{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "tarski-real-closed-field-quantifier-elimination-1948",
    "title": "Tarski: quantifier elimination and decidability of elementary real algebra",
    "summary": "Tarski gave an effective procedure converting first-order real-algebra formulas built from polynomial equations, inequalities and logical quantifiers into equivalent quantifier-free formulas. For closed sentences this yields a terminating truth-decision method; coordinate translation gives a corresponding decision method for elementary Euclidean geometry.",
    "boundary": "The language ranges over real numbers with ordered-ring operations and encodable constants, not arbitrary sets/functions, unrestricted integer quantifiers or every theorem of mathematics. Parameters can remain free in a quantifier-free output; this is not a decision oracle for arbitrary uncomputably specified real coefficients. Adding an integer predicate or sine is explicitly outside the decidable language. The report does not supply a practical uniform runtime guarantee, an implementation or a resolution of real exponentiation. Fixed-size instances do not prove a variable-size universal conjecture.",
    "area": "Mathematical logic and real algebra",
    "topic": "Mathematics",
    "year": 1948,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "tarski-real-closed-field-quantifier-elimination-1948",
    "target": {
      "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
      "version": 1,
      "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
    },
    "humans": [
      "Alfred Tarski"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-tarski-real-algebra-decision-1948",
        "title": "A Decision Method for Elementary Algebra and Geometry",
        "url": "https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf",
        "date": "1948-08-01"
      }
    ],
    "points": "420000",
    "published_grade": {
      "id": "tarski-real-closed-field-quantifier-elimination-1948",
      "entry_id": "tarski-real-closed-field-quantifier-elimination-1948",
      "band": "S",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
        "version": 1,
        "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
      },
      "what_changed": "Tarski gave an effective procedure converting first-order real-algebra formulas built from polynomial equations, inequalities and logical quantifiers into equivalent quantifier-free formulas. For closed sentences this yields a terminating truth-decision method; coordinate translation gives a corresponding decision method for elementary Euclidean geometry.",
      "scope": "Tarski gave an effective procedure converting first-order real-algebra formulas built from polynomial equations, inequalities and logical quantifiers into equivalent quantifier-free formulas. For closed sentences this yields a terminating truth-decision method; coordinate translation gives a corresponding decision method for elementary Euclidean geometry. The language ranges over real numbers with ordered-ring operations and encodable constants, not arbitrary sets/functions, unrestricted integer quantifiers or every theorem of mathematics. Parameters can remain free in a quantifier-free output; this is not a decision oracle for arbitrary uncomputably specified real coefficients. Adding an integer predicate or sine is explicitly outside the decidable language. The report does not supply a practical uniform runtime guarantee, an implementation or a resolution of real exponentiation. Fixed-size instances do not prove a variable-size universal conjecture.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
        "version": 1,
        "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Alfred Tarski",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "A Decision Method for Elementary Algebra and Geometry",
            "url": "https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "A Decision Method for Elementary Algebra and Geometry",
            "url": "https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The language ranges over real numbers with ordered-ring operations and encodable constants, not arbitrary sets/functions, unrestricted integer quantifiers or every theorem of mathematics. Parameters can remain free in a quantifier-free output; this is not a decision oracle for arbitrary uncomputably specified real coefficients. Adding an integer predicate or sine is explicitly outside the decidable language. The report does not supply a practical uniform runtime guarantee, an implementation or a resolution of real exponentiation. Fixed-size instances do not prove a variable-size universal conjecture.",
        "domain": "Mathematical logic and real algebra"
      },
      "sources": [
        {
          "citation": "A Decision Method for Elementary Algebra and Geometry",
          "url": "https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf"
        }
      ],
      "statement": "# Tarski: quantifier elimination and decidability of elementary real algebra\n\n## Can a computer remove the word exists?\n\nSome mathematical questions ask whether a real number with a certain property exists. Tarski showed that a large, precisely defined class of such questions can be transformed into equivalent statements without quantifiers. A quantifier is a logical instruction such as there exists or for every. The permitted ingredients are polynomial equations, inequalities and logical combinations over real numbers. This is an algorithmic result about a language, not a promise to decide every possible mathematical statement.\n\n## A tiny example reveals the idea\n\nConsider the invented question: does there exist a real x for which x squared plus a equals zero? If a is positive, a square cannot cancel it. If a is zero, x=zero works. If a is negative, a square root of minus a works. The complete answer is therefore a is less than or equal to zero. The variable x and the phrase there exists have disappeared, while the parameter a remains in the answer.\n\n## The domain is part of the problem\n\nNow ask whether x squared equals two. Over the real numbers the answer is yes. Over the integers the answer is no. Allowing an unrestricted integer predicate inside the language changes the decision problem. Likewise, adding functions such as sine is not a harmless spelling change to a polynomial question. Tarski's method depends on exactly what expressions and quantified objects are allowed. A correct answer for one language does not automatically transfer to another.\n\n## Terminating does not mean fast\n\nA decision method must finish after a finite number of steps on every permitted input. That is different from finishing quickly enough for a large practical problem. The 1948 report establishes a broad existence and construction result, without a convenient uniform runtime promise for users. Later algorithmic improvements have their own contributions. The report was prepared for publication by J. C. C. McKinsey; that publishing role does not change the original scientific byline.\n\n## Build a small collection of exact questions\n\nAsk your agent to explain five simple real-polynomial existence questions, including a parameterized quadratic and an impossible negative square. Require an equivalent condition rather than just a yes or no for one sampled value. Keep any numerical sampling separate from a universal argument. This exercise makes the attraction of quantifier elimination concrete: a complicated-looking family can sometimes be replaced by an explicit condition that another person can directly inspect.\n\n## Scientific scope\n\nThe language ranges over real numbers with ordered-ring operations and encodable constants, not arbitrary sets/functions, unrestricted integer quantifiers or every theorem of mathematics. Parameters can remain free in a quantifier-free output; this is not a decision oracle for arbitrary uncomputably specified real coefficients. Adding an integer predicate or sine is explicitly outside the decidable language. The report does not supply a practical uniform runtime guarantee, an implementation or a resolution of real exponentiation. Fixed-size instances do not prove a variable-size universal conjecture.\n\n## Sources for this explanation\n\n- [A Decision Method for Elementary Algebra and Geometry](https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Tarski gave an effective procedure converting first-order real-algebra formulas built from polynomial equations, inequalities and logical quantifiers into equivalent quantifier-free formulas. For closed sentences this yields a terminating truth-decision method; coordinate translation gives a corresponding decision method for elementary Euclidean geometry.",
      "title": "Tarski: quantifier elimination and decidability of elementary real algebra"
    },
    "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
      "version": 1,
      "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:962453aa57022a7c1078af4b39c407ba93018d194cbc26f8c1ca428e73f7d303"
  },
  "target": {
    "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
    "version": 1,
    "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "tarski-real-closed-field-quantifier-elimination-1948",
    "entry_id": "tarski-real-closed-field-quantifier-elimination-1948",
    "revision": 1,
    "title": "Tarski: quantifier elimination and decidability of elementary real algebra",
    "context": "Mathematical logic and real algebra",
    "track": "Exact theorem and reusable mathematical method",
    "band": "S",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Tarski gave an effective procedure converting first-order real-algebra formulas built from polynomial equations, inequalities and logical quantifiers into equivalent quantifier-free formulas. For closed sentences this yields a terminating truth-decision method; coordinate translation gives a corresponding decision method for elementary Euclidean geometry.",
    "scope": "Tarski gave an effective procedure converting first-order real-algebra formulas built from polynomial equations, inequalities and logical quantifiers into equivalent quantifier-free formulas. For closed sentences this yields a terminating truth-decision method; coordinate translation gives a corresponding decision method for elementary Euclidean geometry. The language ranges over real numbers with ordered-ring operations and encodable constants, not arbitrary sets/functions, unrestricted integer quantifiers or every theorem of mathematics. Parameters can remain free in a quantifier-free output; this is not a decision oracle for arbitrary uncomputably specified real coefficients. Adding an integer predicate or sine is explicitly outside the decidable language. The report does not supply a practical uniform runtime guarantee, an implementation or a resolution of real exponentiation. Fixed-size instances do not prove a variable-size universal conjecture.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The language ranges over real numbers with ordered-ring operations and encodable constants, not arbitrary sets/functions, unrestricted integer quantifiers or every theorem of mathematics. Parameters can remain free in a quantifier-free output; this is not a decision oracle for arbitrary uncomputably specified real coefficients. Adding an integer predicate or sine is explicitly outside the decidable language. The report does not supply a practical uniform runtime guarantee, an implementation or a resolution of real exponentiation. Fixed-size instances do not prove a variable-size universal conjecture.",
    "references": [
      {
        "title": "A Decision Method for Elementary Algebra and Geometry",
        "url": "https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
      "version": 1,
      "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "420000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "tarski-real-closed-field-quantifier-elimination-1948",
    "target": {
      "record_id": "urn:uuid:e41f4675-2b98-46cb-abf4-a9ffb85872c5",
      "version": 1,
      "content_hash": "sha256:8f674e8e5d43f78d6e3fd85c9efdaad295e20ed2ae157a6c2840a62ea692aba2"
    },
    "sections": [
      {
        "title": "Can a computer remove the word exists?",
        "text": "Some mathematical questions ask whether a real number with a certain property exists. Tarski showed that a large, precisely defined class of such questions can be transformed into equivalent statements without quantifiers. A quantifier is a logical instruction such as there exists or for every. The permitted ingredients are polynomial equations, inequalities and logical combinations over real numbers. This is an algorithmic result about a language, not a promise to decide every possible mathematical statement."
      },
      {
        "title": "A tiny example reveals the idea",
        "text": "Consider the invented question: does there exist a real x for which x squared plus a equals zero? If a is positive, a square cannot cancel it. If a is zero, x=zero works. If a is negative, a square root of minus a works. The complete answer is therefore a is less than or equal to zero. The variable x and the phrase there exists have disappeared, while the parameter a remains in the answer."
      },
      {
        "title": "The domain is part of the problem",
        "text": "Now ask whether x squared equals two. Over the real numbers the answer is yes. Over the integers the answer is no. Allowing an unrestricted integer predicate inside the language changes the decision problem. Likewise, adding functions such as sine is not a harmless spelling change to a polynomial question. Tarski's method depends on exactly what expressions and quantified objects are allowed. A correct answer for one language does not automatically transfer to another."
      },
      {
        "title": "Terminating does not mean fast",
        "text": "A decision method must finish after a finite number of steps on every permitted input. That is different from finishing quickly enough for a large practical problem. The 1948 report establishes a broad existence and construction result, without a convenient uniform runtime promise for users. Later algorithmic improvements have their own contributions. The report was prepared for publication by J. C. C. McKinsey; that publishing role does not change the original scientific byline."
      },
      {
        "title": "Build a small collection of exact questions",
        "text": "Ask your agent to explain five simple real-polynomial existence questions, including a parameterized quadratic and an impossible negative square. Require an equivalent condition rather than just a yes or no for one sampled value. Keep any numerical sampling separate from a universal argument. This exercise makes the attraction of quantifier elimination concrete: a complicated-looking family can sometimes be replaced by an explicit condition that another person can directly inspect."
      }
    ],
    "references": [
      {
        "title": "A Decision Method for Elementary Algebra and Geometry",
        "url": "https://archive.org/download/a-decision-method-for-elementary-algebra-and-geometry-by-tarski-images/A%20decision%20method%20for%20elementary%20algebra%20and%20geometry%20by%20Tarski.pdf",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}