3,772 research outputs found
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page
Schematic Cut elimination and the Ordered Pigeonhole Principle [Extended Version]
In previous work, an attempt was made to apply the schematic CERES method [8]
to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof
encapsulating the infinitary pigeonhole principle) [5]. However the derived
schematic refutation for the characteristic clause set of the proof could not
be expressed in the formal language provided in [8]. Without this formalization
a Herbrand system cannot be algorithmically extracted. In this work, we provide
a restriction of the proof found in [5], the ECA-schema (Eventually Constant
Assertion), or ordered infinitary pigeonhole principle, whose analysis can be
completely carried out in the framework of [8], this is the first time the
framework is used for proof analysis. From the refutation of the clause set and
a substitution schema we construct a Herbrand system.Comment: Submitted to IJCAR 2016. Will be a reference for Appendix material in
that paper. arXiv admin note: substantial text overlap with arXiv:1503.0855
Generating Schemata of Resolution Proofs
Two distinct algorithms are presented to extract (schemata of) resolution
proofs from closed tableaux for propositional schemata. The first one handles
the most efficient version of the tableau calculus but generates very complex
derivations (denoted by rather elaborate rewrite systems). The second one has
the advantage that much simpler systems can be obtained, however the considered
proof procedure is less efficient
A New Arithmetically Incomplete First- Order Extension of Gl All Theorems of Which Have Cut Free Proofs
Reference [12] introduced a novel formula to formula translation tool (“formulators”) that enables syntactic metatheoretical investigations of first-order modal logics, bypassing a need to convert them first into Gentzen style logics in order to rely on cut elimination and the subformula property. In fact, the formulator tool, as was already demonstrated in loc. cit., is applicable even to the metatheoretical study of logics such as QGL, where cut elimination is (provably, [2]) unavailable. This paper applies the formulator approach to show the independence of the axiom schema _A ! _8xA of the logics M3 and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics (compare with the more complex proofs in [2, 8]).This research was partially supported by NSERC grant No. 8250
Inducing syntactic cut-elimination for indexed nested sequents
The key to the proof-theoretic study of a logic is a proof calculus with a
subformula property. Many different proof formalisms have been introduced (e.g.
sequent, nested sequent, labelled sequent formalisms) in order to provide such
calculi for the many logics of interest. The nested sequent formalism was
recently generalised to indexed nested sequents in order to yield proof calculi
with the subformula property for extensions of the modal logic K by
(Lemmon-Scott) Geach axioms. The proofs of completeness and cut-elimination
therein were semantic and intricate. Here we show that derivations in the
labelled sequent formalism whose sequents are `almost treelike' correspond
exactly to indexed nested sequents. This correspondence is exploited to induce
syntactic proofs for indexed nested sequent calculi making use of the elegant
proofs that exist for the labelled sequent calculi. A larger goal of this work
is to demonstrate how specialising existing proof-theoretic transformations
alleviate the need for independent proofs in each formalism. Such coercion can
also be used to induce new cutfree calculi. We employ this to present the first
indexed nested sequent calculi for intermediate logics.Comment: This is an extended version of the conference paper [20
Elimination of Cuts in First-order Finite-valued Logics
A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information
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