Complexity, expressivity, syntax and semantics

I will summarise the philosophical motivations behind two research topics; 1. complexity/computability and 2. logic (structural proof theory), and discuss how they may help us understand what makes some problems harder than others, or equivalently, some knowledge more difficult to attain than others (my broad research goals).
I. Complexity/computability
Computational complexity and computability theory are a subfield of theoretical computer science in which we mathematically study the 'hardness' of problems. We do so by classifying algorithms or a collection of pre-defined rules that some solver can apply without ingenuity by how much time and memory space they require.
II. Structural proof theory
Even whilst maintaining the basic idea that a well-formed sentence, or a proposition, is either true or false, one can still make a conscious choice about what kind of principles to permit in deriving a new statement from assumptions. Structural proof theory formalises this as a logical-deduction system to study their effect on what the logic can and cannot do.
III. What I do, more specifically
I take advantage of equivalences between some computational complexity classes and logic, the latter of which, I hope, can serve as an interface to connect, via semantics, complexity with wider mathematics to elucidate something that can tell us what makes some computation inherently costly.
IV. 'Computational view' of science
I would love to discuss if time permits, how we may apply the idea of complexity to illuminate how information transfers from one thing to another in physical, biological and social systems.