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Techniques for Proving the Completeness of a Proof System

Techniques for Proving the Completeness of a Proof System. Hongseok Yang Seoul National University Cristiano Calcagno Imperial College. Completeness Question. Completeness: Is a proof system powerful enough to verify all true facts? Proof system for classical propositional logic:

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Techniques for Proving the Completeness of a Proof System

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  1. Techniques for Proving the Completeness of a Proof System Hongseok Yang Seoul National University Cristiano Calcagno Imperial College

  2. Completeness Question • Completeness: Is a proof system powerful enough to verify all true facts? • Proof system for classical propositional logic: ` P)(Q)R) ` (P)(Q)Q’)))((P)Q))(P)Q’)) ` ::P)P ` P ` P)Q ` Q • Truth: P holds (denoted ² P) iff P always evaluates to true by the “table method.” • Completeness Theorem: if ² P, then ` P. • Exercise: Prove ` ((q)r))q))q.

  3. Reasons for Studying Completeness • Sometimes it is easier to show the truth of a formula than to derive the formula. • The completeness result shows that nothing is missing in a proof system. • The completeness result formalizes what a proof system achieves. • With a completeness result, a paper about a proof system has more chances to get accepted. 

  4. Goal of My Talk • To present common techniques for showing the completeness, so that you can apply them to your own problem. • In particular, to explain the following concepts: • maximally consistent set • truth lemma • Lindenbaum lemma • If time permits, I will briefly explain what I’m working on with Calcagno in Imperial college.

  5. Simple Modal Logic P := q | :P | P)P | P • Proof system: usual rules in classical logic with the following additional ones for the modality: ` P ` P `(P)Q))(P)Q) • Example: • student, ðphd, ððprofessor • deadlock, ðdeadlock, :ð:deadlock

  6. Semantics A model M is a triple (M, R:M$M, I:Symb!P(M)). • Interpretation of Simple Modal Logic M,m ² q iff m 2 I(q) M,m ²:P iff M,m 2 P M,m ² P)Q iff if M,m ² P, then M,m ² Q M,m ²P iff for all n, if R(m,n), then M,n ² P • Example: • M=years, • R(n,m) iff m=n+1, • I(phd)={2001,…}, I(student)={1982,…,2001}

  7. Completeness Question • P is valid (denoted ² P) iff for all models (M,R,I) and all m in M, (M,R,I),m ² P. • Completeness: If ² P, then ` P. • General guide: consider the contrapositive! • Contrapositive: if 0 P, then 2 P. • Guide: for each P such that 0 P, construct a model (M,R,I) with m in M such that (M,R,I),m ²:P.

  8. Strategy for Constructing a Required Model • Build a model M = (M,R,I) such that • [term model] each m in M is a set of formulas; • [truth lemma] for all m and Q, m ² Q iff Q is in m; • there exists n in M containing :P. • This model is what we are looking for. (Why?) • How to build such a model?

  9. Inferring Requirements from the Truth Lemma • Let M=(M,R,I) be a model such that each m in M is a set of formulas. • Try to use induction to show the truth lemma for M: for all m in M and Q, m²Q iff Q is in m. • What conditions do R and I satisfy?

  10. Inferring Requirements from the Truth Lemma • Let M=(M,R,I) be a model such that each m in M is a set of formulas. • Try to use induction to show the truth lemma for M: for all m in M and Q, m²Q iff Q is in m. • What conditions do R and I need to satisfy? • Q is not in m iff :Q is in m. • If both Q and Q)Q’ are in m, then Q’ is in m. • If R(m,n) and ðQ is in m, then Q is in n. • m is in I(q) iff q is in m.

  11. Maximally Consistent Set • A set m of formulas is maximally consistent iff • for all Q, only one of Q and :Q is in m; and • if Q0,Q1,...,Qn2m and `Q0)Q1)…)Qn)Q’, then Q’ in m. • By turning the conditions into the definition (almost directly), we can construct the required model: • M consists of maximally consistent sets of formulas; • R(m,n) iff for all ðQ in m, Q in n; • m2I(p) iff p2m. • The proof for the truth lemma “almost” works.

  12. Lindenbaum Lemma • Still need to show two facts: • If m ²Q, then Q is in m. • There exists m in M such that m ²:P. • Lindenbaum Lemma: Let {Q0,Q1,…,Qn} be a set of formulas. If 0 Q0)Q1) …)Qn)false, then there is a maximally consistent set m s.t. Q0, Q1, …, Qn2 m. • Try to show the two properties with Lindenbaum Lemma.

  13. Summary We constructed “canonical” model (M,R,I): • M consists of maximally cons. sets of formulas. • R(m,n) iff for all P in m, P is in n. • m 2 I(p) iff p in m. The model satisfies the following properties: • truth lemma: m ² P iff P is in m. • If 0:P, then there is m such that P 2 m. These two properties lead to the completeness.

  14. My Work with Calcagno Interested in the completeness of Boolean BI wrt PCM models. • Conceptual implication: supports that BBI is a logic for computational resources. • “Practical” implication: shows that the proof rules in separation logic are powerful enough. Roughly, the question is similar to asking whether our model logic are complete wrt injective models. • (M,R,I) is injective iff R(m,n)ÆR(m,n’) ) n=n’ We found a method to transform the “canonical” model to an injective one, while preserving the satisfiability of formulas.

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