EXP. LOGIC: M.Ziegler PSPACE. NPcomplete. School of Computing PSPACE CH #P PH. Martin Ziegler 박세원신승우조준희 ( 박찬수 ) complete. co- P NP. Re a ) Computation

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1 EXP PSPACE complete PSPACE CH #P PH conpcomplete NPcomplete co- NP P NP P L NP School of Computing Martin Ziegler 박세원신승우조준희 ( 박찬수 ) Complexity and Re a ) Computation Please ask questions!

2 Informal Logic a) cdonald: The sum of two even primes is a square b) 2 mothers and 2 daughters together buy 3 hats, yet each receives her own. c) Smart phones are prohibited during exams: A students' phones are collected initially and only half of them returned. Tautology Still, nobody complains! Satisfiability d) All pink unicorns can fly! Inconsistency e) Epimenides the Cretan said: All Cretans are liars! f) Can you correctly answer this very question? Kobayashi-Maru g) Is "no" the only correct answer to this question? Boolean Logic Truth values and operations like,, Example (expression): ( ( ) ( )) x y z x x x y x y (x y) ( x y) ((x y) ( x y)) ((x z) ( x z)) x y x y x y (x z) ( x z)

3 Propositional "Programming" k n x k := x x 2 x n k l n (x k x l ) "at least one" "at most one" x x 2 x 3 x 4 "x = & x 2 = & x 3 = & x 4 =" h b,,b n (x,,x n ) "x =b & x 2 =b 2 & & x n =b n " b:f(b)= h b (x) = f(x) "x y" = "y x" "x y" = "(x y) ( x y)" Can express every Boolean function using only,, Satisfiability and Tautology A propositional formula ϕ(x,,x n ) is satisfiable if there exists an assignment of its variables x,,x n (with s and s) that makes ϕ evaluate to true. A set Φ of formulae is satisfiable if there exists a (joint) assignment to all occurring variables that makes every ϕ Φ evaluate to true. A tautology ϕ evaluates to true for every assignment. Examples: a) x y satisfiable, no tautology b) x x y satisfiable and tautology c) x x neither satisfiable nor tautology d) { x y, x y, x y, x y } not satisfiable Observe: ϕ is not tautology iff ϕ is satisfiable

4 Sequent Calculus Formalize "expressions implies other expression(s)": Def: Let Φ,Ψ denote sets of propositional formulae ϕ,ψ. Write "Φ Ψ" if every assignment of variables making all ϕ Φ true, also renders at least one ψ Ψ true. Examples: a) { x y } { x } b) { ϕ ψ } { ϕ } c) { ϕ ψ } { ϕ, ψ } d) { x y, x y } { x } e) {} ψ iff ψ tautology f) Φ {} iff Φ not satisfiable Observation (sound rules): d) Φ Ψ,ϕ,ψ Φ Ψ,ϕ ψ e) Φ,ϕ,ψ Ψ Φ,ϕ ψ Ψ Abbr Φ,ϕ for Φ {ϕ} f) Φ,ϕ Ψ & Φ,ψ Ψ Φ,ϕ ψ Ψ g) Φ Ψ,ϕ & Φ Ψ,ψ Φ Ψ,ϕ ψ a) Φ,ϕ Ψ,ϕ b) Φ Ψ,ϕ Φ, ϕ Ψ c) Φ,ϕ Ψ Φ Ψ, ϕ Examples of Formal Proofs "Theorem:" x (x y) y Syntactic proof by deduction/application of rules: a) x,y y c) x y, y a) x x, y "Theorem:" ψ ψ "Theorem:" ψ ψ "Theorem:" ϕ ψ ψ ϕ Sound Rules: d) Φ Ψ,ϕ,ψ Φ Ψ,ϕ ψ e) Φ,ϕ,ψ Ψ Φ,ϕ ψ Ψ f) Φ,ϕ Ψ & Φ,ψ Ψ Φ,ϕ ψ Ψ g) Φ Ψ,ϕ & Φ Ψ,ψ Φ Ψ,ϕ ψ semantic proof: truth table g) x x y, y d) x (x y) y a) ψ ψ b) ψ, ψ {} c) ψ ψ a) ψ ψ c) {} ψ, ψ b) ψ ψ a) Φ,ϕ Ψ,ϕ b) Φ Ψ,ϕ Φ, ϕ Ψ c) Φ,ϕ Ψ Φ Ψ, ϕ

5 Sound and Complete Proof System "Theorem:" x (x y) y vs. x,y. x (x y) y (Meta)Theorem (proven by structural induction): Φ ψ is true iff Additional rules for Predicate Logic (Quantifiers): Φ Ψ,ψ(x,) ψ(x,) Φ Ψ, y.ψ(x,y) Φ Ψ,ψ(x,) ψ(x,) Φ Ψ, y.ψ(x,y) d) Φ Ψ,ϕ,ψ Φ Ψ,ϕ ψ e) Φ,ϕ,ψ Ψ Φ,ϕ ψ Ψ f) Φ,ϕ Ψ & Φ,ψ Ψ Φ,ϕ ψ Ψ g) Φ Ψ,ϕ & Φ Ψ,ψ Φ Ψ,ϕ ψ it can be derived from the rules. Such a derivation can be found algorithmically! a) Φ,ϕ Ψ,ϕ b) Φ Ψ,ϕ Φ, ϕ Ψ c) Φ,ϕ Ψ Φ Ψ, ϕ First-Order Logic So far just Boolean operations vs. x,y. x (x y) y and variables ranging over Boolean values and. Examples: a) ( {,},,,,,, = ) Booleans b) (,,, +,,, < ) Reals as a ring c) (,,, +,,, = ) Complex numbers d) (,,, +,, < ) Peano Arithmetic e) (,,, +, < ) Presburger Arithmetic f) ( 2 2,, I, +,, = ) Real square matrices g) (, < ) Rationals as linearly ordered set (Constants are functions of arity, + and of 2.) (Meta)Definition: A structure is a set X with functions f k :X σ k X & relations R l Xτ l of aritiesσ k,τ k

6 Data Structures are Structures E.g. a stack S storing elements from D, with methods new S, push:s D S, pop:s D considered as functions on structure X := S D. Property/Axiom: pop(push(s,d))=d pop(new) may return any element of D which can be prevented: if (s!=new) d:=pop(s) Alternatively, consider method/relation empty S. (Meta)Definition: A structure is a set X with functions f k :X σ k X & relations R l Xτ l of aritiesσ k,τ k Expressions and Formulae Examples: a) =( {,},,,,,, = ) b) (,,, +,,, < ) x y. x + y = c) (,,, +,,, = ) x. ( x = y. x y = ) d) (,,, +,, < ) x y. x = y y f) ( 2 2,, I, +,, = ) x y. x y = y x h) (S,new,push,pop,=) y z.. ( x y z y = z = ) a) expr. x and (x y) y, formula x. x = (x y) y An expression is (syntactially valid) composed from functions; a formula is a Boolean/quantified combination of relations among expressions. (Meta)Definition: A structure is a set X with functions f k :X σ k X & relations R l Xτ l of aritiesσ k,τ k

Axiomatizing/Abstract Data Types Examples: a) =( {,},,,,,, = ) b) (,,, +,,, < ) x y. x + y = c) (,,, +,,, = ) x. ( x = y. x y = ) d) (,,, +,, < ) x y. x = y y f) ( 2 2,, I, +,, = ) x y.

current on/off, stack=array/single/double linked list, un/bin-ary User must only rely on axioms of abstract data type: sufficient to capture its properties up to isomorphism.

(Tarski-Seidenberg): There is an infinite decidable family Φ of axioms, such that the below claim holds for. The theory of real numbers is decidable!

7 Axiomatizing/Abstract Data Types Examples: a) =( {,},,,,,, = ) b) (,,, +,,, < ) x y. x + y = c) (,,, +,,, = ) x. ( x = y. x y = ) d) (,,, +,, < ) x y. x = y y f) ( 2 2,, I, +,, = ) x y. x y = y x h) (S,new,push,pop,=) y z.. ( x y z y = z = ) Hide implementation details: / vs. current on/off, stack=array/single/double linked list, un/bin-ary User must only rely on axioms of abstract data type: sufficient to capture its properties up to isomorphism. (Meta)Definition: A structure is a set X with functions f k :X σ k X & relations R l Xτ l of aritiesσ k,τ k Hilbert, Gödel, Tarski ( =(, {,},, +,,,, <, ),, (,, = ),, +,, < ) Theorem (Tarski-Seidenberg): There is an infinite decidable family Φ of axioms, such that the below claim holds for. The theory of real numbers is decidable! Theorem: There exists no (even semi-) decidable family Φ of axioms, such that the below claim holds for. st Idea: Take as Φ all valid formulae!? The theory of integers is undecidable decidable! User Hilbert must Program only rely(92): on axioms identify of abstract setφ data type: sufficient of axioms to for capture natural real itsnumbers properties such up to that: isomorphism. ψ iff ψ can be derived from Φ syntactically and such derivation can be found algorithmically.

Giuseppe Peano and Kurt Gödel 2 nd Idea: Peano's Axiomsfor (,, S, +, ) i) n: S(n) successor/increment ii) n,m: n=m S(n) S(m) iii) n m: n= n=s(m) iv) n: n=n+

First-order Logic: only quantification over elements, not over subsets/relations/functions! st Idea: Take as Φ all valid formulae!

Consequences and Conclusion (,,, +,, < ) (,,, +,, < ) The theory of integers is undecidable decidable! (Algebr.) theory of real numbers is decidable!

8 Giuseppe Peano and Kurt Gödel 2 nd Idea: Peano's Axiomsfor (,, S, +, ) i) n: S(n) successor/increment ii) n,m: n=m S(n) S(m) iii) n m: n= n=s(m) iv) n: n=n+ v) n,m: n+s(m)=s(n+m) vi) n: n = vii) n,m: n S(m) = n m + n f()= n. f(s(n))=f(n)+n n. 2 f(n) = n S(n)? First-order Logic: only quantification over elements, not over subsets/relations/functions! st Idea: Take as Φ all valid formulae! Theorem: There exists no semi-decidable family Φ of axioms,, such that it holds: ψ iff ψ can be derived from Φ syntactically. Consequences and Conclusion (,,, +,, < ) (,,, +,, < ) The theory of integers is undecidable decidable! (Algebr.) theory of real numbers is decidable! (,,, +,, < ) The theory of rationals is undecidable. There are correct algorithms over (,,, +,, < ) whose total correctness however cannot be verified! Correctness of loop invariants and algorithms over (,,, +,, < ) is decidable! Other infinite structures? It depends or is open!

Theoretical Computer Science for Numerics IEEE float/double: fast but awkward semantics, violates distributive law

Programming language for real computation: imperative abstr.

practical acceptance Folklore: Don t test for equality! realram/bss-machine: So, how about inequality "<"?

9 Theoretical Computer Science for Numerics IEEE float/double: fast but awkward semantics, violates distributive law Interval arithmetic: error propagation Multiprecision arithmetic: how choose initial precision? Programming language for real computation: imperative abstr. data type REAL computable semantics non-extensionality formal verification Stream computing/recursive Analysis: No practical acceptance Folklore: Don t test for equality! realram/bss-machine: So, how about inequality "<"? uncomputable semantics x= (x<) (x>) 이계식, 김선영, 박세원 non-extensional semantics Thanks for your attention! L O G I C Logic Set Theory Analysis Geometry (linear) Algebra Combinatorics Stochastics Number Theory Theory of Computing

10 Levels of Understanding. reproduce 2. apply 3. transfer 4. extend What is thought is not said What is said is not heard What is heard is not understood What is understood is not believed What is believed is not yet advocated What is advocated is not yet acted on What is acted on is not yet completed Bloom's Taxonomy Konrad Lorenz (Nobel Prize 973)

First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms

First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO