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Beyond the Turing threshold: Productive grammars generate essentially undecidable languages

A theoretical paper designs formal grammars that emulate Post's productive sets, generating languages that are provably beyond Turing decidability.

The paper elaborates on Emil Post's productive sets, which are not even semi-computable, and builds formal grammars that emulate their construction over natural numbers. The resulting languages are shown to be essentially undecidable, placing them beyond Turing decidability. This is pure computability and formal language theory with limited direct security relevance.

arXiv cs.CR · 6d agoResearch1

Microsoft September 2026 Patch Tuesday fixes 966 flaws, 2 zero-days

Microsoft's September 2026 Patch Tuesday fixes a record 966 flaws, including two Windows zero-days actively exploited to gain SYSTEM privileges.

Microsoft's September 2026 Patch Tuesday addresses a record 966 vulnerabilities, including 105 rated Critical, 81 of them remote code execution bugs. Two zero-days were actively exploited: a Windows Update Stack link-following flaw and a Windows ALPC heap-based buffer overflow, both allowing local elevation to SYSTEM privileges. The ALPC flaw was reported by Volexity and Proofpoint researchers, while the Update Stack flaw was credited to Romain Deperne and the Microsoft Threat Intelligence Centre. Microsoft shared no details on how the flaws were exploited in attacks.

BleepingComputer · 7d agoAdvisory in the wildCVE-2026-69805CVE-2026-58649CVE-2026-69806+27 CVEs1

Bidding war for defunct Spirit Airlines’ employee data will not die

AI firm Micro1 bid $12.5 million for Spirit Airlines' 600 million employee email and chat records, challenging Google's $10 million auction win.

After Spirit Airlines entered bankruptcy, Google won an auction for the airline's data trove with a $10 million bid, beating Mercor's $7.5 million offer, but Micro1 has now counter-offered $12.5 million. The data includes about 600 million email and chat records from 17,000 employees, 17 million OneDrive files, 20.5 million SharePoint items, and over 30 million customer service calls, valuable for AI training. Former employees' unions, including the Association of Flight Attendants-CWA representing 5,500 flight attendants, have taken legal action to block the sale over privacy concerns.

CSO Online · 12d agoAI industry