Ground Joinability and Connectedness in the Superposition Calculus

Duarte, André Ricardo Barbosa; Korovin, Konstantin

doi:10.1007/978-3-031-10769-6_11

Cited by 5 publications

(2 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Sorting algorithms, however, often require induction axioms that are more complex than instances of structural induction (5). Such axioms are typically instances of the computation/recursion induction schema, arising from divide-and-conquer strategies as introduced in Section 3.1.…”

Section: Computation Induction In Saturationmentioning

confidence: 99%

“…For example, the precedence of function quicksort is higher than of symbols nil, cons, append, f ilter < and f ilter ≥ , ensuring that quicksort function terms are expanded to their functional definitions. We further apply recent results of encompassment demodulation [5] to improve equality reasoning within saturation (-drc encompass). We use induction on data types (-ind struct), including complex data type terms (-indoct on).…”

Section: Implementation and Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

Saturating Sorting without Sorts

Georgiou,

Hajdu,

Kovács

EPiC Series in Computing

View full text Add to dashboard Cite

We present a first-order theorem proving framework for establishing the correctness of functional programs implementing sorting algorithms with recursive data structures. We formalize the semantics of recursive programs in many-sorted first-order logic and integrate sortedness/permutation properties within our first-order formalization. Rather than focus- ing on sorting lists of elements of specific first-order theories, such as integer arithmetic, our list formalization relies on a sort parameter abstracting (arithmetic) theories and hence concrete sorts. We formalize the permutation property of lists in first-order logic so that we automatically prove verification conditions of such algorithms purely by superpositon- based first-order reasoning. Doing so, we adjust recent efforts for automating induction in saturation. We advocate a compositional approach for automating proofs by induction re- quired to verify functional programs implementing and preserving sorting and permutation properties over parameterized list structures. Our work turns saturation-based first-order theorem proving into an automated verification engine by (i) guiding automated inductive reasoning with manual proof splits and (ii) fully automating inductive reasoning in satu- ration. We showcase the applicability of our framework over recursive sorting algorithms, including Mergesort and Quicksort.

show abstract

Section: Computation Induction In Saturationmentioning

confidence: 99%

Section: Implementation and Experimentsmentioning

confidence: 99%

Saturating Sorting without Sorts

Georgiou,

Hajdu,

Kovács

EPiC Series in Computing

View full text Add to dashboard Cite

show abstract

Ground Joinability and Connectedness in the Superposition Calculus

Duarte

Korovin

2022

Automated Reasoning

View full text Add to dashboard Cite

Problems in many theories axiomatised by unit equalities (UEQ), such as groups, loops, lattices, and other algebraic structures, are notoriously difficult for automated theorem provers to solve. Consequently, there has been considerable effort over decades in developing techniques to handle these theories, notably in the context of Knuth-Bendix completion and derivatives. The superposition calculus is a generalisation of completion to full first-order logic; however it does not carry over all the refinements that were developed for it, and is therefore not a strict generalisation. This means that (i) as of today, even state of the art provers for first-order logic based on the superposition calculus, while more general, are outperformed in UEQ by provers based on completion, and (ii) the sophisticated techniques developed for completion are not available in any problem which is not in UEQ. In particular, this includes key simplifications such as ground joinability, which have been known for more than 30 years. In fact, all previous completeness proofs for ground joinability rely on proof orderings and proof reductions, which are not easily extensible to general clauses together with redundancy elimination. In this paper we address this limitation and extend superposition with ground joinability, and show that under an adapted notion of redundancy, simplifications based on ground joinability preserve completeness. Another recently explored simplification in completion is connectedness. We extend this notion to “ground connectedness” and show superposition is complete with both connectedness and ground connectedness. We implemented ground joinability and connectedness in a theorem prover, iProver, the former using a novel algorithm which we also present in this paper, and evaluated over the TPTP library with encouraging results.

show abstract

On the (In-)Completeness of Destructive Equality Resolution in the Superposition Calculus

Waldmann

2024

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Bachmair’s and Ganzinger’s abstract redundancy concept for the Superposition Calculus justifies almost all operations that are used in superposition provers to delete or simplify clauses, and thus to keep the clause set manageable. Typical examples are tautology deletion, subsumption deletion, and demodulation, and with a more refined definition of redundancy joinability and connectedness can be covered as well. The notable exception is Destructive Equality Resolution, that is, the replacement of a clause $$x \not \approx t \vee C$$ x ≉ t ∨ C with $$x \notin \textrm{vars}(t)$$ x ∉ vars ( t ) by $$C\{x \mapsto t\}$$ C { x ↦ t } . This operation is implemented in state-of-the-art provers, and it is clearly useful in practice, but little is known about how it affects refutational completeness. We demonstrate on the one hand that the naive addition of Destructive Equality Resolution to the standard abstract redundancy concept renders the calculus refutationally incomplete. On the other hand, we present several restricted variants of the Superposition Calculus that are refutationally complete even with Destructive Equality Resolution.

show abstract

Ground Joinability and Connectedness in the Superposition Calculus

Cited by 5 publications

References 25 publications

Saturating Sorting without Sorts

Saturating Sorting without Sorts

Ground Joinability and Connectedness in the Superposition Calculus

On the (In-)Completeness of Destructive Equality Resolution in the Superposition Calculus

Contact Info

Product

Resources

About