16 research outputs found
A new face of the branching recurrence of computability logic
This letter introduces a new, substantially simplified version of the
branching recurrence operation of computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html), and proves its equivalence to the
old, "canonical" version
Separating the basic logics of the basic recurrences
This paper shows that, even at the most basic level, the parallel, countable
branching and uncountable branching recurrences of Computability Logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) validate different principles
A logical basis for constructive systems
The work is devoted to Computability Logic (CoL) -- the
philosophical/mathematical platform and long-term project for redeveloping
classical logic after replacing truth} by computability in its underlying
semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article
elaborates some basic complexity theory for the CoL framework. Then it proves
soundness and completeness for the deductive system CL12 with respect to the
semantics of CoL, including the version of the latter based on polynomial time
computability instead of computability-in-principle. CL12 is a sequent calculus
system, where the meaning of a sequent intuitively can be characterized as "the
succedent is algorithmically reducible to the antecedent", and where formulas
are built from predicate letters, function letters, variables, constants,
identity, negation, parallel and choice connectives, and blind and choice
quantifiers. A case is made that CL12 is an adequate logical basis for
constructive applied theories, including complexity-oriented ones
The taming of recurrences in computability logic through cirquent calculus, Part I
This paper constructs a cirquent calculus system and proves its soundness and
completeness with respect to the semantics of computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html). The logical vocabulary of the system
consists of negation, parallel conjunction, parallel disjunction, branching
recurrence, and branching corecurrence. The article is published in two parts,
with (the present) Part I containing preliminaries and a soundness proof, and
(the forthcoming) Part II containing a completeness proof