Lecture 2 Propositional Logic & SAT
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1 CS 5110/6110 Rigorous System Design Spring 2017 Jan-17 Lecture 2 Propositional Logic & SAT Zvonimir Rakamarić University of Utah
2 Announcements Homework 1 will be posted soon Propositional logic: Chapter 1 of our textbook You can download it for free as a PDF
3 Syntax of Propositional Logic (PL) truth_symbol ::= > (true),? (false) variable ::= p, q, r, atom ::= truth_symbol variable literal ::= atom :atom formula ::= literal :formula formula Æ formula formula Ç formula formula formula formula $ formula
4 Examples of PL Formulae F : > F : p F : :p F : (p Æ q) (p Ç :q) F : (p Ç :q Ç r) Æ (q Ç :r) F : (:p Ç q) $ (p q) F : p $ (q r)
5 Semantics Semantics provides meaning to a formula Defines mechanism for evaluating a formula Formula evaluates to truth values true/1 and false/0 Formula F evaluated in two steps 1) Interpretation I assigns truth values to propositional variables I : {p false, q true } 2) Compute truth value of F based on I using e.g. truth table formula F + interpretation I = truth value
6 Notation Let F be a formula and I an interpretation I [ F ] denotes evaluation of F under I If I [ F ] = true then we say that F is true in I I satisfies F I is a model of F and write I ² F If I [ F ] = false we write I 6j= F
7 Example F : (p Æ q) (p Ç :q) I : {p 1, q 0} (i.e., I [ p ] = 1, I [ q ] = 0) p q :q p Æ q p Ç :q F F evaluates to true under I or I [ F ] = true or I ² F
8 Satisfiability and Validity F is satisfiable iff (if and only if) there exists I such that I ² F Otherwise, F is unsatisfiable F is valid iff for all I, I ² F Otherwise, F is invalid We write ² F if F is valid Duality between satisfiablity and validity: F is valid iff :F is unsatisfiable Note: only holds if logic is closed under negation
9 Equivalence Two formulae F 1 and F 2 are equivalent, denoted by F 1, F 2, iff they have the same models
10 Decision Procedure for Satisfiability Algorithm that in some finite amount of computation decides if given PL formula F is satisfiable NP-complete problem Modern decision procedures for PL formulae are called SAT solvers Naïve approach Enumerate truth table Modern SAT solvers DPLL algorithm Davis-Putnam-Logemann-Loveland Operates on Conjunctive Normal Form (CNF)
11 Normal Forms Negation Normal Form (NNF) Only allows :, Æ, Ç Negation only in literals Disjunctive Normal Form (DNF) Disjunction of conjunction of literals: WV Conjunctive Normal Form (CNF) Conjunction of disjunction of literals: i V i j W j l i ;j l i ;j
12 Negation Normal Form To transform F into F in NNF recursively apply the following equivalences: ::F 1, F 1 :>,? :?, > :(F 1 Æ F 2 ), :F 1 Ç :F 2 :(F 1 Ç F 2 ), :F 1 Æ :F 2 F 1 F 2, :F 1 Ç F 2 F 1 $ F 2, (F 1 F 2 ) Æ (F 2 F 1 )
13 Example F : p $ (q r)
14 Conjunctive Normal Form To transform F into F in CNF first transform F into NNF and then recursively apply the following equivalences: (F 1 Æ F 2 ) Ç F 3, (F 1 Ç F 3 ) Æ (F 2 Ç F 3 ) F 1 Ç (F 2 Æ F 3 ), (F 1 Ç F 2 ) Æ (F 1 Ç F 3 ) (Note: a disjunction of literals is called a clause.)
15 Example F : p $ (q r)
16 Exponential Blow-Up Such a naïve transformation can blow-up exponentially (in formula size) for some formulae For example: transforming from DNF into CNF
17 Tseitin Transformation [1968] Used in practice No exponential blow-up CNF formula size is linear wrt original formula Does not produce an equivalent CNF However, given F, the following holds for the computed CNF F : F is equisatisfiable to F Every model of F can be translated (i.e., projected) to a model of F Every model of F can be translated (i.e., completed) to a model of F No model is lost or added in the conversion
18 Tseitin Transformation Main Idea Introduce a fresh variable e i for every subformula G i of F e i represents the truth value of G i Assert that every e i and G i pair are equivalent Assertions expressed as CNF Conjoin all such assertions in the end
19 Example F : p $ (q r) $ e 0 p e 1 q r
20 SAT Solver Input Format Based around DIMACS c c start with comments c p cnf
21 Classical DPLL Searching for a model M for a given CNF formula F Incrementally try to build a model M Maintain state during search State is a pair M F F is a set of clauses and it doesn t change during search M is a sequence of literals No literals appear twice and no contradiction Order does matter Decision literals marked with l d
22 Abstract Transition System Contains a set of rules of the form M F M F denoting that search can move from state M F to state M F
23 DPLL Rules Extending M Propagate M G,C Ç l M,l G,C Ç l if M ² :C and l not in M Decide M F M,l d F if l or :l in F and l not in M
24 DPLL Rules Adjusting M Fail M G,C fail if M ² :C and M contains no decision literals Backtrack M,l d,n G,C M,:l G,C if M,l d,n ² :C and N contains no decision literals
25 Propagate M G,C Ç l M,l G,C Ç l if M ² :C and l not in M Decide M F M,l d F if l or :l in F and l not in M Fail M G,C fail if M ² :C and M contains no decision literals Backtrack M,l d,n G,C M,:l G,C if M,l d,n ² :C and N contains no decision literals
26 DPLL Example 1 ; :pçqçr, p, :qçr, :qç:r, qçr, qç:r
27 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q
28 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p)
29 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q
30 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q)
31 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q
32 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r)
33 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q
34 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s)
35 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q
36 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t)
37 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q
38 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q (Propagate :u)
39 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q (Propagate :u) p d,q,r d,s,t d,:u :pçq, :rçs, :tç:u, uç:tç:q
40 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q (Propagate :u) p d,q,r d,s,t d,:u :pçq, :rçs, :tç:u, uç:tç:q (Backtrack)
41 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q (Propagate :u) p d,q,r d,s,t d,:u :pçq, :rçs, :tç:u, uç:tç:q (Backtrack) p d,q,r d,s,:t :pçq, :rçs, :tç:u, uç:tç:q
42 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q (Propagate :u) p d,q,r d,s,t d,:u :pçq, :rçs, :tç:u, uç:tç:q (Backtrack) p d,q,r d,s,:t :pçq, :rçs, :tç:u, uç:tç:q (Decide u)
43 DPLL Example 2 ; :pçq, :rçs, :tç:u, uç:tç:q (Decide p) p d :pçq, :rçs, :tç:u, uç:tç:q (Propagate q) p d,q :pçq, :rçs, :tç:u, uç:tç:q (Decide r) p d,q,r d :pçq, :rçs, :tç:u, uç:tç:q (Propagate s) p d,q,r d,s :pçq, :rçs, :tç:u, uç:tç:q (Decide t) p d,q,r d,s,t d :pçq, :rçs, :tç:u, uç:tç:q (Propagate :u) p d,q,r d,s,t d,:u :pçq, :rçs, :tç:u, uç:tç:q (Backtrack) p d,q,r d,s,:t :pçq, :rçs, :tç:u, uç:tç:q (Decide u) p d,q,r d,s,:t,u d :pçq, :rçs, :tç:u, uç:tç:q
44 Modern SAT Solvers DPLL + improvements Backjumping Dynamic variable ordering Learning conflict clauses Random restarts
45 Next Lecture First-order logic
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