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arxiv: 2606.23516 · v1 · pith:7YJTJAJZnew · submitted 2026-06-22 · 💻 cs.LO

Towards an Automated Reasoning Tool for Complexity Analysis of Automated Reasoners

Louis Rustenholz , Manuel V. Hermenegildo , Pedro Lopez-Garcia , Alessio Mansutti , F\'elix Ridoux , Niki Vazou This is my paper

Pith reviewed 2026-06-26 06:27 UTC · model grok-4.3

classification 💻 cs.LO
keywords complexity analysisautomated reasoningabstract interpretationrecurrence equationsSMT solversquantifier eliminationoperator semantics
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The pith

A tool automates complexity analysis of automated reasoners by extracting recurrence equations via higher-order abstract interpretation on operator semantics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents the theory for a tool that automates the mechanical steps of complexity analysis for algorithms in automated reasoning, such as super-exponential quantifier elimination procedures. The user supplies key metrics and lemmas corresponding to the creative parts of a pen-and-paper proof. The tool then applies a higher-order abstract interpretation built on operator semantics to compile the symbolic program into generalized recurrence equations on numerical objects like interval-valued functions. These recurrences are solved by pre- and postfixpoint techniques that leverage SMT solvers and ideas from termination analysis to discover and verify bounds.

Core claim

The central claim is that the novel higher-order abstract interpretation technique built on operator semantics enables optimal abstract compilation of symbolic programs to purely numerical recursive representations such as recurrence equations on interval-valued functions, after which pre/postfixpoint-based methods with SMT solvers can discover and verify candidate bounds.

What carries the argument

Higher-order abstract interpretation on operator semantics, which performs abstract compilation from symbolic programs to numerical recursive forms.

If this is right

  • The pipeline separates human-provided creative input (metrics and lemmas) from automated extraction and solving steps.
  • Pre/postfixpoint techniques combined with SMT solvers can go beyond direct use of computer algebra systems for bound discovery.
  • Ideas from termination analysis research can be incorporated to improve the solving phase.
  • The approach targets algorithms whose manual complexity proofs involve super-exponential growth.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same extraction technique could be tested on recursive definitions arising outside automated reasoning, such as in symbolic computation libraries.
  • Tighter integration between the SMT-based bound verification and existing termination tools might produce sharper complexity results.
  • If the abstract compilation is optimal, the resulting bounds could sometimes be tighter than those obtained by hand.

Load-bearing premise

The higher-order abstract interpretation on operator semantics produces correct and optimal numerical recursive representations from the symbolic programs being analyzed.

What would settle it

An example algorithm whose manually derived complexity bound differs from the recurrence equations extracted by the tool or from the bounds verified by the prefixpoint method.

read the original abstract

We present the theory underpinning a complexity analysis tool (under development) that aims at automating tedious parts of the analysis of complex algorithms originating from the field of automated reasoning. Examples are given by super-exponential quantifier elimination procedures in real and integer arithmetics. Our tool implements the following pipeline. * Together with the algorithm to be analysed, the user (expert, e.g. the algorithm designer) can provide key metrics to track, and lemmas to improve the analysis. In pen-and-paper proofs, these correspond to the "non-tedious" and "creative" parts of the complexity analysis, that require human ingenuity. * The second step consists in the extraction of (generalised) recurrence equations. Here, we rely on a novel higher-order abstract interpretation technique, built on the concept of operator semantics. It enables (optimal) abstract compilation of symbolic programs to different kinds of purely numerical recursive representations, such as recurrence equations on interval-valued functions or numerical logic programs. * Finally, our tool solves the recurrence equations. We propose to go beyond the direct usage of computer algebra systems (CAS), and use pre/postfixpoint-based techniques to discover and verify candidate bounds on the solution. This approach makes use, in turn, of recent progress in SMT solvers, and can also be improved by techniques originating in termination analysis research.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents the theory for a complexity analysis tool (under development) targeting algorithms from automated reasoning, such as super-exponential quantifier elimination in real and integer arithmetic. It describes a pipeline in which the user supplies key metrics and lemmas, a novel higher-order abstract interpretation on operator semantics extracts generalized recurrence equations (on interval-valued functions or numerical logic programs), and pre/postfixpoint techniques combined with SMT solvers solve for bounds.

Significance. If the higher-order abstract interpretation can be shown to soundly and optimally compile symbolic programs to recurrence equations, the approach could automate substantial portions of complexity analysis that currently require human ingenuity, complementing existing work on termination and resource analysis. The use of SMT-based prefixpoint search for bound discovery is a plausible direction, but the manuscript supplies no derivations, soundness arguments, or examples that would allow evaluation of these claims.

major comments (2)
  1. [Abstract] Abstract, second bullet: the central claim that a 'novel higher-order abstract interpretation technique, built on the concept of operator semantics' enables optimal abstract compilation to recurrence equations is unsupported; the manuscript provides neither a definition of operator semantics, the abstract transfer functions, a soundness or optimality theorem, nor a single concrete extraction step on a toy program.
  2. [Abstract] Abstract, third bullet: the description of solving recurrence equations via pre/postfixpoint techniques with SMT solvers remains at the level of a high-level proposal; no indication is given of how candidate bounds are generated, how the SMT encoding is constructed, or how termination-analysis techniques are integrated.
minor comments (2)
  1. [Title] The title refers to 'Automated Reasoners' while the abstract focuses on algorithms originating from automated reasoning; a brief clarification of scope would help.
  2. [Abstract] The manuscript would benefit from references to prior work on abstract interpretation for complexity or resource analysis (e.g., in the context of term rewriting or functional programs).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript describing the theory for an automated complexity analysis tool under development. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, second bullet: the central claim that a 'novel higher-order abstract interpretation technique, built on the concept of operator semantics' enables optimal abstract compilation to recurrence equations is unsupported; the manuscript provides neither a definition of operator semantics, the abstract transfer functions, a soundness or optimality theorem, nor a single concrete extraction step on a toy program.

    Authors: We agree that the abstract states the central claim at a high level without embedding the supporting formal elements. The manuscript is positioned as an overview of the theory and pipeline for ongoing tool development rather than a fully expanded formal paper. To address the concern, we will revise by inserting a new subsection that (i) defines operator semantics, (ii) specifies the higher-order abstract transfer functions, (iii) states the soundness and optimality theorems, and (iv) walks through a concrete extraction example on a small symbolic program. This will make the novelty claim directly verifiable from the text. revision: yes

  2. Referee: [Abstract] Abstract, third bullet: the description of solving recurrence equations via pre/postfixpoint techniques with SMT solvers remains at the level of a high-level proposal; no indication is given of how candidate bounds are generated, how the SMT encoding is constructed, or how termination-analysis techniques are integrated.

    Authors: We acknowledge that the solving step is described concisely in the abstract. The body of the manuscript sketches the pre/postfixpoint approach but does not yet detail the practical mechanisms. In revision we will expand the corresponding section to explain (i) how candidate bounds are proposed (template-based generation combined with SMT-guided search), (ii) the precise SMT encoding used to certify prefixpoints, and (iii) the integration points with termination-analysis techniques such as ranking-function synthesis or dependency-pair methods. These additions will turn the proposal into a concrete, evaluable method. revision: yes

Circularity Check

0 steps flagged

No circularity: high-level pipeline description with no self-referential derivations or fitted quantities

full rationale

The paper presents an architectural outline for a complexity-analysis tool consisting of user-supplied metrics/lemmas, extraction of generalized recurrences via a described higher-order abstract interpretation on operator semantics, and postfixpoint/SMT solving. No equations, transfer functions, fitted parameters, or self-citations appear in the provided text that would make any claimed extraction or bound equivalent to its inputs by construction. The derivation chain therefore remains non-circular and self-contained at the level of a proposed methodology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no concrete free parameters, axioms, or invented entities; the description relies on standard background from abstract interpretation and SMT without introducing new fitted quantities or entities.

pith-pipeline@v0.9.1-grok · 5792 in / 1107 out tokens · 20597 ms · 2026-06-26T06:27:13.182022+00:00 · methodology

discussion (0)

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Reference graph

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