Dear Mr Das,
You may refer to Dictionary of Nyaya Terms of Prof V N Jha, published by Centre of Advanced Study in Sanskrit, Pune which is in English. If you can refer to Prof Bhimacharya Jhalkikar's Nyayakosha published by Bhandarkar Oriental Research Institute, Pune which is in Sanskrit will be great and useful.
Here is the meaning of these words for your immediate reference.
1) Annyay - I am not able to understand exactly what is the word you are referring to.
I assume it might be Anvaya and there are five meanings for that.
a) relations between two positive entities - भावयोः साहचर्यं यत् अन्वयं तत् प्रचक्षते |
b) the presence of what is produced in the locus of cause - कारणाधिकरणे कार्यस्य सत्त्वम् |
c) sentence - meaning
d) relation - पदार्थयोरन्वयो द्विविधः |
e) invariable concomitance of probans in general with the probandum in general - साध्यसामान्येन साध्यसामान्यस्य व्याप्तिरन्वयः
2) vyatireka (in vyatireka-vyapti) = pervasion of difference or dissimilitude, a comprehensive argument derived from negation or non-existence of certain qualities (refer to: https://sanskrit.inria.fr/MW/252.html#H_vyatireka/ opposite to anvaya
3) Vyapti =
a) invariable concomitance - व्याप्तिः अविनाभावः नित्यसाहचर्यमित्यर्थः |
b) a natural relation between sadhya and sadhana - स्वभावतः साध्येन साधनस्य व्याप्तिरविनाभावः
c) invariable concomitance is one type of a causal relationship between a proban and a probandum - वह्निधूमयोः कार्यकारणभावलक्षणैव व्याप्तिः
d) invariable and inseparable connection
4) Ativyapti = over-application - लक्ष्यवृत्तित्वे सति अलक्ष्यवृत्तित्वम् अतिव्याप्तिः
5) Avyapti = Narrow-application - लक्ष्यैकदेश-अवृत्तित्वम् अव्याप्तिः
Thank you.
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Regards,
Dr Arjuna S R
Assistant Professor
Department of Philosophy
Manipal Academy of Higher Education, Manipal