k-Provability in $$\hbox {PA}$$

We study the decidability of k-provability in $$\hbox {PA}$$ PA —the relation ‘being provable in $$\hbox {PA}$$ PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of $$\hbox {PA}$$ PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.

A Resolution Calculus for Second-order Logic with Eager Unification

The efficiency of the first-order resolution calculus is impaired when lifting it to higher-order logic. The main reason for that is the semi-decidability and infinitary natureof higher-order unification algorithms, which requires the integration of unification within the calculus and results in a non-efficient search for refutations.We present a modification of the constrained resolution calculus (Huet'72) which uses an eager unification algorithm while retaining completeness. Thealgorithm is complete with regard to bounded unification only, which for many cases, does not pose a problem in practice.

Hybrid Unification in the Description Logic EL

Unification in Description Logics (DLs) has been proposed as an inferenceservice that can, for example, be used to detect redundancies in ontologies.For the DL EL, which is used to define several largebiomedical ontologies, unification is NP-complete. However, the unification algorithms for EL developeduntil recently could not deal with ontologies containing general concept inclusions (GCIs).In a series of recent papers we have made some progress towards addressing this problem, but the ontologies thedeveloped unification algorithms can deal with need to satisfy a certain cycle restriction.In the present paper, we follow a different approach. Instead of restricting the input ontologies,we generalize the notion of unifiers to so-called hybrid unifiers. Whereas classical unifiers can be viewed as acyclic TBoxes,hybrid unifiers are cyclic TBoxes, which are interpreted together with the ontology of the input using a hybrid semantics thatcombines fixpoint and declarative semantics. We show that hybrid unification in EL is NP-complete

Hierarchical Combination of Unication Algorithms (Extended Abstract)

A critical question in unification theory is how to obtaina unification algorithm for the combination of non-disjointequational theories when there exists unification algorithmsfor the constituent theories. The problem is known to bedifficult and can easily be seen to be undecidable in thegeneral case. Therefore, previous work has focused onidentifying specific conditions and methods in which theproblem is decidable.We continue the investigation in this paper, building onprevious combination results and our own work.We are able to develop a novel approach to the non-disjointcombination problem. The approach is based on a new set ofrestrictions and combination method such that if the restrictionsare satisfied the method produces an algorithm for the unificationproblem in the union of non-disjoint equational theories.

Anti-Unification: Algorithms and Applications

The anti-unification problem of two terms t<sub>1</sub> and t<sub>2</sub> is concerned with finding a term t which generalizes both t<sub>1</sub> and t<sub>2</sub>. That is, the input terms should be substitution instances of the generalization term. Interesting generalizations are the least general ones. The purpose of anti-unification algorithms is to compute such least general generalizations.Research on anti-unification has been initiated more than four decades ago, with the pioneering works by Gordon~D.~Plotkin and John~C.~Reynolds. Since then, a number of algorithms and their modifications have been developed, addressing the problem in first-order or higher-order languages, for syntactic or equational theories, over ranked or unranked alphabets, with or without sorts/types, etc. Anti-unification has found applications in machine learning, inductive logic programming, case-based reasoning, analogy making, symbolic mathematical computing, software maintenance, program analysis, synthesis, transformation, and verification. Some of these algorithms and applications will be reviewed in the talk. We will also consider recent developments in unranked and higher-order generalization computation.

Bounded Higher-order Unification using Regular Terms

We present an algorithm for the bounded unification of higher-order terms.The algorithm extends G. P. Huet's pre-unification algorithm with rules for the generation and folding of regular terms.The concise form of the algorithm allows the reuse of the pre-unification correctness proof. Furthermore, the regular termscan be restricted in order to decide the unifiability problem.Finally, the algorithm avoids re-computation of terms in a non-deterministic search which leads to a better performance in practice when compared to other boundedunification algorithms.

Recent Advances in Unification for the EL Family

Unification in description logics (DLs) has been proposed as a means to detect redundancies in ontologies. The DL EL, even though quite inexpressive, can be used to formulate large medical ontologies like SNOMED CT. Previously, several unification algorithms for EL were developed that can only deal with acyclic terminologies. In this paper, we give an overview over our recent efforts to generalize these algorithms to allow for general concept inclusions, transitive roles, and role hierarchies. For our new algorithms to be complete, the ontology needs to satisfy a certain cycle restriction.

The use of types in designing unification algorithms: two case studies

We discuss the use of type systems in a non-strict sensewhen designing unification algorithms. We first give anew (rule-based) algorithm for an equational theory which representsa property of El-Gamal signature schemes and show howa type system can be used to prove termination of the algorithm.Lastly, we reproduce termination result for theory of partialexponentiation given earlier.

Proof-relevant unification: Dependent pattern matching with only the axioms of your type theory

Dependently typed languages such as Agda, Coq, and Idris use a syntactic first-order unification algorithm to check definitions by dependent pattern matching. However, standard unification algorithms implicitly rely on principles such asuniqueness of identity proofsandinjectivity of type constructors. These principles are inadmissible in many type theories, particularly in the new and promising branch known as homotopy type theory. As a result, programs and proofs in these new theories cannot make use of dependent pattern matching or other techniques relying on unification, and are as a result much harder to write, modify, and understand. This paper proposes a proof-relevant framework for reasoning formally about unification in a dependently typed setting. In this framework, unification rules compute not just a unifier but also a corresponding soundness proof in the form of anequivalencebetween two sets of equations. By rephrasing the standard unification rules in a proof-relevant manner, they are guaranteed to preserve soundness of the theory. In addition, it enables us to safely add new rules that can exploit the dependencies between the types of equations, such as rules for eta-equality of record types and higher dimensional unification rules for solving equations between equality proofs. Using our framework, we implemented a complete overhaul of the unification algorithm used by Agda. As a result, we were able to replace previousad-hocrestrictions with formally verified unification rules, fixing a substantial number of bugs in the process. In the future, we may also want to integrate new principles with pattern matching, for example, the higher inductive types introduced by homotopy type theory. Our framework also provides a solid basis for such extensions to be built on.