 'wip.h' content is the code sample in section 'Deterministic Automaton
+in C'.
+
+Remarks
+-------
+
+The automata formalism allows modeling discrete event systems (DES) in
+multiple formats, suitable for different applications/users.
+
+For example, the formal description using set theory is better suitable
+for automata operations, while the graphical format for human interpretation;
+and computer languages for machine execution.
+
+References
+----------
+
+Many textbooks cover automata formalism. For a brief introduction see::
+
+  O'Regan, Gerard. Concise guide to software engineering. Springer,
+  Cham, 2017.
+
+For a detailed description, including operations, and application on Discrete
+Event Systems (DES), see::
+
+  Cassandras, Christos G., and Stephane Lafortune, eds. Introduction to discrete
+  event systems. Boston, MA: Springer US, 2008.
+
+For the C representation in kernel, see::
+
+  De Oliveira, Daniel Bristot; Cucinotta, Tommaso; De Oliveira, Romulo
+  Silva. Efficient formal verification for the Linux kernel. In:
+  International Conference on Software Engineering and Formal Methods.
+  Springer, Cham, 2019. p. 315-332.

Documentation/trace/rv/index.rst

···
    :glob:
 
    runtime-verification.rst
+   deterministic_automata.rst

tools/verification/dot2/automata.py

···
 # Copyright (C) 2019-2022 Red Hat, Inc. Daniel Bristot de Oliveira <[e-mail]>
 #
 # Automata object: parse an automata in dot file digraph format into a python object
+#
+# For further information, see:
+#   Documentation/trace/rv/deterministic_automata.rst
 
 import ntpath
 

tools/verification/dot2/dot2c

···
 #  de Oliveira, D. B. and Cucinotta, T. and de Oliveira, R. S.
 #  "Efficient Formal Verification for the Linux Kernel." International
 #  Conference on Software Engineering and Formal Methods. Springer, Cham, 2019.
+#
+# For further information, see:
+#   Documentation/trace/rv/deterministic_automata.rst
 
 if __name__ == '__main__':
     from dot2 import dot2c

tools/verification/dot2/dot2c.py

···
 #  de Oliveira, D. B. and Cucinotta, T. and de Oliveira, R. S.
 #  "Efficient Formal Verification for the Linux Kernel." International
 #  Conference on Software Engineering and Formal Methods. Springer, Cham, 2019.
+#
+# For further information, see:
+#   Documentation/trace/rv/deterministic_automata.rst
 
 from dot2.automata import Automata