Updating the Knowledge Compilation Map Simone Bova (TU Wien) - PowerPoint PPT Presentation

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Updating the Knowledge Compilation Map Simone Bova (TU Wien) Dagstuhl Seminar on Recent Trends in Knowledge Compilation September 17–22 , 2017, Dagstuhl (Germany)

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Outline Knowledge Compilation Map Updates Extensions

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Outline Knowledge Compilation Map Updates Extensions

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Knowledge Compilation Represent knowledge so to facilitate reasoning. Example Represent DNFs by OBDDs to count models. x w ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( w ∧ y ) ∨ ( w ∧ z ) ≡ y z ⊥ ⊤

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Knowledge Compilation Represent knowledge so to facilitate reasoning. Example Represent DNFs by OBDDs to count models. x w ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( w ∧ y ) ∨ ( w ∧ z ) ≡ y z ⊥ ⊤

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Knowledge Compilation Represent knowledge so to facilitate reasoning. Example Represent DNFs by OBDDs to count models. x 9 = ( 6 + 12 ) / 2 w 6 = ( 0 + 12 ) / 2 ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( w ∧ y ) ∨ ( w ∧ z ) ≡ y 12 = ( 8 + 16 ) / 2 z 8 = ( 0 + 16 ) / 2 ⊥ ⊤ 0 16

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Knowledge Compilation Represent propositional knowledge so to facilitate reasoning. Find a class R of circuits expressing a class of Boolean functions � �� � Represent propositional knowledge . . . such that certain logical tasks on the circuits in R are computationally feasible. � �� � . . . so to facilitate reasoning.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Logical Tasks on Circuits Given circuits C and C ′ in a circuit class R . Counting: Count the models of C . Entailment: Decide if C entails C ′ . . . . . . . Negation: Find a circuit in R computing ¬ C . Conjunction: Find a circuit in R computing C ∧ C ′ . Disjunction: Find a circuit in R computing C ∨ C ′ . . . . . . . A task is computationally feasible on R if it is polytime tractable wrt the input size ie, wrt the size of the circuits in R .

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness vs Tractability Tradeoff succinctness/tractability in choosing a representation language: Truthtables: Tractable , but useless because too verbose . Circuits: Succinct , but useless because too hard . Darwiche and Marquis (2002) systematically investigate a hierarchy of representation languages wrt their succinctness/tractability tradeoffs: A. Darwiche and P. Marquis. A Knowledge Compilation Map. J. Artif. Intell. Res. , 17, 229-264, 2002.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Representation Languages NNF DNNF CNF DNF dDNNF PI IP MODS FBDD OBDD Figure: Inclusions.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Representation Languages Negation Normal Forms ( NNF ) Boolean circuits having unbounded fanin AND and OR gates with negations pushed to the input gates. Decomposable NNFs ( DNNF ) NNFs where each AND gate has subcircuits using disjoint sets of variables. Deterministic DNNFs ( dDNNF ) DNNFs where each OR gate has pairwise inconsistent subcircuits. Prime Implicate Forms ( PI ) CNFs where entailed clauses are already entailed by a single clause in the CNF and no clause in the CNF is entailed by another. Models, MODS DNFs where each disjunct uses the same variables. Free Binary Decision Diagrams, FBDDs : Read-once branching programs.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Determinism and Decomposability ∨ ∧ ∧ ∧ x 1 x 4 ∨ ∨ ⊤ ∨ ∧ ∧ ∧ ∧ ∧ x 3 x 4 x 3 x 1 x 2 x 1 x 1 x 2 ⊥ ⊥

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Counting via Determinism and Decomposability + × × × x 1 x 4 + + ⊤ + × × × × × x 3 x 4 x 3 x 1 x 2 x 1 x 1 x 2 ⊥ ⊥

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Representation Languages Negation Normal Forms ( NNF ) Boolean circuits having unbounded fanin AND and OR gates with negations pushed to the input gates. Decomposable NNFs ( DNNF ) NNFs where each AND gate has subcircuits using disjoint sets of variables. Deterministic DNNFs ( dDNNF ) DNNFs where each OR gate has pairwise inconsistent subcircuits. Prime Implicate Forms ( PI ) CNFs where entailed clauses are already entailed by a single clause in the CNF and no clause in the CNF is entailed by another. Models, MODS DNFs where each disjunct uses the same variables. Free Binary Decision Diagrams, FBDDs : Deterministic read-once branching programs.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness Relation Let S , T ⊆ NNF. S is (polysize) compilable into T (or T is at least as succinct as S) if there exists a polynomial p : N → N such that for all C ∈ S there exists D ∈ T equivalent to C such that size ( D ) ≤ p ( size ( C )) . Write S � T if S is compilable into T, and S � � T otherwise.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Knowledge Compilation in Practice Want to count models of circuit C (offline) modulo a sequence of partial assignments g 1 , g 2 , g 3 , . . . (online): C | g 1 , C | g 2 , C | g 3 , . . . Ow, lookup in the KC map a maximally succinct language supporting counting mod assignments. Represent C by a form C ∗ in the chosen language: C ∗ , C ∗ | g 1 , C ∗ | g 2 , C ∗ | g 3 , . . . High compilation cost is eventually amortized by low querying costs: t 0 t 1 · · · t j · · · · · · · · · t i · · · t i + 1 · · · C | g 1 · · · C | g j · · · · · · · · · C | g i · · · C | g i + 1 C ∗ C ∗ | g 1 C ∗ | g 2 C ∗ | g i C ∗ | g i + 1 C ∗ | g i + 2 · · · · · · · · · · · ·

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Outline Knowledge Compilation Map Updates Extensions

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness Relation Darwiche and Marquis (2002) summarize the status of the succinctness relation also appealing to previous work including • Quine (1959), • Chandra and Markowsky (1978), • Bryant (1986), • Wegener (1987), • Gergov and Meinel (1994), • Gogic, Kautz, Papadimitriou, and Selman (1995), • Selman and Kautz (1996), • Cadoli and Donini (1997), and • Darwiche (1999).

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness Relation NNF DNNF Let S and T be languages in the diagram. DNF dDNNF CNF If ( S , T ) in transitive closure of → , then S � T. Else: IP FBDD PI ? • If S ��� T, then S � T (unknown). • Else S � � T. OBDD MODS

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness Relation NNF NNF DNNF DNNF DNF dDNNF CNF DNF dDNNF CNF IP FBDD PI IP FBDD PI OBDD OBDD MODS MODS Figure: S ��� T means S � T unknown. S �→ T means S � � T unless PH collapses.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness Relation | 2002–2016 NNF DNNF Let S and T be languages in the diagram. DNF dDNNF CNF If ( S , T ) in transitive closure of → , then S � T. Else: IP FBDD PI ? • If S ��� T, then S � T. • If S �→ T, then S � � T unless PH collapses. • Else S � � T. OBDD MODS

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness | Updates | 1 | PI � � DNNF NNF NNF DNNF DNNF DNF dDNNF CNF DNF dDNNF CNF IP FBDD PI IP FBDD PI OBDD OBDD MODS MODS Figure: S �→ T means S � � T.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness | Updates | 2 | DNNF � � dDNNF NNF NNF DNNF DNNF DNF dDNNF CNF DNF dDNNF CNF IP FBDD PI IP FBDD PI OBDD OBDD MODS MODS Figure: S �→ T means S � � T.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Succinctness | Updates NNF NNF DNNF DNNF DNF dDNNF CNF DNF dDNNF CNF IP FBDD PI IP FBDD PI OBDD OBDD MODS MODS

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Knowledge Compilation Meets Communication Complexity Theorem (B, Capelli, Mengel, and Slivovsky) Let D be a DNNF (resp, dDNNF) computing a function F. The size of D is an upper bound on the size of a smallest rectangle cover (resp, disjoint rectange cover) of F. ∗ A Z -rectangle is a function R ( Z ) of the form R ≡ S ( X ) ∧ T ( Y ) for functions S and T with X and Y forming a partition of Z . { R i } i ∈ I is a rectangle cover of a Boolean function F ( Z ) if each R i is a Z -rectangle and F ≡ � i ∈ I R i . A rectangle cover { R i } i ∈ I is disjoint if R i ∧ R j ≡ ⊥ ( i � = j ∈ I ). ∗ Balanced.

K NOWLEDGE C OMPILATION M AP U PDATES E XTENSIONS Rectangle | R ( x , y , z , w ) ≡ S ( x , y ) ∧ × T ( z , w ) R ( x , y , z , w ) ¯ x ¯ y ¯ z ¯ w 0 ¯ x ¯ y ¯ zw 0 ¯ x ¯ yz ¯ w 0 ¯ x ¯ yzw 0 ¯ xy ¯ z ¯ w 0 ¯ xy ¯ z ¯ ¯ ¯ z ¯ zw 1 w zw w zw ¯ xyz ¯ x ¯ ¯ w 1 y 0 0 0 0 ¯ xyzw 0 ≡ xy ¯ 0 0 1 1 x ¯ y ¯ z ¯ x ¯ w 0 y 0 1 1 0 x ¯ y ¯ zw xy 0 0 0 0 1 x ¯ yz ¯ w 1 x ¯ yzw 0 xy ¯ z ¯ w 0 xy ¯ zw 0 xyz ¯ w 0 xyzw 0