extended overview marc bezem university of bergen l.
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(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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cl as a fragment of fol
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
examples
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)
rationale
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
inductive definition of x t d
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)

examples of derivations
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!
metaproperties
Metaproperties
  • Soundness
  • Completeness wrt Tarskian semantics
  • Constructivity
  • Reduction of FOL to CL
  • Semidecidability (even without functions!)
  • CL with = is Finite Model Complete
  • Automation (SATCHMO!)
samples of atp
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)
semantics and completeness
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

example of non constructivism digression
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?
tarskian semantics
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 …
semantics for constructivism digr
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, …
semantics for cl
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 …
completeness wrt tarskian models
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
further reading
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)