1 / 15

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

Sophia
Télécharger la présentation

Theory and Practice of Coherent Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related