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    3-SAT can be reduced to INDEPENDENT SET in polynomial time — Carmelics
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    Home/Modality & Possibility
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    3-SAT can be reduced to INDEPENDENT SET in polynomial time

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

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    Reasons For

    1 perspective
    Reason for
    ?
    • 1.A graph G_φ with n triangles can be constructed from a 3-CNF formula φ with n clauses in polynomial time (O(n²) edges from 3n vertices)
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    • 2.If φ is satisfiable, then a satisfying valuation makes at least one literal true per clause, yielding an independent set of size n in G_φ by selecting one node per triangle
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    • 3.If G_φ has an independent set of size n, then by construction it contains exactly one node per triangle, and edges between contradictory literals across triangles prevent contradictions, so a satisfying valuation for φ can be constructed
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The reduction assumes a classical, bivalent semantics where every literal is either true or false, excluding paraconsistent or many-valued logical frameworks.
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    • 2.In a many-valued logic (e.g., Łukasiewicz's three-valued system), a literal can take an intermediate value, so 'at least one literal true per clause' fails to uniquely determine a valid independent set node.
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    • 3.Therefore, the bijection between satisfying valuations and independent sets of size n breaks down in any non-bivalent semantic framework, limiting the reduction's logical generality.
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    Reason against 2 of 2
    ?
    • 1.The argument conflates syntactic polynomial-time constructibility of G_φ with a semantic guarantee that the graph faithfully encodes all and only satisfying assignments.
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    • 2.Quine's thesis that the boundary between logical truth and empirical fact is indeterminate implies that the 'edges between contradictory literals' encode a notion of contradiction that is framework-relative, not absolute.
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    • 3.If the identification of contradictory literal pairs presupposes a fixed, non-negotiable logical syntax, the reduction embeds a substantive metaphysical assumption about negation that is not itself established by the polynomial-time construction.
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    Related

    A graph G_φ with n triangles can be constructed from a 3-CNF formula φ with n cl...If G_φ has an independent set of size n, then by construction it contains exactl...If the identification of contradictory literal pairs presupposes a fixed, non-ne...If φ is satisfiable, then a satisfying valuation makes at least one literal true...
    +5 moreShow less
    In a many-valued logic (e.g., Łukasiewicz's three-valued system), a literal can ...Quine's thesis that the boundary between logical truth and empirical fact is ind...The argument conflates syntactic polynomial-time constructibility of G_φ with a ...The reduction assumes a classical, bivalent semantics where every literal is eit...Therefore, the bijection between satisfying valuations and independent sets of s...

    Similar

    3-SAT polynomial-time reduces to INDEPENDENT SET94%3-SAT is polynomial-time reducible to INDEPENDENT SET86%Composing polynomial time reductions yields another polynomial time re...83%f is a polynomial-time many-one reduction of Y to BHP80%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    The graph \(G_{\phi}\) for the formula \((p_1 \vee p_2 \vee p_3) \wedge (\neg p_1 \vee p_2 \vee \neg p_3) \wedge (p_1 \vee \neg p_2 \vee \neg p_3)\). A reduction of \(3\text{-}\sc{SAT}\) to \(\sc{INDEPENDENT}\ \sc{SET}\) can now be described as follows: Let \(\phi\) be a \(3\text{-}\sc{CNF}\) formula consisting of \(n\) clauses as depicted above. We construct a graph \(G_{\phi} = \langle V,E \rangle\) consisting of \(n\)-triangles \(T_1,\ldots,T_n\) such that the nodes of \(T_i\) are respect
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit