Disproving in First-Order Logic with Definitions, Arithmetic and Finite Domains

181 Pages·2017·0.96 MB·English

by  Josh Bax

Disproving in First-Order Logic
with Definitions, Arithmetic and
Finite Domains
Joshua Bax
A thesis submitted for the degree of
Doctor of Philosophy
The Australian National University
November 2017
(cid:13)c Joshua Bax 2017

Except where otherwise indicated, this thesis consists of my original work.
Joshua Bax
23 November 2017

For my parents.

Acknowledgments
First of all, thanks to my supervisor Peter Baumgartner. You were always around to
answermyquestionsandthisthesisisintheend,theresultofourmanydiscussions.
Also I’d like to thank the other members of my panel, Professor John Slaney and
Professor Phil Kilby. Thanks also to Uwe Waldmann, for allowing me to share in
your research.
ThankstomyofficematesandfellowPhDstudentsJanKuesterandMohammad
Abdulaziz for helping me to understand this business, and for shooting down many
stupid ideas late on Friday afternoons. Bruno Wolzenlogel-Paleo was also included
in this triage of ideas.
Thanks to Geoff Suttcliffe and Christian Suttner for their efforts on the TPTP
library, as well the CASC competition. Without any of those this would have been a
brief thesis. Thanks to the CADE conference organizers for monetary support also.
Thanks to Data61 née NICTA, for generous funding, and thanks to the many
people there who provided both inspiration and support. I’m proud to have you all
as colleagues.
Lastly, thanks to my parents and An Ran for their unwavering support through-
out. Ireallycouldn’thavedoneitwithoutyou. ThosewhomImayhavemissed, rest
assured you have my gratitdue.
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Abstract
This thesis explores several methods which enable a first-order reasoner to conclude
satisfiability of a formula modulo an arithmetic theory. The most general method
requires restricting certain quantifiers to range over finite sets; such assumptions
are common in the software verification setting. In addition, the use of first-order
reasoning allows for an implicit representation of those finite sets, which can avoid
scalability problems that affect other quantified reasoning methods. These new tech-
niques form a useful complement to existing methods that are primarily aimed at
proving validity.
The Superposition calculus for hierarchic theory combinations provides a basis
for reasoning modulo theories in a first-order setting. The recent account of ‘weak
abstraction’andrelatedimprovementsmakeanimplementationofthecalculusprac-
tical. Also,forseverallogicaltheoriesofinterestSuperpositionisaneffectivedecision
procedure for the quantifier free fragment.
The first contribution is an implementation of that calculus (Beagle), including
anoptimizedimplementationofCooper’salgorithmforquantifiereliminationinthe
theory of linear integer arithmetic. This includes a novel means of extracting values
for quantified variables in satisfiable integer problems. Beagle won an efficiency
awardatCADEAutomatedtheoremproverSystemCompetition(CASC)-J7,andwon
the arithmetic non-theorem category at CASC-25. This implementation is the start
point for solving the ‘disproving with theories’ problem.
Some hypotheses can be disproved by showing that, together with axioms the
hypothesis is unsatisfiable. Often this is relative to other axioms that enrich a base
theory by defining new functions. In that case, the disproof is contingent on the
satisfiability of the enrichment.
Satisfiability in this context is undecidable. Instead, general characterizations of
definitionformulas,whichdonotalterthesatisfiabilitystatusofthemainaxioms,are
given. These general criteria apply to recursive definitions, definitions over lists, and
to arrays. This allows proving some non-theorems which are otherwise intractable,
and justifies similar disproofs of non-linear arithmetic formulas.
When the hypothesis is contingently true, disproof requires proving existence of
a model. If the Superposition calculus saturates a clause set, then a model exists,
but only when the clause set satisfies a completeness criterion. This requires each
instance of an uninterpreted, theory-sorted term to have a definition in terms of
theory symbols.
The second contribution is a procedure that creates such definitions, given that
a subset of quantifiers range over finite sets. Definitions are produced in a counter-
example driven way via a sequence of over and under approximations to the clause
set. Two descriptions of the method are given: the first uses the component solver
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modularly, but has an inefficient counter-example heuristic. The second is more
general, correcting many of the inefficiencies of the first, yet it requires tracking
clauses through a proof. This latter method is shown to apply also to lists and to
problems with unbounded quantifiers.
Together, these tools give new ways for applying successful first-order reasoning
methods to problems involving interpreted theories.