1. Canonical Constructive Systems Ori Lahav under the supervision of Prof. Arnon Avron 28 July 2009

2. The Problem What is a constructive connective? Our Approach Proof-theoretically, a constructive connective is defined by a set of canonical logical rules in single-conclusion cut-free Gentzen systems. We identify the largest family of connectives that can be characterized in this way, and provide Kripke-style semantics for this family. 2

3. Strict vs. Non-Strict Non-strict sequential systemStrict sequential system Also negative sequents (sequents of the form  ( are used in derivations. Only definite sequents (sequents of the form  ) are used in derivations. 3 The strict version is more natural to induce an tcr. Gentzen’s original single-conclusion sequential system (LJ) was non-strict. Previous papers (Bowen 71, Kaminsky 88 and Ciabattoni and Terui 06) consider non-strict systems. We begin this presentation with the strict version, and postpone the non- strict case.

4. ApplicationRule p 1  p 2 /  p 1  p 2 Introduction  p 1 p 2  / p 1  p 2  Elimination Example: Strict Canonical Rules for Implication 4

5. Definition: Strict Canonical Rule A strict canonical introduction rule of ◊ is an expression of the form:  1  q 1, …,  m  q m /  ◊(p 1,…, p n ) where  i  q i are definite Horn-clauses over p 1,…, p n. To use this rule: ◦ Choose a substitution . ◦ Choose left context . ◦ Apply the rule: ,  )  1 )  ) q 1 ) … ,  )  m )  ) q m )  )◊(p 1,…, p n )) 5

6. Definition: Strict Canonical Rule A strict canonical elimination rule of ◊ is an expression of the form:  1  q 1, …,  m  q m,  1 , …,  k  / ◊(p 1,…, p n )  where  i  q i and  i  are Horn clauses over p 1,…, p n. To use this rule: ◦ Choose a substitution . ◦ Choose left context . ◦ Choose right context . ◦ Apply the rule: ,  )  1 )  ) q 1 ) … ,  )  m )  ) q m ) ,  )  1 )  … ,  )  k )  ,  )◊(p 1,…, p n ))  6

7. ApplicationRule  p 2 /  p 1 p 2Introduction  p 1 p 2  / p 1 p 2 Elimination Example: Strict Canonical Rules for Gurevich's “Semi-Implication” 7

8. Strict Canonical (Single-Conclusion) Systems  Axioms Cut Weakening Structural Rules Strict Canonical Introduction and Elimination Rules Logical Rules  ,   ,  ,  8

9. Consistency of a Canonical System A canonical system is called consistent (or non- trivial) iff we cannot prove p 1  p 2 in it. Example: “Tonk” [Prior] Question: Can we find a simple criterion for consistency of strict canonical systems? An answer to this question in the multiple- conclusion framework is that the system should be coherent [Avron Lev 01,05]. 9

10. Coherence of Canonical Systems A set of canonical rules for ◊ is called coherent iff whenever it includes both S 1 /  ◊(p 1,…, p n ) and S 2 / ◊(p 1,…, p n )  then S 1  S 2 is classically inconsistent. 10

11. Implication: p 1  p 2 /  p 1  p 2  p 1 p 2  / p 1  p 2  Semi-Implication:  p 2 /  p 1 p 2  p 1 p 2  / p 1 p 2   p1 p2  p2p2 coherent  p1 p2  p1p2p1p2 coherent Examples: Coherence 11

12. “Tonk”:  p 2 /  p 1 T p 2 p 1  / p 1 T p 2  This is what is wrong with “Tonk”. p1p1p2p2 NOT coherent Examples: Coherence 12

13. Coherence and Consistency Theorem: Coherence is necessary and sufficient for consistency of strict canonical system. A strict canonical system is called constructive iff every connective has a coherent set of rules. 13

14. Cut Elimination in Canonical Systems Cut Elimination Whenever s is provable (without assumptions), then there exists a cut-free proof of s. Strong Cut Elimination Whenever s is provable from a set of sequents R, then there exists a proof of s from R, in which the only cuts used are on formulas (not subformulas!) from R. 14

15. Semantics A Generalized (Kripke) Frame is a triple W =  W, <, v  where: ◦ W, <  is a nonempty partially ordered set. ◦ v: W  wffs  { t, f } is a persistent function. (i.e. if v(a,  )=t then for every b≥a v(b,  )=t). 15

16. Semantics Let W =  W, <, v  be a generalized frame: ◦ A sequent  is locally true in a  W if either v(a,  )=f for some , or v(a,  )=t for some . ◦ A sequent is true in a  W if it is locally true in every b ≥ a. ◦ W is a model of a sequent if it is locally true in every a  W. 16

17. Semantics Let G be a strict canonical constructive system. A generalized frame is G-legal iff it respects its rules: The conclusion is locally true in a, whenever the premises are true in a. Respect Strict Introduction Rules The conclusion is locally true in a, whenever the definite premises are true in a and the negative premises are locally true in a. Respect Strict Elimination Rules 17

18. v(a,  )=f if v(a,  )=t and v(a,  )=f  p1 p2  / p1  p2  v(a,  )=t if v(b,  )=f or v(b,  )=t for every b ≥ a p1  p2 /  p1  p2  exactly the well-known Kripke semantics for intuitionistic implication. Example: Semantics of Implication 18

19. v(a,   )=f if v(a,  )=t and v(a,  )=f  p1 p2  / p1 p 2  v(a,   )=t if v(a,  )=t  p2 /  p1 p 2 Free when v(a,  )=f and there is no b≥a such that v(b,  )=t and v(b,  )=f.  Non-deterministic semantics. Example: Semantics of “Semi-Implication” 19

20. Main Results Strong Soundness and Completeness: A sequent s is provable from a set of sequents R in G iff every G-legal generalized frame which is a model of R is also a model of s. 20

21. Main Results General Strong Cut Elimination: Every strict canonical constructive system admits strong cut-elimination. Decidability: Every strict canonical constructive system is decidable. Modularity: The characterization of a constructive connective is independent of the system in which it is included. 21

22. Corollary Consistency Cut Elimination Strong Cut Elimination Coherence Equivalences for Strict Canonical Systems 22

23. A non-strict canonical introduction rule of ◊ is an expression of the form:  1  E 1, …,  m  E m /  ◊(p 1,…, p n ) where  i  E i are definite Horn-clauses over p 1,…, p n. To use this rule: ◦ Choose a substitution . ◦ Choose left context . ◦ Apply the rule: ,  )  1 )  ) E 1 ) … ,  )  m )  ) E m )  )◊(p 1,…, p n )) 23 Definition: Non-Strict Canonical Rule

24. A non-strict canonical elimination rule of ◊ is an expression of the form:  {  1  E 1, …,  m  E m }, {  1 , …,  k  }  / ◊(p 1,…, p n )  where  i  E i and  i  are Horn clauses over p 1,…, p n. To use this rule: ◦ Choose a substitution . ◦ Choose left context . ◦ Choose right context E. ◦ Apply the rule: ,  )  1 )  ) E 1 ) … ,  )  m )  ) E m ) ,  )  1 )  E … ,  )  k )  E ,  )◊(p 1,…, p n ))  E 24

25. ApplicationRule p 1  /   p 1 Introduction  {  p 1 }, { }  /  p 1  Elimination Example: Non-Strict Canonical Rules for Negation 25

26. ApplicationRule p 1  /  p 1 |p 2 p 2  /  p 1 |p 2 Introduction  {  p 1,  p 2 }, { }  / p 1 |p 2  Elimination Example: Non-Strict Canonical Rules for Bowen’s “Not-Both” 26

27. Non-Strict Canonical Systems  Axioms Cut Weakening Structural Rules Non-Strict Canonical Introduction and Elimination Rules Logical Rules  E ,  E  ,  E ,  E 27   

28. Semantics Let G be a non-strict canonical constructive system. A generalized frame is G-legal iff it respects its rules: The conclusion is locally true in a, whenever the premises are true in a. Respect Non- Strict Introduction Rules The conclusion is locally true in a, whenever the definite hard premises are true in a and the negative soft premises are locally true in a. Respect Non- Strict Elimination Rules 28

29. v(a,  |  )=f if v(b,  )=t and v(b,  )=t for some b ≥ a  {  p1,  p2 }, { }  / p1|p2  v(a,  |  )=t if v(b,  )=f for every b ≥ a p1  /  p1|p2 Free in cases like: Example: Semantics of “Not-Both” 29 v(a,  |  )=t if v(b,  )=f for every b ≥ a p2  /  p 1 |p2  Non- deterministic semantics.

30. Main Results Strong Soundness and Completeness General Strong Cut Elimination Decidability The previous equivalences ( ) do not hold. 30 Consistency Cut Elimination Strong Cut Elimination Coherence

31. ApplicationRule p 1  /  o p 1 Introduction  { }, {p 1  }  / o p 1  Elimination 31 The Previous Equivalences Do Not Hold Not coherent Consistent Admits cut-elimination Does not admit strong cut-elimination

32. Strong Consistency of a Canonical System A canonical system is called strongly consistent iff we cannot prove the empty sequent (  ) from  p 1 and p 2  in it. 32 Strong Consistency Strong Cut Elimination Coherence Equivalences for Non- Strict Canonical Systems

33. Finally, A constructive connective is a connective defined by a set of rules in some canonical constructive system. A constructive connective is defined by a coherent set of rules. 33