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Theory and Practice of Coherent Logic

(extended overview) Marc Bezem University of Bergen. Theory and Practice of Coherent Logic . CL as a fragment of FOL. Coherent formula: C => D, where C = A 1 / …/ An (n ≥0, Ai atoms) and D = E 1 / …/ Em (m ≥0), where each

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Theory and Practice of Coherent Logic

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  1. (extended overview) Marc Bezem University of Bergen Theory and Practice of Coherent Logic

  2. CL as a fragment of FOL • Coherent formula: C => D, where • C = A1 /\ …/\ An (n≥0, Ai atoms) and • D = E1 \/ …\/ Em (m≥0), where each • Ej = (∑ x1 …xk ) Cj (k≥0 may vary with j, each Cj a conjunction of atoms) • No function symbols (yet), only constants • Coherent theory = set of coherent formulas

  3. Examples • Lattices (meet is associative, Horn clause, ternary relation): x∩y=u /\ u∩z=v /\ y∩z=w => x∩w=u • Projective unicity (resolution clause): p|l /\ p|m /\ q|l /\ q|m => p=q \/ l=m • Diamond property (coherent clause): a →b /\ a→c => (∑ d) (b→d /\ c→d) • In general: A1 /\ ... /\ An => ((∑x) A11 /\ .../\ A1i) \/ ... \/ ((∑y) Ak1 /\ ... /\ Akj)

  4. Rationale • Horn clauses: DCG and Prolog • Resolution: ATP • coherent logic: ATP and ? • Why skolemize, e.g., p(x,y) => (∑z) p(x,z) ? • Direct proofs • Constructive logic • Natural proof theory/objects

  5. Inductive definition of X├(T) D • (base) X |- D if X |= D • (step) X,C1 |- D, … , X,Cn |- D (℅) -------------------------------- X |- D X a finite set of facts (= closed atoms) D closed coherent disjunction (parameters in D must occur in X) X |= D iff D = ... \/ (∑x) C \/ … and X contains all facts in C[x:=a], for some disjunct in D and for suitable parameters a (℅) there exists a closed instance C0=>D0 of an axiom in T with C0included in X (X contains all facts in C0) and D0 = …\/ (∑x) Ci \/…, each Ci a fresh instance of Ci (1≤i≤n)

  6. Examples of derivations • T={true=>p, p=>q}, Ø |- q • T={p\/q, p=>r, q=>r}, Ø |- r • T={p, p=>q, q=>false}, Ø |- r • T={(∑ x)p(x), p(x)=>q}, Ø |- q • T={s(a,b), s(x,y)=>(∑ z) s(y,z)}, Ø |- (∑ x y)(s(a,x)/\s(x,y)) • Forward ground reasoning!

  7. Metaproperties • Soundness • Completeness wrt Tarskian semantics • Constructivity • Reduction of FOL to CL • Semidecidability (even without functions!) • CL with = is Finite Model Complete • Automation (SATCHMO!)

  8. Samples of ATP • Various *.in (input) files enclosed • To be processed with CL.pl • Yielding files: • *.out (output) • *.prf (intermediate proof format) • *.v (Coq proof object)

  9. Semantics and completeness • coherent logic: no proof by contradiction (= EM, TND, A \/ ~A, ~ ~A => A) • Digression: constructivism in mathematics • p \/ q stronger than ~(~p /\ ~q) • (∑ x) p(x) stronger than ~(A x) ~p(x) • more strict on ontology of objects • EM only in specific cases, f.e., for integers (A x)(x=0 \/ x≠0), but not for reals

  10. Example of non-constructivism (digression) • Do there exist irrational real numbers x and y such that xy is rational ? • Greek constructivists: √2 is irrational • Non-constructivist: take x = y = √2. If xy is rational, then I’m done. If xy is not rational, then I’m also done: (xy) y = xy∙y = x2 = 2 is rational. Next problem, please. • Constructivist: what do you mean?

  11. Tarskian semantics • Truth values from a complete Boolean algebra, without loss of generality ({0,1}, max, min, not(x) =1-x), [|p\/q|]= max([|p|] [|q|]) etc. • Thus p\/q is true iff p is true or q is true (Girard: ``what a discovery!´´) • Sound but not complete for constructive logic, not sound for some forms of constructive mathematics • Constructive logic is more expressive (\/, ∑) and requires a more refined semantics …

  12. Semantics for constructivism (digr.) • Algebraic: complete Heyting algebras (plural!) • Topological: open sets as truth values • Kripke semantics: tree-structured Tarski models (graph-stuctured for modal logic), creative subject • Curry-Howard interpretation: [|φ|] is the set of proofs of φ • Kleene, Beth, Joyal, … • Different aspects, counter models, metatheory, …

  13. Semantics for CL • Tarskian (non-constructive completeness) • Beth-Joyal-Coquand (fully constructive, extra information, but highly non-trivial) • Curry-Howard (for proof objects) • Other semantics unexplored …

  14. Completeness wrt Tarskian models • Given D true in all models of T, how do you find a proof ? Try them all ! • Breadth-first derivability on the blackboard • Herbrand models along the branches • König’s Lemma to get the tree finite • Finite tree => breadth-first proof => |- proof • Details in paper

  15. Further reading • M.A. Bezem and T. Coquand, Automating Coherent Logic. In: G. Sutcliffe and A. Voronkov, editors, Proceedings LPAR-12, LNCS 3835, pages 246--260, 2005. • H. De Nivelle and J. Meng, Geometric Resolution: A Proof Procedure based on Finite Model Search. To appear in: Proceedings IJCAR 2006. (CL with =, refutation complete and finite model complete)

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