ABSTRACT. State-of-the-art refutation systems for SAT are largely based on the derivation of clauses meeting some redundancy criteria, ensuring their addition to a formula does not alter its satisfiability. However, there are strong propositional reasoning techniques whose inferences are not easily expressed in such systems. This paper extends the redundancy framework beyond clauses to Boolean constraints in general, and develops efficiently checkable refutation systems using redundancy properties for Binary Decision Diagrams (BDDs). Using a form of reverse unit propagation for conjunctions of BDDs, these systems capture Gaussian elimination reasoning over XOR constraints encoded in a formula, without the need for clausal translations or extension variables. Notably, these systems generalize those based on the strong Propagation Redundancy (PR) property, without an increase in complexity.

Multi-Dimensional Interpretation Methods for Termination of Term Rewriting

ABSTRACT. Interpretation methods remain fundamental technique in termination analysis for term rewriting throughout its history. From time to time remarkable instances of interpretation methods appeared, such as polynomial interpretations, matrix interpretations, arctic interpretations, and their variants. In this paper we introduce a general framework, the multi-dimensional interpretation method, that subsumes and generalizes many of these variants. Employing the notion of derivers, thesoundness of the proposed method is proved in an elegant way. We implement the proposed method in termination prover NaTT and verify itspower through experiments.

Finding Good Proofs for Description Logic Entailments Using Recursive Quality Measures

ABSTRACT. Logic-based approaches to AI have the advantage that their
behavior can in principle be explained to a user. If, for instance, a
Description Logic reasoner derives a consequence that triggers some
action of the overall system, then one can explain such an entailment by
presenting a proof of the consequence in an appropriate calculus. How
comprehensible such a proof is depends not only on the employed calculus,
but also on the properties of the particular proof, such as its overall size,
its depth, the complexity of the employed axioms and proof steps, etc.
For this reason, we want to determine the complexity of generating proofs
that are below a certain threshold w.r.t. a given measure of proof quality.
Rather than investigating this problem for a fixed proof calculus and a
fixed measure, we aim for general results that hold for wide classes of
calculi and measures. In previous work, we first restricted the attention
to a setting where proof size is used to measure the quality of a proof.
We then extended the approach to a more general setting, but important
measures such as proof depth were not covered. In the present paper, we
provide results for a class of measures called recursive, which yields lower
complexities and also encompasses proof depth. In addition, we close
some gaps left open in our previous work, thus providing a comprehensive
picture of the complexity landscape.

Computing Optimal Repairs of Quantified ABoxes w.r.t. Static EL TBoxes

ABSTRACT. The application of automated reasoning approaches to Description Logic (DL) ontologies may produce certain consequences that either are deemed to be wrong or should be hidden for privacy reasons. The question is then how to repair the ontology such that the unwanted consequences can no longer be deduced. An optimal repair is one where the least amount of other consequences is removed. Most of the previous approaches to ontology repair are of a syntactic nature in that they remove or weaken the axioms explicitly present in the ontology, and thus cannot achieve semantic optimality. In previous work, we have addressed the problem of computing optimal repairs of (quantified) ABoxes, where the unwanted consequences are described by concept assertions of the lightweight DL EL.

In the present paper, we improve on the results achieved so far in two ways. First, we allow for the presence of terminological knowledge in the form of an EL TBox. This TBox is assumed to be static in the sense that it cannot be changed in the repair process. Second, the construction of optimal repairs described in our previous work is best case exponential. We introduce an optimized construction that is exponential only in the worst case. First experimental results indicate that this reduces the size of the computed optimal repairs considerably.

Generalized Completeness for SOS Resolution and its Application to a New Notion of Relevance

ABSTRACT. We prove the SOS strategy for first-order resolution to be refutationally
complete on a clause set $N$ and set-of-support $S$ if and only
if there exists a clause in $S$ that occurs in a resolution refutation
from $N\cup S$. This strictly generalizes
and sharpens the original completeness result requiring $N$ to
be satisfiable. The generalized SOS completeness result
supports automated reasoning on a new notion of relevance aiming at
capturing the support of a clause in the refutation
of a clause set.
A clause $C$ is \emph{relevant} for refuting a clause set $N$ if
$C$ occurs in any refutation of $N$. The clause $C$ is \emph{semi-relevant},
if it occurs in some refutation, i.e., if there exists an
SOS refutation with set-of-support $S = \{C\}$ from $N\setminus\{C\}$.
A clause that does not occur in any refutation from $N$ is \emph{irrelevant}, i.e.,
it is not semi-relevant.
Our new notion of relevance separates clauses in a proof that are ultimately needed from clauses that may be replaced by different clauses.
In this way it provides insights in refutations beyond existing notions towards proof explanation such as the notion of an unsatisfiable core.

ABSTRACT. AVATAR is an elegant and effective way to split clauses in a saturation prover using a SAT solver. But is it refutationally complete? And how does it relate to other splitting architectures? To answer these questions, we present a unifying framework that extends a saturation calculus (e.g., superposition) with splitting and embeds the result in a prover guided by a SAT solver. The framework also allows us to study locking, a subsumption-like mechanism based on the current propositional model. Various architectures are instances of the framework, including AVATAR, labeled splitting, and SMT with quantifiers.

ABSTRACT. Integers are ubiquitous in programming and therefore also in applications of program
analysis and verification. Such applications often require some sort of inductive reasoning. In this paper we analyze the challenge of automating inductive reasoning with integers. We introduce inference rules for integer induction within the saturation framework of first-order theorem proving. We implemented these rules in the Vampire theorem prover and evaluated our work against comparable state-of-the-art reasoners. Our results demonstrate the strength of our approach by solving new problems inspired by program analysis and mathematical properties of integers.

Superposition with First-Class Booleans and Inprocessing Clausification

ABSTRACT. We present a complete superposition calculus for first-order logic with an interpreted Boolean type. Our motivation is to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, such as higher-order logic, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing. Working directly on formulas, our calculus avoids the costly axiomatic encoding of the theory of Booleans into first-order logic and offers various ways to interleave clausification with other derivation steps. We evaluate our calculus using the Zipperposition theorem prover, and observe that, with no tuning of heuristic parameters, our approach is on a par with the state-of-the-art approach.

ABSTRACT. We recently designed two proof calculi as stepping stones towards superposition for full higher-order logic: Boolean-free lambda-superposition and superposition for first-order logic with interpreted Booleans. Stepping on these stones, we finally reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics. In addition to the complexity of combining the calculus's two predecessors, new challenges arise from the interplay between lambda-terms and Booleans. Our implementation in Zipperposition outperforms all other higher-order theorem provers and is on a par with an earlier, pragmatic prototype of Booleans in Zipperposition.

ABSTRACT. Superposition is among the most successful calculi for first-order logic. Its
extension to higher-order logic introduces new challenges such as infinitely
branching inference rules, new possibilities such as native reasoning with
formulas, and the need to curb the explosion of specific higher-order rules. We
describe techniques that address these issues and extensively evaluate their
implementation in the Zipperposition theorem prover. Largely thanks to them,
Zipperposition won the higher-order division of the CASC-J10 competition.

Dual Proof Generation for Quantified Boolean Formulas with a BDD-Based Solver

ABSTRACT. Existing proof-generating quantified Boolean formula (QBF) proof solvers must construct a different type of proof depending on whether the formula is false (refutation) or true (satisfaction). We show that a QBF solver based on ordered binary decision diagrams (BDDs) can emit a single dual proof as it operates, supporting either outcome. This form consists of a sequence of equivalence-preserving clause addition and deletion steps in an extended resolution framework. For a false formula, the proof will terminate with the empty clause, indicating conflict. For a true one, it will terminate with all clauses deleted, indicating tautology. Both the length of the proof and the time required to check it are proportional to the total number of BDD operations performed. We evaluate the capabilities of our solver with a scalable benchmark based on a two-player tiling game.

Reliable Reconstruction of Fine-Grained Proofs in a Proof Assistant

ABSTRACT. We present a fast and reliable reconstruction of proofs generated by the SMT solver veriT in Isabelle. The fine-grained proof format makes the reconstruction simple and efficient. For typical proof steps, such as arithmetic reasoning and skolemization, our reconstruction can avoid expensive search. By skipping proof steps that are irrelevant for Isabelle, the performance of proof checking is improved. Our method increases the success rate of Sledgehammer by halving the failure rate and reduces the checking time by 13%. We provide a detailed evaluation of the reconstruction time for each rule. It reveals that the runtime is influenced by both simple rules that appear very often and common complex rules.