Compact Propositional Encoding of First Order Theories

Compact Propositional Encoding ofFirst Order Theories Eyal Amir University of Illinois at Urbana-Champaign (Joint with Deepak Ramachandran and Igor Gammer) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Example: Pigeonhole Principle p1 H1 p2 H2 p3 H3 p4 H4 Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Pigeonhole: Classical Encoding H1(pi) H2(pi)  H3(pi)  H4(pi)  [H1(pi) H2(pi)] p,q (pq  Hi(p)  Hi(q)) Every i: O(n2) props Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

A Reformulation Hi+(p) : pigeon p is not in a hole in the subtree rooted at i. H1+ p1 p3 H2+ H3+ p4 p2 1 2 3 4 Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

A Reformulation Hi+(p) : pigeon p is not in a hole in the subtree rooted at i. p[H1+(p) (H2+(p) H3+(p)) H1+ p1 p3 p[H2+(p) (-H1(p) -H2(p)) p[H1+(p) (-H3(p)-H4(p)) p4 p2 1 2 3 4 Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

A Reformulation Hi+(p) : pigeon p is not in a hole in the subtree rooted at i. p[H1+(p)  H2+(p)] p1 p3 p[H1(p) H2(p)] p[H3(p)H4(p)] p4 p2 1 2 3 4 Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

A Reformulation Hi+(p) : pigeon p is not in a hole in the subtree rooted at i. H1+ p1 p3 H2+ H3+ p,q[Hi(p)Hi(q) (p=q)] p4 p2 Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

A Reformulation Hi+(p) : pigeon p is not in a hole in the subtree rooted at i. p[H1+(p)] p1 p3 H2+ H3+ O(n·logn) props p4 p2 Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Propositional Encodings: Motivation • How do we do this in general? – scientific curiosity • Applications: • Formal verification, BMC – specification in FOL • AI: planning, diagnosis, spatial reasoning, question answering from knowledge – specified in FOL • Useful with many objects, predicates, time steps, etc. • Can use efficient SAT solvers Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Outline • Algorithm for compact prop. encoding • Analysis • Interpolants – correctness of algorithm • A different Treewidth – number of props. • Towards automatic partitioning • Related work & open questions Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Naïve Propositional Encoding • Tasks converted into propositional logic • P(a)  Pa x P(x)  PaPb Pc (Naive Propositionalization) • Problem: Very many propositions: |P||C|k for |P| predicates of arity k, and |C| constant symbols • Note: Assumes Domain Closure (DCA) • constant symbols name all domain objects Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Our Results • New method for propositioal encoding • Size depends on optimal tree decomposition (e.g., O(|P|+|C|) sometimes; O(n logn) instead of O(n2) for Pigeonhole Principle) • Applications: • Machine scheduling problem • Board-Level Routing Problem for FPGA’s Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Our New Algorithm Rough algorithm (given theory T): • Partition theory T into T1…Tr (*1) • Encode every Ti in propositional logic Ti’ separately (create proposition Pa for constant a and predicate P in L(Ti)) (*2) • Return Ui≤rTi’ Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Partitioned Theory T … Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Partitioned Theory T1 T2 … … Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Propositional Encoding T1 T2 … … Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

# props. in partitioned encoding 1 # props. in naïve encoding #partitions  Partitioned Propositional Encoding Algorithm: • Partition theory T into T1…Tr • Propositionalize each Ti separately • Return the propositional encoding Ui≤rTi |Pi||Ci| props.  an exponential speed-up for SAT solver How to make this sound+complete? Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Experiments: Board-Level Routing #prop. vars per problem SAT Running time (sec.) per problem Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Experiments: Machine Scheduling #prop. vars per problem SAT Running time (sec.) per problem Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Key to Exact Encoding • Craig’s interpolation theorem (First-Order Logic): • If T1 T2, then there is a formula C including only symbols from L(T1)  L(T2) such that T1 C and C T2 ^ ^ ^ T1 T2 ^ P(x) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

T1T2 has no model, iff T1 T2 ^ Key to Exact Encoding • Craig’s interpolation theorem (First-Order Logic): • If T1 T2, then there is a formula C including only symbols from L(T1)  L(T2) such that T1 C and C T2 ^ ^ ^ T1 T2 ^ The only interpolants (the C’s) can be xP(x),xP(x),… (+6 others) P(x) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

T1T2 has no model, iff T1 T2 ^ Key to Exact Encoding • Key idea: If our prop. encoding of T1 [T2] implies all possible interpolants in L(T1)L(T2), then the combined encoding is SAT iff T (in FOL) is SAT. T1 T2 ^ The only interpolants (the C’s) can be xP(x),xP(x),… (+6 others) P(x) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

The only interpolants (the C’s) can be xP(x),xP(x),… (+6 others) Propositional Encoding T1 T2 … … Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Correctness of Encoding Alg. • Encoding of each partition needs to include encodings of all possible interpolants (detailed next) • When there are more than two partitions, the partitions are arranged in a tree decomposition (in next section) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

The only interpolants (the C’s) can be xP(x),xP(x),… (+6 others) Propositional Encoding Size T1 T2 … … Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Propositional encoding of Monadic FOL Factor = single-variable existentially quantified formula Theorem : Can convert any monadic FOL formula to a conjunction of disjunctions of factors Example: Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Propositionalizing a Partition in Monadic FOL • Convert Ti to factorized form • Replace factors in Τi with prop. symbols • Add meaning of new propositions ε(Ti) = This approach creates |Pi||Ci|+3|Pi| propositional symbols. Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Automatic Partitioning and Propositional Encoding • Optimization criteria • Tree of partitions of axioms that satisfies the running intersection property • Minimize |Pi||Ci|+3|Pi| (contrast with treewidth, where we minimize |Pi|) • Can use current algorithms for treewidth and decomposition as a starting point • Sometime need reformulation of axioms Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Automatic Partitioning • Find a tree decomposition G of minimum width: • Every node in G is a set of symbols from T • G satisfies the running intersection property: if P appears in both v,u in G, then P appears in all the nodes on the path connecting them • Choose G with minimum width (width(G) is the size of its largest node v) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Related Work • [McMillan 2005] – some ground (QF) FOL theories have ground interpolants • May use his results for similar results on compact prop. for those theories • McMillan’s talk yesterday – may use our compact prop. method to generate all interpolants Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Related Work • [Pnueli, Rodeh, Strichman, Siegel ’99, ’02] – equational theories (no predicates), and weak(er?) bound where R = resulting universe size, and others are problem dependent (worst case n!, for n variables/constants) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Conclusion • Created efficient propositional encodings: • Monadic FOL – Sound + Complete • Ramsey class – Sound, Incomplete without DCA • Applies to general FOL (with DCA) • Speedup in SAT solving • Exploits the structure of real world problems • Future work: Classes beyond Monadic FOL • Right now: ground FOL with equality • 2-var fragment • Some software on http://reason.cs.uiuc.edu/eyal Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

THE END Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Problems with Naïve Prop. • Problem 1: Very large number of propositions: |P||C|k for |P| predicates of arity k, and |C| constant symbols • Problem 2: Assumes Domain Closure Assumption (DCA) – only elements are those with constant symbols Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

General FOL (with DCA) Use DCA repeatedly until we get a formula in Monadic FOL • Replace x1Qx2…Qxn(x1,…, xn) with cCQx2…Qxn(x1,…, xn) • Replace x1Qx2…Qxn(x1,…, xn) with cCQx2…Qxn(x1,…, xn) Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Automatic Partitioning • Begin with logical Knowledge Base in FOL • Construct symbol graph • vertex = symbol • edge = axiom uses both symbols • Find a tree decomposition • Partition axioms using clusters in tree decomposition Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Automatic Partitioning key  locked  can_open can_open Ù try_open  open open Ù fetch  broom key  open try_open  open  broom  fetch broom Ù dry  can_clean can_clean  clean broom Ù let_dry  dry time  let_dry drier  let_dry time  drier Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Automatic Partitioning key locked can_open try_open can_clean clean open fetch dry time broom let_dry drier Compact Encoding of FOL Eyal Amir, Cambridge Present., May 2006

Compact Propositional Encoding of First Order Theories

Compact Propositional Encoding of First Order Theories

Presentation Transcript

Critique of coercive theories of social order

Propositional Approaches to First-Order Theorem Proving

Propositional logic (continue …) Propositional logic versus first-order logic

OSHL: A Propositional Prover with Semantics for First-Order Logic

Limitations of First-Order Logic

First order theories

Order encoding

Theories of First Language Acquisition

Propositional and First Order Reasoning

Resolution in Propositional and First-Order Logic

Propositional Logic And First Order Logic.

Propositional Encoding - Decision Procedure

Checking Validity of Quantifier-Free Formulas in Combinations of First-Order Theories

Propositional logic versus first-order (predicate) logic The universe of discourse

Propositional and First-Order Logic

Propositional and First-Order Logic

First order theories

Spontaneous Order Theories

Response of First-Order Circuits

First Order Predicate Logic Limitation of Propositional Logic

Propositional and First-Order Logic

Chapter 1: Propositional and First-Order Logic