5 research outputs found
Computers and relevant logic : a project in computing matrix model structures for propositional logics
I present and discuss four classes of algorithm
designed as solutions to the problem of generating matrix
representations of model structures for some non-classical
propositional logics. I then go on to survey the output
from implementations of these algorithms and finally exhibit
some logical investigations suggested by that output.
All four algorithms traverse a search tree depthfirst.
In the case of the first and fourth methods the
tree is fixed by imposing a lexicographic order on possible
matrices, while the second and third create their search tree
dynamically as the job progresses. The first algorithm is a
simple "backtrack" with some pruning of the tree in response
to refutations of possible matrices. The fourth, the most
efficient we have for time, maximises the amount of pruning
while keeping the same basic form. The second, which uses
a large number of special properties of the logics in question,
and so requires some logical and algebraic knowledge on the
part of the programmer, finds the matrices at the tips of
branches only, while the third, due to P.A. Pritchard, is far
easier to program and tests a matrix at every node of the search
tree.
The logics with which I am concerned are in the "relevant"
group first seriously investigated by A.R. Anderson and N.D.
Belnap (see their Entailment: the logic of relevance and
necessity, 1975). The most surprising observation in my
preliminary survey of the numbers of matrices validating such
systems is that the typical models are not much like the models
normally taken as canonical for the logics. In particular the proportion of inconsistent models (validating some cases of the
scheme 'A & ~A') is much higher than might have been expected.
Among the logical investigations already suggested by the
quasi-empirical data now available in the form of matrices are
some work on the system R-W, including my theorem, proved in
chapter 2.3, that with the law of excluded middle it suffices
to trivialise naive set theory, and the little-noticed subject
of Ackermann constants (sentential constants) in these logics.
The formula which collapses naive set theory in R-W plus
A v ~A
is the most damaging set-theoretic antinomy known. The theorem
that there are at least 3088 Ackermann constants in the logic R
(chapter 2.4) could not reasonably have been proved without the
aid of a computer.
My major conclusion is that this work on applications of
computers in logical research has reached a point where we are
able not only to relieve logicians of some drudgery, but to
suggest theorems and insights of new and possibly important
kinds
Automated Deduction – CADE 28
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Automated Reasoning
This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book
Efficient local search for Pseudo Boolean Optimization
Algorithms and the Foundations of Software technolog