I have not focussed on सिद्धि so far - but this is a key process through which all words can be derived from. We can use a state-machine formulation for the derivation of the words from the Paninian rules. Let's take the word दद्ध्यत्र. Here, the root words are दधि अत्र. The inputs given to us are the two independent words - which could be reduced or expanded (a पद could not be joined or transformed without conditions). With सु or तिबादि - a particular instance (of the function if you will) would be used.
1. दधि अत्र
2. Transformation - दध् यत्र (through इको यणचि)
3. Expansion through अनचि च. Here ध् is यर्-वर्ण next-to i.e. परे the अनच् य् therefore a द्वित्व of ध् is implied. Here, all actions (expansions/transformations/reductions) are subject to स्थानिवदादेशो अनल्विधौ, अचः परस्मिन् पूर्वविधौ. A स्थानी behaviour would mean that rules would be applied with इ considered in-place. If so, इ is not अनच् and अनचि च would not be applied. While स्थानी is avoided with अनल्विधौ, अचः परस्मिन् exception applies here स्थानी is restated (पूर्वविधि). But again, 1.1.57 न पदान्तद्विर्वचनवरेयलोपस्वरसवर्णानुस्वारदीर्घजश्चर्विधिषु exception on exception applies and स्थानी is prevented and the result is दध् ध्यत्र
This is a rather complicated illustration - mostly because स्थानी status is mediated through 3 rules. Before applying a rule like अनचि च, one needs to consider whether स्थानी is to be used or not. The application conditions evaluated in the order they appear in the ashtadhyayi.
4. We have transformation through झलां जश् झशि - which results in दद्ध्यत्र.
In terms of the state-machine, we have only gone through an expansion (द्वित्व) and two transformations. Another example is पचेरन् which is पच् with लिङ्.
पच् लिङ्
1. First is the expansion (whether expansion happens before reduction depends on the order of rules in ashtadhyayi) through सीयुट् which leads to पच् सीयुट् लिङ्
2. Next the instantiation of झ is used leading to पच् सीयुट् झ
3. Another expansion through शप् follows (we put शप् just after the root - a better criterion would be later specified) so that we have पच् शप् सीयुट् झ. हलन्त्यम् and लशक्वतद्धिते are applied for on any आदेश (among similar others from इत् संज्ञा प्रकरण). This is not a reduction but merely application of शप् so that we have पच् अ सीयुट् झ
4. Reduction follows for सीयुट् (this could reduced while application as well - if the स्थानी discussion is not needed) with लिङः सलोपोअनन्तस्य - which deletes the स्. हलन्त्यम् and अनुनासिक इत् - leave us with पच् अ ईय् झ
5. Transfomation झस्य रन् is applied so that we have पच् अ ईय् रन्
6. Reduction follows again due to लोपो व्योर्वलि - पच ई रन्
7. Reduction with अदेङ् गुणः leads to पचेरन्
The next पद is चक्रतुः
1. The starting point is कृ अतुस्
2. लिट् च and सार्वधातुकार्धधातुकयोः imply a गुण but that is barred indirectly through असंयोगात् लिट् कित् (In non-संयोग context, लिट् acts as कित्) and क्क्ङिति च (bars गुण for कित्). We may choose to use these rules for application rather than reducing and expanding the states.
3. Transformation follows with इको यणचि leading to क्र् अतुस्
4. A द्वित्व expansion is implied due to लिटि धातोरनभ्यासस्य and एकाचो द्वे प्रथमस्य. In the application of this rule, स्थानी (pre-application) status is resumed through द्विर्वचने अचि - so what we repeat is कृ (not क्र्). Therefore, we have कृ क्र् अतुस्.
5. Transformation due to उरत् and उरण् रपरः leads to कर् क्र् अतुस् (chapter 6)
6. Application of अभ्यास permits the reduction due to हलादि शेषः and leaves क क्र् अतुस् . The application of अभ्यास also implies कुहोश्चुः (chapter 7) and results in चक्रतुस्
7. Transformation through ससजुषो रुः leads to चक्रतुर्
8. Transformation through विरामो अवसानम् and खरवसानयोर्विसर्जनीयः leads to चक्रतुः