Revisiting Horn Approximations to Clausal Theories Henry Kautz University of Rochester Rochester, New York, USA
Not Unique Unique
Rest of talk about GLBs only
GLBs are always small and can be found by searching through the space of Horn strengthenings of the theory
Compiling First-Order Theories • GLBs for FOL with existential quantifiers do not always exist • Example: • Source = FOL • Target = Variable-free sentences • Let there be an infinite number of constants c1, c2, ...
Clausal FOL • First order clause = • matrix is a clause • Literals may contain variables • Variables are implicitly universally quantified • GLBs for first order clausal theories exist and can be found by Generate_GLB for restrictions of first order clauses to • Horn • Restricted set of predicates • Both
Restricting Set of Function Symbols (new) • Original definition of clause strengthening:
Restricting Set of Function Symbols (new) • Original definition of clause strengthening: Every literal in C θ subsumes some literal in C C’ θ does as well, but C’ θ is logically weaker than C θ
Example: Target is function-free p ( X ) ⊃ q ( f ( X )) ∨ r ( f ( X )) p ( X ) ⊃ q ( Y ) ∨ r ( Y ) p ( X ) ⊃ q ( f ( X )) ∨ r ( g ( X )) p ( X ) ⊃ q ( Y ) ∨ r ( Z )
Example: Target is function-free Horn p ( X ) ⊃ q ( f ( X )) ∨ r ( f ( X )) GLB 1: p ( X ) ⊃ q ( Y ) GLB 2: p ( X ) ⊃ r ( Y ) p ( X ) ⊃ q ( f ( X )) ∨ r ( g ( X )) Same!
Acyclic Horn Theories • The target languages considered up to this point are defined by syntactic properties of individual clauses • Def: a set of Horn clauses is acyclic iff • Its predicates can be numbered such that for every clause, the predicate in the positive literal (if any) has a higher number than any predicate in a negative literal • In each clause, every variable in positive literal appears in a negative literal • Why of interest? • In the FO case, Horn theories can be undecidable, but non-recursive Horn theories are always decidable.
Note on Acyclic Horn • BDI: a new decidable clause class. Manuel Lamotte-Schubert, Christoph Weidenbach. Journal of Logic and Computation, Volume 27, Issue 2, 1 March 2017, Pages 441-468, https://doi.org/10.1093/logcom/exu074. • L. Georgieva, U. Hustadt and R.A. Schmidt. A new clausal class decidable by hyperresolution. In Automated Deduction CADE-18, Vol. 2392 of Lecture Notes in Computer Science, A. Voronkov, ed., pp. 260-274. Springer, 2002. • See Datalog • Jaio de Silva: recommends • ACM Computing Surveys 2001 article by Gottlieb • papers by Bjorner and Ryval Chenko
Decidable Fragments of FO Prolog • Slides on ASP by Eiter: " complete • With no function symbols, pure prolog is Σ " • Which is almost the same as NP complete (?) • With function symbols, decidable fragments: ω-restricted programs, finitary programs, finitely recursive programs, FDNC programs ... see e.g. [Baselice et al., 2007], [Bonatti and Baselice, 2008], [Calimeri et al., 2008] (DLV-complex), [Simkus and Eiter, 2007] • Mantas Simkus and Thomas Eiter. FDNC: Decidable non-monotonic disjunctive logic programs with function symbols. • Piero Bonatti and Sabrina Baselice. Composing normal programs with function symbols. In Proceedings 24th International Conference on Logic Programming (ICLP 2008), Lecture Notes in Computer Science (LNCS). Springer, 2008.
Finding Acyclic Horn Lower Bounds • Search though ways to transform the source theory to acylic Horn by • Deleting all but one positive literal in each clause • Deleting a minimal number of negative literals from the clauses such that the resulting theory is non-recursive • I.e.: identify and break cycles in the “call graph” of the set of clauses • Return any LB found this way that does not strictly entail some other LB • Obviously correct – is logically stronger • Question 1: is the result always a greatest lower bound? • Or could there be some other weird way of constructing a logically weaker acyclic Horn LB?
Solution? • Let B be a acyclic Horn LB • Let B’ be B together with all its resolvants • B’ must also be non-recursive Horn • By the subsumption theorem for resolution, each clause in the target theory is subsumed by some clause from B’ • Let B’’ be the set of such clauses from B’ • B’’ is also a acyclic Horn LB • Generate_GLB will find a logically weakest such set B’’ • Generate_GLB will find a acyclic Horn GLB
Compiling Prolog to ASP? • Prolog extends Horn logic with negation as failure • Negation as failure operator may appear in front of literals in the body of a clause • Undecidable • Some programs cannot be given coherent semantics • Answer set programming restricts Prolog to stratified theories • Decidable • Every program has stable set semantics • Question 2: Can ASP GLBs of Prolog programs be found by a version of Generate_GLB?
Recommend
More recommend
