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diff --git a/Documentation/trace/rv/deterministic_automata.rst b/Documentation/trace/rv/deterministic_automata.rst new file mode 100644 index 0000000000..d0638f95a4 --- /dev/null +++ b/Documentation/trace/rv/deterministic_automata.rst @@ -0,0 +1,184 @@ +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. |