  The syntactic nature of logic and computation separates them from other
fields of mathematics. Nevertheless, syntax has been the only way to adequately
capture the dynamics of proofs and programs such as cut-elimination, and the
finiteness and the atomicity of syntax are preferable for foundational aims as
seen in Hilbert's program. Another issue is that a uniform basis for logic and
computation has been missing, and this problem hampers a coherent view on them.
For instance, formal proofs in proof theory are far from (ordinary) proofs of
the validity of a formula in model theory. Our goal is to solve these
fundamental problems by rebuilding central concepts in logic and computation
such as formal systems, validity (in such a way that it coincides with the
existence of proofs), cut-elimination and computability uniformly in terms of
finite graphs based on game semantics. Unlike game semantics, however, we do
not rely on anything infinite or extrinsic to the graphs. A key idea that
enables our finitary, autonomous approach is the shift from graphs in game
semantics to dynamic ones. The resulting combinatorics establishes a single,
syntax-free, finitary framework that recasts formal systems admitting proofs
with cuts, validity, the finest computational steps of cut-elimination and
higher-order computability. This subsumes fully complete semantics of
intuitionistic linear logic, which solves a problem open for thirty years, and
even extends the full completeness to proofs with cuts. As a byproduct, our
dynamic graphs give rise to Hopf algebras, which opens up new applications of
algebras to logic and computation.
