40,836 research outputs found
Rejection in Łukasiewicz's and Słupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Łukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Łukasiewicz and developed by
his student Słupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
A Meta-Logic of Inference Rules: Syntax
This work was intended to be an attempt to introduce the meta-language for
working with multiple-conclusion inference rules that admit asserted
propositions along with the rejected propositions. The presence of rejected
propositions, and especially the presence of the rule of reverse substitution,
requires certain change the definition of structurality
SpecSatisfiabilityTool: A tool for testing the satisfiability of specifications on XML documents
We present a prototype that implements a set of logical rules to prove the
satisfiability for a class of specifications on XML documents. Specifications
are given by means of constrains built on Boolean XPath patterns. The main goal
of this tool is to test whether a given specification is satisfiable or not,
and justify the decision showing the execution history. It can also be used to
test whether a given document is a model of a given specification and, as a
by-product, it permits to look for all the relations (monomorphisms) between
two patterns and to combine patterns in different ways. The results of these
operations are visually shown and therefore the tool makes these operations
more understandable. The implementation of the algorithm has been written in
Prolog but the prototype has a Java interface for an easy and friendly use. In
this paper we show how to use this interface in order to test all the desired
properties.Comment: In Proceedings PROLE 2014, arXiv:1501.0169
Some subsystems of constant-depth Frege with parity
We consider three relatively strong families of subsystems of AC0[2]-Frege proof systems, i.e., propositional proof systems using constant-depth formulas with an additional parity connective, for which exponential lower bounds on proof size are known. In order of increasing strength, the subsystems are (i) constant-depth proof systems with parity axioms and the (ii) treelike and (iii) daglike versions of systems introduced by Krajíček which we call PKcd(⊕). In a PKcd(⊕)-proof, lines are disjunctions (cedents) in which all disjuncts have depth at most d, parities can only appear as the outermost connectives of disjuncts, and all but c disjuncts contain no parity connective at all.
We prove that treelike PKO(1)O(1)(⊕) is quasipolynomially but not polynomially equivalent to constant-depth systems with parity axioms. We also verify that the technique for separating parity axioms from parity connectives due to Impagliazzo and Segerlind can be adapted to give a superpolynomial separation between daglike PKO(1)O(1)(⊕) and AC0[2]-Frege; the technique is inherently unable to prove superquasipolynomial separations.
We also study proof systems related to the system Res-Lin introduced by Itsykson and Sokolov. We prove that an extension of treelike Res-Lin is polynomially simulated by a system related to daglike PKO(1)O(1)(⊕), and obtain an exponential lower bound for this system.Peer ReviewedPostprint (author's final draft
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
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
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