  The relevance of polynomial formula classes to deductive efficiency motivated
their search, and currently, a great number of such classes is known.
Nonetheless, they have been exclusively sought in the setting of clausal form
and propositional logic, which is of course expressively limiting for real
applications. As a consequence, a first polynomial propositional class in
non-clausal (NC) form has recently been proposed. Along these lines and towards
making NC tractability applicable beyond propositional logic, firstly, we
define the Regular many-valued Horn Non-Clausal class, or RH, obtained by
suitably amalgamating both regular classes: Horn and NC. Secondly, we
demonstrate that the relationship between (1) RH and the regular Horn class is
that syntactically RH subsumes the Horn class but that both classes are
equivalent semantically; and between (2) RH and the regular non-clausal class
is that RH contains all NC formulas whose clausal form is Horn. Thirdly, we
define Regular Non-Clausal Unit-Resolution, or RUR-NC , and prove both that it
is complete for RH and that checks its satisfiability in polynomial time. The
latter fact shows that our intended goal is reached since RH is many-valued,
non-clausal and tractable. As RH and RUR-NC are, both, basic in the DPLL
scheme, the most efficient in propositional logic, and can be extended to some
other non-classical logics, we argue that they pave the way for efficient
non-clausal DPLL-based approximate reasoning.
