105 research outputs found
Positive Unit Hyperresolution Tableaux and Their Application to Minimal Model Generation
Minimal Herbrand models of sets of first-order clauses are useful in several areas of computer science, e.g. automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are
inappropriate because they generate nonminimal Herbrand models and can
be inefficient. This article describes an approach for generating the minimal
Herbrand models of sets of first-order clauses. The approach builds upon
positive unit hyperresolution (PUHR) tableaux, that are in general smaller
than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The first one expands PUHR tableaux
depth-first relying on a complement splitting expansion rule and on a form
of backtracking involving constraints. A Prolog implementation, named
MM-SATCHMO, of this procedure is given and its performance on benchmark suites is reported. The second minimal model generation procedure
performs a breadth-first, constrained expansion of PUHR (complement)
tableaux. Both procedures are optimal in the sense that each minimal model
is constructed only once, and the construction of nonminimal models is interrupted as soon as possible. They are complete in the following sense
The depth-first minimal model generation procedure computes all minimal
Herbrand models of the considered clauses provided these models are all
finite. The breadth-first minimal model generation procedure computes all
finite minimal Herbrand models of the set of clauses under consideration.
The proposed procedures are compared with related work in terms of both
principles and performance on benchmark problems
MizAR 60 for Mizar 50
As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60% of the Mizar theorems in the hammer setting. We also automatically prove 75% of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically
Learning-Assisted Automated Reasoning with Flyspeck
The considerable mathematical knowledge encoded by the Flyspeck project is
combined with external automated theorem provers (ATPs) and machine-learning
premise selection methods trained on the proofs, producing an AI system capable
of answering a wide range of mathematical queries automatically. The
performance of this architecture is evaluated in a bootstrapping scenario
emulating the development of Flyspeck from axioms to the last theorem, each
time using only the previous theorems and proofs. It is shown that 39% of the
14185 theorems could be proved in a push-button mode (without any high-level
advice and user interaction) in 30 seconds of real time on a fourteen-CPU
workstation. The necessary work involves: (i) an implementation of sound
translations of the HOL Light logic to ATP formalisms: untyped first-order,
polymorphic typed first-order, and typed higher-order, (ii) export of the
dependency information from HOL Light and ATP proofs for the machine learners,
and (iii) choice of suitable representations and methods for learning from
previous proofs, and their integration as advisors with HOL Light. This work is
described and discussed here, and an initial analysis of the body of proofs
that were found fully automatically is provided
ProofWatch: Watchlist Guidance for Large Theories in E
Watchlist (also hint list) is a mechanism that allows related proofs to guide
a proof search for a new conjecture. This mechanism has been used with the
Otter and Prover9 theorem provers, both for interactive formalizations and for
human-assisted proving of open conjectures in small theories. In this work we
explore the use of watchlists in large theories coming from first-order
translations of large ITP libraries, aiming at improving hammer-style
automation by smarter internal guidance of the ATP systems. In particular, we
(i) design watchlist-based clause evaluation heuristics inside the E ATP
system, and (ii) develop new proof guiding algorithms that load many previous
proofs inside the ATP and focus the proof search using a dynamically updated
notion of proof matching. The methods are evaluated on a large set of problems
coming from the Mizar library, showing significant improvement of E's standard
portfolio of strategies, and also of the previous best set of strategies
invented for Mizar by evolutionary methods.Comment: 19 pages, 10 tables, submitted to ITP 2018 at FLO
Deduction by combining semantic tableaux and integer programming
. In this paper we propose to extend the current capabilities of automated reasoning systems by making use of techniques from integer programming. We describe the architecture of an automated reasoning system based on a Herbrand procedure (enumeration of formula instances) on clauses. The input are arbitrary sentences of first-order logic. The translation into clauses is done incrementally and is controlled by a semantic tableau procedure using unification. This amounts to an incremental polynomial CNF transformation which at the same time encodes part of the tableau structure and, therefore, tableau-specific refinements that reduce the search space. Checking propositional unsatisfiability of the resulting sequence of clauses can either be done with a symbolic inference system such as the Davis-Putnam procedure or it can be done using integer programming. If the latter is used a number of advantages become apparent. Introduction In this paper we propose to extend the current capabilit..
Premise Selection for Mathematics by Corpus Analysis and Kernel Methods
Smart premise selection is essential when using automated reasoning as a tool
for large-theory formal proof development. A good method for premise selection
in complex mathematical libraries is the application of machine learning to
large corpora of proofs. This work develops learning-based premise selection in
two ways. First, a newly available minimal dependency analysis of existing
high-level formal mathematical proofs is used to build a large knowledge base
of proof dependencies, providing precise data for ATP-based re-verification and
for training premise selection algorithms. Second, a new machine learning
algorithm for premise selection based on kernel methods is proposed and
implemented. To evaluate the impact of both techniques, a benchmark consisting
of 2078 large-theory mathematical problems is constructed,extending the older
MPTP Challenge benchmark. The combined effect of the techniques results in a
50% improvement on the benchmark over the Vampire/SInE state-of-the-art system
for automated reasoning in large theories.Comment: 26 page
Proceedings of the Joint Automated Reasoning Workshop and Deduktionstreffen: As part of the Vienna Summer of Logic – IJCAR 23-24 July 2014
Preface
For many years the British and the German automated reasoning communities have successfully run independent series of workshops for anybody working in the area of automated reasoning. Although open to the general
public they addressed in the past primarily the British and the German communities, respectively. At the occasion of the Vienna Summer of Logic the two series have a joint event in Vienna as an IJCAR workshop. In the spirit of the two series there will be only informal proceedings with abstracts of the works presented. These are collected in this document. We have tried to maintain the informal open atmosphere of the two series and have welcomed in particular research students to present their work. We have solicited for all work related to automated reasoning and its applications with a particular interest in work-in-progress and the presentation of half-baked ideas.
As in the previous years, we have aimed to bring together researchers from all areas of automated reasoning in order to foster links among researchers from various disciplines; among theoreticians, implementers and users alike, and among international communities, this year not just the British and German communities
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