Declarative Knowledge Processing
1. Introduction
Outline
1. Introduction
2. Syntax
Declarative Knowledge Processing
Additional material: Propositional Logic
3. Semantics
4. Reasoning Problems
4.1 SAT, the satisfiability problem
4.2 Other reasoning problems
Magdalena Ortiz
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and their complexity
5.2 Complexity of Reasoning
Knowledge Base Systems Group
Institute of Information Systems
[email protected]
6. SAT in the real world
WS 2012/2013
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Declarative Knowledge Processing
1. Introduction
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Declarative Knowledge Processing
Propositional logic
1. Introduction
Modeling problems in Propositional Logic
as a formalisms for declarative knowledge processing
We start with classical propositional logic, the simplest possible logic
Propositional logic is the simples and most basic logic.
Recall that, for each formalism, we study:
We use propositional symbols (aka Boolean variables) to represent
propositions, i.e., possible facts that are either true or false.
1
Background and motivation
2
The syntax and semantics of the formalism
3
Main reasoning problems
4
Reasoning techniques and algorithms
5
Computational complexity and implementation of reasoners
p
r
u
w
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Paul is outside
It is raining
Paul has an umbrella
Paul is wet
overheat1
valveOpen1
alarm
Engine 1 is overheated
The safety valve of engine 1
is open
The alarm is activated
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Declarative Knowledge Processing
1. Introduction
Declarative Knowledge Processing
Modeling problems in Propositional Logic
Syntax of Propositional Logic
Boolean combinations of these symbols allow us to build more
complex statements
p ∧ r ∧ ¬u,
2. Syntax
We start form a countable set V of propositional variables (or simply
propositions)
p ∧ r ∧ ¬u → w
Propositional formulas are defined inductively
overheat1 ∧ (¬valveOpen1 ∨ ¬alarm)
• Every p ∈ V is a formula
We can infer properties of the described domain by reasoning about
these statements
• > (verum, top) and ⊥ (falsum, bottom) are formulas
• If φ1 and φ2 are formulas, then so are
Is Paul wet?
(¬φ1 )
(φ1 ∧ φ2 )
(φ1 ∨ φ2 )
negation
conjunction
disjunction
Is the system safe?
We can use (φ1 → φ2 ) as a shortcut for (¬φ1 ∨ φ2 ) and (φ1 ↔ φ2 ) as a
shortcut for (φ1 → φ2 ) ∧ (φ2 → φ1 )
Recall:
p
r
Paul is outside
it is raining
u
w
Paul has an umbrella
Paul is wet
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Declarative Knowledge Processing
2. Syntax
Declarative Knowledge Processing
Outline
3. Semantics
Outline
1. Introduction
1. Introduction
2. Syntax
2. Syntax
3. Semantics
3. Semantics
4. Reasoning Problems
4.1 SAT, the satisfiability problem
4.2 Other reasoning problems
4. Reasoning Problems
4.1 SAT, the satisfiability problem
4.2 Other reasoning problems
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and their complexity
5.2 Complexity of Reasoning
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and their complexity
5.2 Complexity of Reasoning
6. SAT in the real world
6. SAT in the real world
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Declarative Knowledge Processing
3. Semantics
Declarative Knowledge Processing
Interpretations
3. Semantics
Models, Logical Implication
We consider two truth values: true and false
A interpretation (aka truth assignment) is a mapping I that assigns
either true or false to each propositional variable, i.e.,
An interpretation I is a model of a formula φ if I(φ) = true. In this
case we write
I |= φ
For a set of formulas Γ (aka theory), we say I is a model of Γ if
I(φ) = true for every φ ∈ Γ
I : V 7→ {true, false},
We say that a formula φ follows from a theory Γ, if every model of Γ
is a model of φ. In this case we write
and such that I(>) = true and I(⊥) = false.
Γ |= φ
An interpretation I is inductively extended to all formulas (next slide)
Questions: • What can we say about Γ |= φ if φ ∈ Γ?
• What is the intuitive meaning of ∅ |= φ?
Once we fix an interpretation (i.e., values for the propositional
variables), the value of all formulas is unambiguously determined
We write |= φ instead of ∅ |= φ, and say that φ is valid (true in all
interpretations)
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Declarative Knowledge Processing
3. Semantics
Declarative Knowledge Processing
Interpretations (ctd.)
true
=
false
true
I(φ ∧ ψ)=
false
true
I(φ ∨ ψ)=
false
4. Reasoning Problems
Outline
Given an interpretation I, the semantics of formulas is inductively defined
as follows:
I(¬φ)
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if I(φ) = false,
otherwise.
1. Introduction
2. Syntax
3. Semantics
if I(φ) = true and I(ψ) = true,
otherwise.
4. Reasoning Problems
4.1 SAT, the satisfiability problem
4.2 Other reasoning problems
if I(φ) = true or I(ψ) = true,
otherwise.
Questions:
• What is the meaning of the defined connectives → and ↔?
• Given a formula φ, does the value of φ in I depend on all
the propositional symbols in V? Why?
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and their complexity
5.2 Complexity of Reasoning
Exercises: You should be able to determine the semantics (truth value) of
any given formula in any given interpretation
6. SAT in the real world
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Declarative Knowledge Processing
4. Reasoning Problems
4.1 SAT, the satisfiability problem
SAT, the satisfiability problem
Declarative Knowledge Processing
5. Reasoning in Propositional Logic
Outline
The most important reasoning problem in Propositional Logic is the
satisfiability problem
1. Introduction
2. Syntax
Formula satisfiability
A formula φ is satisfiable if there exists a model for φ
3. Semantics
SAT, the satisfiability problem
4. Reasoning Problems
4.1 SAT, the satisfiability problem
4.2 Other reasoning problems
Input: A formula φ
Question: Is φ satisfiable?
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and their complexity
5.2 Complexity of Reasoning
SAT is a decision problem, i.e., the answer is either yes or no
6. SAT in the real world
We call UNSAT the complementary problem, i.e., to decide if φ is
not satisfiable
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Declarative Knowledge Processing
4. Reasoning Problems
4.2 Other reasoning problems
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Declarative Knowledge Processing
Other Reasoning Problems
5. Reasoning in Propositional Logic
Solving different Reasoning Problems
We have mentioned different reasoning problems
Validity
Problem
Modelhood
Satisfiability
Validity
Logical Consequence
Input: A formula φ
Question: Does |= φ hold?
Logical Consequence
Input: A theory Γ, a formula φ
Question: Does Γ |= φ hold?
Input
I, φ
φ
φ
Γ, φ
Question
Does I |= φ?
Is there some I such that I |= φ?
Does I |= φ for every I?
Does I |= φ for every model I of Γ?
Questions: • How do we solve modelhood? (it’s very easy!)
• Why do the other problems seem to be harder?
• Do we need an algorithm for each problem?
Modelhood (aka satisfaction)
Input: An interpretation I, a formula φ
Question: Does I |= φ hold?
Validity and logical consequence can be reduced to UNSAT.
Note: These are all decision problems, but there are other reasoning problems
that are not decision problems (finding a model, counting models, etc.)
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Hence, an algorithm for (UN)SAT suffices.
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Declarative Knowledge Processing
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and complexity
A non-deterministic algorithm for SAT
Declarative Knowledge Processing
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and complexity
A deterministic algorithm for SAT
Assume we have an algorithm selectProp(φ) that returns a propositional variable
occurring in φ.
Assume we have an algorithm models(I, φ) that decides modelhood.
Let φ be a given propositional formula, and let {p1 , . . . , pn } be the set of
all propositional variables in φ. Then SAT(φ) decides the satisfiability of φ:
We denote by φ[p/>] (resp. φ[p/⊥]) the result of replacing in φ each instance of
the propositional letter p by > (resp. ⊥) and simplifying the resulting formula.
Algorithm SAT(φ)
Algorithm SAT(φ)
1
guess an assignment I : {p1 , . . . , pn } 7→ {true, false};
2
return models(I, φ);
Soundness is trivial: if SAT(φ) says yes, then φ has a model I.
Completeness is also trivial: if there is a model for φ, then there is a
way to guess an interpretation I such that models(I, φ) says yes, and
hence SAT(φ) says yes.
1
if φ == > then return true;
2
if φ == ⊥ then return false;
3
p := selectProp(φ);
4
return
(SAT(φ[p/>]) OR SAT(φ[p/⊥])) ;
Exercises: Can you explain why the algorithm is it sound and complete? Why
does it terminate? How much time and space does it require?
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Declarative Knowledge Processing
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and complexity
A non-deterministic algorithm for SAT - complexity
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Declarative Knowledge Processing
5. Reasoning in Propositional Logic
5.2 Complexity of Reasoning
Complexity of SAT
From the non-deterministic algorithm we discussed, we can infer:
It is not hard to show that models(I, φ) can be run in time
polynomial (linear) in the size of φ
Theorem. SAT is in NP
Storing a guessed interpretation I needs only polynomial (linear)
space
Proof sketch: there is a non-deterministic algorithm for SAT
that runs in polynomial time
Is this optimal, or can we solve SAT more efficiently?
Hence, SAT(φ) needs only polynomial time
Theorem [Cook 70, Levin 71]. SAT is NP-hard
However, in practice, we can not really guess the right interpretation!
Proof sketch: For any non-deterministic polynomial time
Turing machine M and input word w, we can build in
polynomial time a formula φM,w such φM,w is satisfiable iff
M has an accepting run on w.
If n propositional variables occur in φ, then there are
2n possible interpretations that have to be tested!
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Declarative Knowledge Processing
5. Reasoning in Propositional Logic
5.2 Complexity of Reasoning
Declarative Knowledge Processing
Complexity of reasoning in propositional logic
6. SAT in the real world
SAT Solvers and applications
Nowadays there are many SAT solvers available
From the two theorems above, it follows that:
e.g., MiniSAT, Chaff, HyperSAT, WalkSAT, precosat, LySAT, SATzilla,
ManySAT . . .
Corollary. SAT is in NP-complete
Although they all need exponential time in the worst case (why?),
some of them can efficiently handle huge formulas with thousands of
propositional variables
We can also easily infer that:
Many of them work on formulas in clausal form, aka Conjunctive
Normal Form (CNF)
The validity problem is coNP-complete
The logical implication problem is coNP-complete
Question:
• What can you say about the complexity of the modelhood problem?
Many SAT solvers employ optimized versions of the famous
Davis-Putnam-Logemann-Loveland algorithm (DPLL)
There is an international SAT Competition and a SAT Race every
year, see http://www.satcompetition.org/
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Declarative Knowledge Processing
6. SAT in the real world
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Declarative Knowledge Processing
Outline
6. SAT in the real world
SAT Solvers and applications (ctd.)
SAT solvers can be applied for solving any NP-complete problem
1. Introduction
They are employed in a huge range of fields
2. Syntax
See Marques-Silva, Practical Applications of Boolean Satisfiability, WODES’08:
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3. Semantics
4. Reasoning Problems
4.1 SAT, the satisfiability problem
4.2 Other reasoning problems
5. Reasoning in Propositional Logic
5.1 SAT Algorithms and their complexity
5.2 Complexity of Reasoning
Noise prediction in integrated circuits
Termination analysis in term-rewriting systems
Model-checking of finite state systems
Design debugging
AI planning
Package management in software distributions
Software model-checking and testing
Haplotype inference in bioinformatics
SAT solving is an extremely active and large research field, with
hundreds of researchers, thousands of publications, own conferences
and journals
6. SAT in the real world
See, e.g., http://www.satlive.org/, http://www.lri.fr/SAT2011
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