1,836,946 research outputs found
Proof Outlines as Proof Certificates: A System Description
We apply the foundational proof certificate (FPC) framework to the problem of
designing high-level outlines of proofs. The FPC framework provides a means to
formally define and check a wide range of proof evidence. A focused proof
system is central to this framework and such a proof system provides an
interesting approach to proof reconstruction during the process of proof
checking (relying on an underlying logic programming implementation). Here, we
illustrate how the FPC framework can be used to design proof outlines and then
to exploit proof checkers as a means for expanding outlines into fully detailed
proofs. In order to validate this approach to proof outlines, we have built the
ACheck system that allows us to take a sequence of theorems and apply the proof
outline "do the obvious induction and close the proof using previously proved
lemmas".Comment: In Proceedings WoF'15, arXiv:1511.0252
Focusing and Polarization in Intuitionistic Logic
A focused proof system provides a normal form to cut-free proofs that
structures the application of invertible and non-invertible inference rules.
The focused proof system of Andreoli for linear logic has been applied to both
the proof search and the proof normalization approaches to computation. Various
proof systems in literature exhibit characteristics of focusing to one degree
or another. We present a new, focused proof system for intuitionistic logic,
called LJF, and show how other proof systems can be mapped into the new system
by inserting logical connectives that prematurely stop focusing. We also use
LJF to design a focused proof system for classical logic. Our approach to the
design and analysis of these systems is based on the completeness of focusing
in linear logic and on the notion of polarity that appears in Girard's LC and
LU proof systems
Classes of representable disjoint NP-pairs
For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist
Disjoint NP-pairs from propositional proof systems
For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist
Automated proof search system for logic of correlated knowledge
The automated proof search system and decidability for logic of correlated
knowledge is presented in this paper. The core of the proof system is the
sequent calculus with the properties of soundness, completeness, admissibility
of cut and structural rules, and invertibility of all rules. The proof search
procedure based on the sequent calculus performs automated terminating proof
search and allows us to achieve decision result for logic of correlated
knowledge
Nondeterministic functions and the existence of optimal proof systems
We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system
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