65 research outputs found
Strategic Issues, Problems and Challenges in Inductive Theorem Proving
Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP
Scavenger 0.1: A Theorem Prover Based on Conflict Resolution
This paper introduces Scavenger, the first theorem prover for pure
first-order logic without equality based on the new conflict resolution
calculus. Conflict resolution has a restricted resolution inference rule that
resembles (a first-order generalization of) unit propagation as well as a rule
for assuming decision literals and a rule for deriving new clauses by (a
first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201
A Proof Strategy Language and Proof Script Generation for Isabelle/HOL
We introduce a language, PSL, designed to capture high level proof strategies
in Isabelle/HOL. Given a strategy and a proof obligation, PSL's runtime system
generates and combines various tactics to explore a large search space with low
memory usage. Upon success, PSL generates an efficient proof script, which
bypasses a large part of the proof search. We also present PSL's monadic
interpreter to show that the underlying idea of PSL is transferable to other
ITPs.Comment: This paper has been submitted to CADE2
Building and Combining Matching Algorithms
International audienceThe concept of matching is ubiquitous in declarative programming and in automated reasoning. For instance, it is a key mechanism to run rule-based programs and to simplify clauses generated by theorem provers. A matching problem can be seen as a particular conjunction of equations where each equation has a ground side. We give an overview of techniques that can be applied to build and combine matching algorithms. First, we survey mutation-based techniques as a way to build a generic matching algorithm for a large class of equational theories. Second, combination techniques are introduced to get combined matching algorithms for disjoint unions of theories. Then we show how these combination algorithms can be extended to handle non-disjoint unions of theories sharing only constructors. These extensions are possible if an appropriate notion of normal form is computable
Removing Algebraic Data Types from Constrained Horn Clauses Using Difference Predicates
We address the problem of proving the satisfiability of Constrained Horn
Clauses (CHCs) with Algebraic Data Types (ADTs), such as lists and trees. We
propose a new technique for transforming CHCs with ADTs into CHCs where
predicates are defined over basic types, such as integers and booleans, only.
Thus, our technique avoids the explicit use of inductive proof rules during
satisfiability proofs. The main extension over previous techniques for ADT
removal is a new transformation rule, called differential replacement, which
allows us to introduce auxiliary predicates corresponding to the lemmas that
are often needed when making inductive proofs. We present an algorithm that
uses the new rule, together with the traditional folding/unfolding
transformation rules, for the automatic removal of ADTs. We prove that if the
set of the transformed clauses is satisfiable, then so is the set of the
original clauses. By an experimental evaluation, we show that the use of the
differential replacement rule significantly improves the effectiveness of ADT
removal, and we show that our transformation-based approach is competitive with
respect to a well-established technique that extends the CVC4 solver with
induction.Comment: 10th International Joint Conference on Automated Reasoning (IJCAR
2020) - version with appendix; added DOI of the final authenticated Springer
publication; minor correction
The use of data-mining for the automatic formation of tactics
This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques
Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints
The monadic shallow linear Horn fragment is well-known to be decidable and
has many application, e.g., in security protocol analysis, tree automata, or
abstraction refinement. It was a long standing open problem how to extend the
fragment to the non-Horn case, preserving decidability, that would, e.g.,
enable to express non-determinism in protocols. We prove decidability of the
non-Horn monadic shallow linear fragment via ordered resolution further
extended with dismatching constraints and discuss some applications of the new
decidable fragment.Comment: 29 pages, long version of CADE-26 pape
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