Does the logic go as follows?
short term rate + expected inflation= long term rates?
Not exactly. There is some confusion. Inflation distinguishes between real and nominal interest rates, whereas the discrepancy between short- and long-term interest rates is derived from the preclusion of arbitrage opportunities. It is neither actual nor expected inflation, but the forward rate, what explains the discrepancy between short- and long-term interest rates.
By way of example, suppose the short- and long-term rates currently are 3% and 5%, with short- and long-term referring to 1 and 2 years, respectively. The interest rate $r_{f}$ for the 2nd year should be such that $(1+0.03)(1+r_{f})=(1+0.05)^2$. Any other rate $r_{f}$ would allow either the borrower or the lender to make profit with no risk (or somewhat equivalently, to make profit without having to invest his own funds).
It is true that a lender requires higher return for long-term loans than for short-term ones because a longer term entails greater exposure to risk. But the inflation is just one of multiple risks that the lender factors in when determining at what rate (of return) he is willing to lend money.
Edited on 7/22/2018 (at 3:29PM EDT) to supplement as per 2nd-to-last comment
There is no robust or widely-adopted method for calculation of long-term rates based on estimated future inflation (and/or short-term rates). Nor is there a mathematical expression describing how --or even whether-- market forces use the estimated future inflation for determining the rate at which lenders are willing to commit funds for a long-term period. I personally doubt that such method or expression could ever exist.
One can only resort to qualitative reasoning to sketch the generic effect that some information tends to have on long-term rates. In the case of expected/estimated future inflation, it might be implicitly (at most, but perhaps not at all) reflected in other factors and preferences that influence those market forces. Uncertainty --as you rightly point out in one of your comments--, risk aversion, and cost of opportunity are three of many other factors and preferences that influence how much a long-term rate would differ from the short-term rate.
Although it is reasonable to conjecture that these estimates, factors, and preferences are positively correlated with the duration of a long-term period, it would be very hard (or I would say futile as well as impossible) to reliably model by how much they cause a long-term rate to differ from the short-term rate.
