Some Topics Related to Propositional Logic
Xishun Zhao
Institute of Logic and Cognition, Sun Yat-sen University
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 1/42 Outline
Propositional Logic The Satisfiability Problem
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline
Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline
Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline
Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline
Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline
Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Expressive Power and Complexity
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline
Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Expressive Power and Complexity Future Work
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 1. Propositional Logic
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 3/42 Propositional Formulas
Propositional Variables: x1,x2, ···.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas
Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas
Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Literal: either a variable x or its negation ¬x.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas
Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Literal: either a variable x or its negation ¬x. Clauses: disjunction of finite literals. Example (x1 ∨¬x2 ∨¬x3).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas
Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Literal: either a variable x or its negation ¬x. Clauses: disjunction of finite literals. Example (x1 ∨¬x2 ∨¬x3). CNF: conjunction of finite clauses. Example: (x1 ∨¬x2 ∨¬x3) ∧ (x1 ∨ x2) ∧ (¬x1 ∨ x3).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Semantics
We consider propositional variable x as a Boolean variable which can take value 0 or 1.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 5/42 Semantics
We consider propositional variable x as a Boolean variable which can take value 0 or 1. We interpret connectives as Boolean functions:
¬x 1 − x x ∧ y min(x, y) x ∨ y max(x, y) x → y max(1 − x, y)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 5/42 Semantics
We consider propositional variable x as a Boolean variable which can take value 0 or 1. We interpret connectives as Boolean functions:
¬x 1 − x x ∧ y min(x, y) x ∨ y max(x, y) x → y max(1 − x, y) Then For any formula F ,foranytruthassignment v : var(F ) →{0, 1}, F has a unique truth value under v,we denote the value by v(F ).
x y (x →¬y) ∧ (¬x → y) 1 0 1
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 5/42 Satisfiability
We say a formula F is satisfiable if there is an assignment v : var(F ) →{0, 1} such that v(F )=1.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 6/42 Satisfiability
SAT Problem: determining whether a given propositional formula is satisfiable. Instance: APropositionalformulaF , Query: Is F satisfiable?
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 6/42 Satisfiability
SAT Problem: determining whether a given propositional formula is satisfiable. Instance: APropositionalformulaF , Query: Is F satisfiable? SAT is the first problem shown to be NP-complete (Cook’s Theorem).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 6/42 Splitting Rule
Let F be a CNF formula, x be a literal of F .DefineF [x =0] (resp F [x =1])istheformulaobtainedfromF by deleting all clauses containing ¬x (resp. x), and removing all occurrences of x (resp. ¬x).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 7/42 Splitting Rule
Let F be a CNF formula, x be a literal of F .DefineF [x =0] (resp F [x =1])istheformulaobtainedfromF by deleting all clauses containing ¬x (resp. x), and removing all occurrences of x (resp. ¬x).
Example: Let F be (x1 ∨¬x2 ∨¬x3) ∧ (x1 ∨ x2) ∧ (¬x1 ∨ x3).
F [x1 =0]is (¬x2 ∨¬x3) ∧ (x2)
F [x1 =1]is (x3)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 7/42 Splitting Rule
Let F be a CNF formula, x be a literal of F .DefineF [x =0] (resp F [x =1])istheformulaobtainedfromF by deleting all clauses containing ¬x (resp. x), and removing all occurrences of x (resp. ¬x).
Example: Let F be (x1 ∨¬x2 ∨¬x3) ∧ (x1 ∨ x2) ∧ (¬x1 ∨ x3).
F [x1 =0]is (¬x2 ∨¬x3) ∧ (x2)
F [x1 =1]is (x3) F is satisfiable ⇐⇒ either F [x =0]or F [x =1]is satisfiable.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 7/42 Unit Clause Rule
Unit clause: consists of only one literal. Example: (x), (¬x)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Unit Clause Rule
Unit clause: consists of only one literal. Example: (x), (¬x)
Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then(x2) is a unit clause.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Unit Clause Rule
Unit clause: consists of only one literal. Example: (x), (¬x)
Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then(x2) is a unit clause. F [x2 =1]=(¬x3)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Unit Clause Rule
Unit clause: consists of only one literal. Example: (x), (¬x)
Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then(x2) is a unit clause. F [x2 =1]=(¬x3) If F contains a unit clause (x),then F is satisfiable ⇐⇒ F [x =1]is satisfiable.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Pure literal rule
Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F .
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 Pure literal rule
Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F .
Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then¬x3 is a pure literal.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 Pure literal rule
Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F .
Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then¬x3 is a pure literal. F [¬x3 =1]=(x2)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 Pure literal rule
Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F .
Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then¬x3 is a pure literal. F [¬x3 =1]=(x2) If x is pure in F ,then F is satisfiable ⇐⇒ F [x =1]is satisfiable.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 DPLL Algorithm
F
Apply unit-clause-rule Apply pure-literal-rule
0 1
0 1 0 1
continue ⊥ ⊥ continue
Whenever we obtain the empty formula ", we know F is satisfiable.
Otherwise F is unsatisfiable.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 10/42 2. Why SAT Is Important?
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 11/42 First-order Logic and SAT
Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 12/42 First-order Logic and SAT
Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model. 0 Skolem Normal Form (Π1): ∀x1∀x2 ···∀xnA ,hereA is quantifier-free.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 12/42 First-order Logic and SAT
Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model. 0 Skolem Normal Form (Π1): ∀x1∀x2 ···∀xnA ,hereA is quantifier-free. 0 For any first-order formula A,thereisaΠ1-formula B such that A has a model ⇐⇒ B has a model.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 12/42 First-order Logic and SAT
Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model. 0 Skolem Normal Form (Π1): ∀x1∀x2 ···∀xnA ,hereA is quantifier-free. 0 For any first-order formula A,thereisaΠ1-formula B such that A has a model ⇐⇒ B has a model. Herbrand Theorem:Themodel-existenceofafirst-order 0 Π1-formula can be reduced to the satisfiability of a (potentially infinite) set of propositional formulas.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 12/42 PversusNP
The PversusNPproblem: Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly?
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 13/42 PversusNP
The PversusNPproblem: Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly? SAT is the first problem shown to be NP-complete (Cook’s Theorem).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 13/42 PversusNP
The PversusNPproblem: Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly? SAT is the first problem shown to be NP-complete (Cook’s Theorem). Thus, if one can prove or disprove that SAT can be solved in polynomial time, then he will win the US$ 1,000,000 prize.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 13/42 PversusNP
The PversusNPproblem: Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly? SAT is the first problem shown to be NP-complete (Cook’s Theorem). Thus, if one can prove or disprove that SAT can be solved in polynomial time, then he will win the US$ 1,000,000 prize. An August 2010 claim of proof that P )= NP, by Vinay Deolalikar, researcher at HP Labs, Palo Alto, received heavy Internet and press attention. In his proof SAT problem plays an important rule.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 13/42 Applications
Every Boolean function can be represented by a propositional formula.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 14/42 Applications
Every Boolean function can be represented by a propositional formula. Aproblemcanbereduced to SAT if there is a transformation T from instances of the problem into propositional formulas such that for any instance α of the problem, α has a solution ⇐⇒ T (α) is satisfiable.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 14/42 Applications
Every Boolean function can be represented by a propositional formula. Aproblemcanbereduced to SAT if there is a transformation T from instances of the problem into propositional formulas such that for any instance α of the problem, α has a solution ⇐⇒ T (α) is satisfiable. NP problems can be reduced to SAT in polynomial-time.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 14/42 Applications
Every Boolean function can be represented by a propositional formula. Aproblemcanbereduced to SAT if there is a transformation T from instances of the problem into propositional formulas such that for any instance α of the problem, α has a solution ⇐⇒ T (α) is satisfiable. NP problems can be reduced to SAT in polynomial-time. Thus, efficient SAT Solvers can be used in many areas.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 14/42 Applications (continued)
In 1957, Alen Newell and Herb Simon tried to applied SAT algorithm to prove theorems in “Principia Mathematica" (Russell & Whitehead).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 15/42 Applications (continued)
In 1960’s, Martin Davis and others applied SAT algorithm (called DP algorithm) to study Hilbert’s 10th problem.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 15/42 Applications (continued)
In 1992, Larrabee & Stephan from Intel applied SAT for hardware verification.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 15/42 Applications (continued)
In 1992, Larrabee & Stephan from Intel applied SAT for hardware verification. Now, more than 6000 problems in science, engineering, economics, as well as military have been reduced to SAT.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 15/42 Application(continued)
Hao Wang, a Chinese American Logician, said: SAT is the first problem in mathematical logic and theoretical computer science.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 16/42 Application(continued)
Edmund Clarke, ACM Turing award winner in 2007, said: Efficient SAT solving is a key technology for 21st century computer science.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 17/42 3. Bottle-necks for SAT Solving and Its Applications
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 18/42 The First Bottle-neck
All existing complete SAT algorithms are not polynomial-time algorithms.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 19/42 The First Bottle-neck
All existing complete SAT algorithms are not polynomial-time algorithms. Now the application of SAT are restricted to solve some instance of propositional formulas coming from practice.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 19/42 The First Bottle-neck
All existing complete SAT algorithms are not polynomial-time algorithms. Now the application of SAT are restricted to solve some instance of propositional formulas coming from practice. There are many instances whose satisfiability can not be solved in reasonable time.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 19/42 The Second Bottle-neck
The reductions from problems to SAT may cause size-explosion.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 20/42 The Second Bottle-neck
The reductions from problems to SAT may cause size-explosion. This greatly restricts the application of SAT Solvers, even if the solver is powerful.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 20/42 The Second Bottle-neck
The reductions from problems to SAT may cause size-explosion. This greatly restricts the application of SAT Solvers, even if the solver is powerful. There are two reasons for size-explosion: The original problem is harder than NP, e.g. PSPACE (it is believed that PSPACE )⊆ NP). The existing reduction is not the “best" one.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 20/42 The Third Bottle-neck
The correctness of all existing SAT Solvers have not been guaranteed.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 21/42 The Third Bottle-neck
The correctness of all existing SAT Solvers have not been guaranteed. R. Brummayer found that in SAT 2007 and SAT 2009 competition, several SAT Solvers have flaws.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 21/42 4. Some Topics
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 22/42 4.1. Minimal Unsatisfiable Formulas
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 23/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
F −{Ci} is satisfiable for each i =1, ··· ,n
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
F −{Ci} is satisfiable for each i =1, ··· ,n Example:
¬aa a F := b ¬b cc ¬c
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
F −{Ci} is satisfiable for each i =1, ··· ,n Example:
¬aa a F := b ¬b cc ¬c
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
F −{Ci} is satisfiable for each i =1, ··· ,n Example:
¬aa a F := b ¬b cc ¬c
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
F −{Ci} is satisfiable for each i =1, ··· ,n Example:
¬aa a F := b ¬b cc ¬c
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 Minimal Unsatisfiable Formulas
ACNFformulaF := {C1, ··· ,Cn} is minimal unsatisfiable (MU) if F is unsatisfiable,
F −{Ci} is satisfiable for each i =1, ··· ,n Example:
¬aa a F := b ¬b cc ¬c
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 24/42 MU(continued)
∀F , F is unsatisfiable =⇒∃G ⊆ F , G is MU.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 25/42 MU(continued)
∀F , F is unsatisfiable =⇒∃G ⊆ F , G is MU. AdeeperunderstandingofMUformulasmightbehelpfulfor find new techniques and heuristics which can increase the efficiency of SAT Solvers. looking for more tractable classes of formulas developing new hard formulas
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 25/42 MU(continued)
∀F , F is unsatisfiable =⇒∃G ⊆ F , G is MU. AdeeperunderstandingofMUformulasmightbehelpfulfor find new techniques and heuristics which can increase the efficiency of SAT Solvers. looking for more tractable classes of formulas developing new hard formulas MU formulas also have some applications in: Believe Revision, Formal Verification.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 25/42 MU(continued)
We have been working on Complexity of MU formulas.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 26/42 MU(continued)
We have been working on Complexity of MU formulas. Structure of MU formulas.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 26/42 MU(continued)
We have been working on Complexity of MU formulas. Structure of MU formulas. Developing powerful algorithms for finding MU subformulas from unsatisfiable formulas.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 26/42 4.2. Quantified Boolean Formulas
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 27/42 Quantified Boolean Formulas
The second bottle-neck: The reductions from problems to SAT may cause size-explosion.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 28/42 Quantified Boolean Formulas
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Main reason: The original problem is (believed) harder than NP.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 28/42 Quantified Boolean Formulas
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Main reason: The original problem is (believed) harder than NP. Many problems are in PSPACE: Planning, model-checking, Reasoning in Modal Logic, non-monotonic reasoning, etc.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 28/42 Quantified Boolean Formulas
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Main reason: The original problem is (believed) harder than NP. Many problems are in PSPACE: Planning, model-checking, Reasoning in Modal Logic, non-monotonic reasoning, etc. QSAT Problem: The Satisfiability Problem of Quantified Boolean formulas(QBF),isareferenceprobleminPSPACE.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 28/42 Quantified Boolean Formulas
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Main reason: The original problem is (believed) harder than NP. Many problems are in PSPACE: Planning, model-checking, Reasoning in Modal Logic, non-monotonic reasoning, etc. QSAT Problem: The Satisfiability Problem of Quantified Boolean formulas(QBF),isareferenceprobleminPSPACE. To avoid size explosion, Many problems are reduced to QSAT, instead of to SAT.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 28/42 Quantified Boolean Formulas
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Main reason: The original problem is (believed) harder than NP. Many problems are in PSPACE: Planning, model-checking, Reasoning in Modal Logic, non-monotonic reasoning, etc. QSAT Problem: The Satisfiability Problem of Quantified Boolean formulas(QBF),isareferenceprobleminPSPACE. To avoid size explosion, Many problems are reduced to QSAT, instead of to SAT. Then we need powerful QSAT Solvers.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 28/42 QCNF
AQCNFformulaΦ has the form Φ=Q1x1 ···Qnxnϕ.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 29/42 QCNF
AQCNFformulaΦ has the form Φ=Q1x1 ···Qnxnϕ.
Qi ∈{∃, ∀} and
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 29/42 QCNF
AQCNFformulaΦ has the form Φ=Q1x1 ···Qnxnϕ.
Qi ∈{∃, ∀} and
ϕ is a CNF formula over {x1, ··· ,xn}.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 29/42 QCNF
AQCNFformulaΦ has the form Φ=Q1x1 ···Qnxnϕ.
Qi ∈{∃, ∀} and
ϕ is a CNF formula over {x1, ··· ,xn}.
Q1x1 ···Qnxn is the prefix of Φ,andϕ is called the matrix of Φ.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 29/42 QCNF
AQCNFformulaΦ has the form Φ=Q1x1 ···Qnxnϕ.
Qi ∈{∃, ∀} and
ϕ is a CNF formula over {x1, ··· ,xn}.
Q1x1 ···Qnxn is the prefix of Φ,andϕ is called the matrix of Φ.
∀x1∃y1∀x2∃y2(x1 ∨¬y1) ∧ (x1 ∨¬x2 ∨ y2) ∧ (x1 ∨ x2 ∨¬y2).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 29/42 Models of QBF
∀x1∃y1∀x2∃y2(x1 ∨¬y1) ∧ (x1 ∨¬x2 ∨ y2) ∧ (x1 ∨ x2 ∨¬y2).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 30/42 Models of QBF
∀x1∃y1∀x2∃y2(x1 ∨¬y1) ∧ (x1 ∨¬x2 ∨ y2) ∧ (x1 ∨ x2 ∨¬y2).
Let f1 be (x1), f2 be (x1 ∨ x2).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 30/42 Models of QBF
∀x1∃y1∀x2∃y2(x1 ∨¬y1) ∧ (x1 ∨¬x2 ∨ y2) ∧ (x1 ∨ x2 ∨¬y2).
Let f1 be (x1), f2 be (x1 ∨ x2).
Replace, in the matrix, y1 by f1, y2 by f2,weget
(x1 ∨¬f1) ∧ (x1 ∨¬x2 ∨ f2) ∧ (x1 ∨ x2 ∨¬f2)
=(x1 ∨¬x1) ∧ (x1 ∨¬x2 ∨ x1 ∨ x2) ∧ (x1 ∨ x2 ∨¬(x1 ∨ x2))
is tautological.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 30/42 Models of QBF
∀x1∃y1∀x2∃y2(x1 ∨¬y1) ∧ (x1 ∨¬x2 ∨ y2) ∧ (x1 ∨ x2 ∨¬y2).
Let f1 be (x1), f2 be (x1 ∨ x2).
Replace, in the matrix, y1 by f1, y2 by f2,weget
(x1 ∨¬f1) ∧ (x1 ∨¬x2 ∨ f2) ∧ (x1 ∨ x2 ∨¬f2)
=(x1 ∨¬x1) ∧ (x1 ∨¬x2 ∨ x1 ∨ x2) ∧ (x1 ∨ x2 ∨¬(x1 ∨ x2))
is tautological.
we call (f1,f2) amodeloftheformula.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 30/42 Minimal False QCNF Formulas
AQCNFformulaΦ:=Qϕ with ϕ = {α1, ··· ,αn} is minimal false (MF) if
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 31/42 Minimal False QCNF Formulas
AQCNFformulaΦ:=Qϕ with ϕ = {α1, ··· ,αn} is minimal false (MF) if Φ is false,
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 31/42 Minimal False QCNF Formulas
AQCNFformulaΦ:=Qϕ with ϕ = {α1, ··· ,αn} is minimal false (MF) if Φ is false,
Φ −{αi} := Q(ϕ −{αi}) is true for each i =1, ··· ,n.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 31/42 Our Work on QBF
Model Theory for QBF (including: model-structure, model-size, simple model existence, relation between model-structure and formula-structure,etc.)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 32/42 Our Work on QBF
Model Theory for QBF (including: model-structure, model-size, simple model existence, relation between model-structure and formula-structure,etc.) Complexity, structure of minimal false formulas.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 32/42 Our Work on QBF
Model Theory for QBF (including: model-structure, model-size, simple model existence, relation between model-structure and formula-structure,etc.) Complexity, structure of minimal false formulas. Our results have been employed in QSAT Solver: Skizzo (M. Benedetti), which plays best for instances from Formal Verification.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 32/42 Our Work on QBF
Model Theory for QBF (including: model-structure, model-size, simple model existence, relation between model-structure and formula-structure,etc.) Complexity, structure of minimal false formulas. Our results have been employed in QSAT Solver: Skizzo (M. Benedetti), which plays best for instances from Formal Verification. Computational Complexity of QBF with Maximal Deficiency. Theoretical Computer Science,(2008)407:448-457.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 32/42 Our Work on QBF
Model Theory for QBF (including: model-structure, model-size, simple model existence, relation between model-structure and formula-structure,etc.) Complexity, structure of minimal false formulas. Our results have been employed in QSAT Solver: Skizzo (M. Benedetti), which plays best for instances from Formal Verification. Computational Complexity of QBF with Maximal Deficiency. Theoretical Computer Science,(2008)407:448-457. Boolean Function as Models, Journal of Automated Reasoning, (2007), 39:49-75.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 32/42 4.3. Expressive Power and Complexity
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 33/42 We need Logic Systems
The second bottle-neck: The reductions from problems to SAT may cause size-explosion.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 34/42 We need Logic Systems
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Another Reason: The existing reductions are not optimal.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 34/42 We need Logic Systems
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Another Reason: The existing reductions are not optimal.
direct reduction SAT problem or QSAT
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 34/42 We need Logic Systems
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Another Reason: The existing reductions are not optimal.
direct reduction SAT problem or QSAT
Direct reduction generally is not the “best" one.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 34/42 We need Logic Systems
The second bottle-neck: The reductions from problems to SAT may cause size-explosion. Another Reason: The existing reductions are not optimal.
direct reduction SAT problem or QSAT
Direct reduction generally is not the “best" one.
suitable SAT problem logic or QSAT
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 34/42 Too many Logic Systems
Generally, logic systems come from classic logic by extending the language of classical logic with additional operators, connectives, and quantifiers to service applications involving time, knowledge, action, etc.;
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 35/42 Too many Logic Systems
Generally, logic systems come from classic logic by extending the language of classical logic with additional operators, connectives, and quantifiers to service applications involving time, knowledge, action, etc.; restricting the language to guarantee better computational properties (e.g., description logic);
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 35/42 Too many Logic Systems
Generally, logic systems come from classic logic by extending the language of classical logic with additional operators, connectives, and quantifiers to service applications involving time, knowledge, action, etc.; restricting the language to guarantee better computational properties (e.g., description logic); changing deductive structure of classical logic create new logics (e.g., intuitionistic, relevance, linear, many-valued).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 35/42 Too many Logic Systems
Generally, logic systems come from classic logic by extending the language of classical logic with additional operators, connectives, and quantifiers to service applications involving time, knowledge, action, etc.; restricting the language to guarantee better computational properties (e.g., description logic); changing deductive structure of classical logic create new logics (e.g., intuitionistic, relevance, linear, many-valued). combining two logics (e.g., relevance modal logic)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 35/42 Equivalence-Reduction
To model a problem, which logic should be used?
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 36/42 Equivalence-Reduction
To model a problem, which logic should be used? The investigation of expressive power often provides a way to addressing the above challenge (D. M. Gabbay).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 36/42 Equivalence-Reduction
To model a problem, which logic should be used? The investigation of expressive power often provides a way to addressing the above challenge (D. M. Gabbay). Developing tools for comparing the relative expressive power of logic systems.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 36/42 Equivalence-Reduction
To model a problem, which logic should be used? The investigation of expressive power often provides a way to addressing the above challenge (D. M. Gabbay). Developing tools for comparing the relative expressive power of logic systems. Equivalence-reduction = Transformation preserving equivalence.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 36/42 Equivalence-Reduction
To model a problem, which logic should be used? The investigation of expressive power often provides a way to addressing the above challenge (D. M. Gabbay). Developing tools for comparing the relative expressive power of logic systems. Equivalence-reduction = Transformation preserving equivalence. We prefer poly-time,orpoly-space equivalence-reduction.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 36/42 Expressive Power and Complexity
SAT for PF and CNF both NP-complete, but there is no poly-pspace equivalence-reduction from PF to CNF.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 37/42 Expressive Power and Complexity
SAT for PF and CNF both NP-complete, but there is no poly-pspace equivalence-reduction from PF to CNF. SAT for answer-set logic program (ASLP) is NP-complete, but there is no poly-space equivalence-transformation from ASLP to PF under the assumption P)⊆NC1/poly.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 37/42 Expressive Power and Complexity
SAT for PF and CNF both NP-complete, but there is no poly-pspace equivalence-reduction from PF to CNF. SAT for answer-set logic program (ASLP) is NP-complete, but there is no poly-space equivalence-transformation from ASLP to PF under the assumption P)⊆NC1/poly. Although QSAT for ∀HORN is poly-time solvable,
∀HORN)≤pspacePF (Zhao, KB]) (hence ∃HORN)≤pspacePF) under the assumption P)⊆NC1/poly.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 37/42 Expressive Power and Complexity
SAT for PF and CNF both NP-complete, but there is no poly-pspace equivalence-reduction from PF to CNF. SAT for answer-set logic program (ASLP) is NP-complete, but there is no poly-space equivalence-transformation from ASLP to PF under the assumption P)⊆NC1/poly. Although QSAT for ∀HORN is poly-time solvable,
∀HORN)≤pspacePF (Zhao, KB]) (hence ∃HORN)≤pspacePF) under the assumption P)⊆NC1/poly. QSAT for QHORN is poly-time solvable,
QHORN=ptime ∃HORN ([Bubeck, KB,Zhao]).
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 37/42 Model-Equivalent Reduction
Informally speaking, a system L1 can be model-equivalently reduced to L2 if
Transformation T L1 L2
poly-time 1-1 correspondence Models Models of F of T(F)
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 38/42 Model-Equivalent Reduction
Informally speaking, a system L1 can be model-equivalently reduced to L2 if
Transformation T L1 L2
poly-time 1-1 correspondence Models Models of F of T(F)
We are investigating expressive power based on poly-time/poly-space model-equivalence-reduction.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 38/42 NP-Logic Systems
NP-Logic system:logicwithsatisfiabilityinNP.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 39/42 NP-Logic Systems
NP-Logic system:logicwithsatisfiabilityinNP. ∃PF has the strongest expressive power among all NP-systems with model-checking problem in NP.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 39/42 NP-Logic Systems
NP-Logic system:logicwithsatisfiabilityinNP. ∃PF has the strongest expressive power among all NP-systems with model-checking problem in NP. NP-Logic Systems and Model-equivalent Reductions, Journal of Computer Science and Technology,(2010),25(6): 1321-1326.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 39/42 NP-Logic Systems
NP-Logic system:logicwithsatisfiabilityinNP. ∃PF has the strongest expressive power among all NP-systems with model-checking problem in NP. NP-Logic Systems and Model-equivalent Reductions, Journal of Computer Science and Technology,(2010),25(6): 1321-1326. Model-equivalent Reductions, In Lecture Notes in Computer Science,3569,pp.355-370,2005,Springer.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 39/42 5. Future Work
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 40/42 Future Work
Many questions remain open.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 41/42 Future Work
Many questions remain open. Address the third Bottle-neck, i.e., the verification of SAT Solvers.
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 41/42 Thank you for your attention!
Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 42/42