Basic definitions Let A be a model (a logical structure). - PDF document

1/23/2010 Basic definitions Let A be a model (a logical structure). • Satisfaction : 𝐵 ⊨ 𝜔 𝜔 is true in 𝐵 . Logical definability over Example : 𝐸, ≤ ⊨ ∃𝑧 ∀𝑦 ,𝑧 ≤ 𝑦- finite models ( D is finite) Let T be a theory (a set of logical sentences). Steven Lindell • Deduction : 𝑈 ⊢ 𝜔 𝜔 is proved from 𝑈 . Haverford College Example : T is the theory of a strict total order. USA 𝑈 ⊢ ∀𝑦 ∀𝑧 ,𝑦 < 𝑧 → 𝑧 ≮ 𝑦- 1/23/2010 ISLA 2010 1 1/23/2010 ISLA 2010 2 Completeness Compactness theorem Theorem [Gödel]: profound correspondence Theorem : A set of first-order sentences has a These are equivalent: model if and only if every finite subset does. Proof : follows from completeness via deduction • semantic validity (truth) 𝑈 ⊨ 𝜔 Corollary : if a sentence has arbitrarily large Every model of 𝑈 is a model of 𝜔 . finite models, then it has an infinite model. Idea : consider the theory consisting of the • syntactic validity (proof) 𝑈 ⊢ 𝜔 original sentence together with sentences that express increasingly large cardinality. There is a proof of 𝜔 from 𝑈 . 1/23/2010 ISLA 2010 3 1/23/2010 ISLA 2010 4 Models of computation Why be concerned? • Growth rates – need to process large data sets Sequential model – single processor (automata) • Data is processed serially n log n n 2 2 n • One-at-a-time access to memory 1 0 1 2 2 1 4 4 5 2 25 32 Concurrent model – multiple processors with 10 3 100 1024 read/write access to a common memory (CRAM) 20 4 400 1048576 50 5 2,500 1.3 10 15 • Data is processed in parallel 100 7 10,000 1.27 10 30 • Simultaneous access (must resolve conflicts) 1000 10 1,000,000 1.07 10 301 10 6 20 1,000,000,000,000 hopelessly large 1/23/2010 ISLA 2010 5 1/23/2010 ISLA 2010 6 1

1/23/2010 Complexity Classes Graphs are universal • Precise definitions only matter for linear cases Given a query in any vocabulary, it can be because models turn out to be all equivalent. translated into an equivalent query over (directed) graphs. A simple example would be a Sequential Space Concurrent Time query over binary words (a linear order with a O (1) constant-space constant-time single monadic predicate). The idea is to O (log n ) Logspace (A)logtime perform a simple translation from one O ( n ) linear-space linear-time vocabulary to the other which is first order O ( n k ) definable in both directions. PSPACE Polytime 1/23/2010 ISLA 2010 7 1/23/2010 ISLA 2010 8 Translating binary strings to graphs Diagram Definition : The diagram of a finite graph is a first- 1 0 1 0  order sentence which describes it completely. < < < ⟼  {0, 1, 2, 3}, <, U   {0, 1, 2, 3}, E  𝑁 ⊨ 𝜀 𝐻 ⇔ 𝑁 ≅ 𝐻 = 𝑏 1 , … , 𝑏 𝑜 , 𝐹 𝜀 𝐻 = ∃𝑦 1 … 𝑦 𝑜 𝑦 𝑗 ≠ 𝑦 𝑘 & ∀𝑧 𝑧 = 𝑦 𝑗  :  u < v  [ u = v  U ( u )] E ( u , v ) 𝑗≠𝑘 1≤𝑙≤𝑜  -1 : x < y  x  y  E ( x , y ) 𝐹(𝑦 𝑗 , 𝑦 𝑘 ) ¬𝐹(𝑦 𝑗 , 𝑦 𝑘 )  -1 : U ( z )  E ( z , z ) 𝐻⊨𝐹(𝑏 𝑗 ,𝑏 𝑘 ) 𝐻⊭𝐹(𝑏 𝑗 ,𝑏 𝑘 ) 1/23/2010 ISLA 2010 9 1/23/2010 ISLA 2010 10 Boolean queries on graphs Definable queries Let K be the collection of finite directed graphs. Logic provides a natural means for classifying I.e. each G =  V G , E G  in K is of the form queries. Let 𝜚 𝐿 = *𝐻 ∈ 𝐿 ∶ 𝐻 ⊨ 𝜚+ be the Boolean query defined by a sentence 𝜚 over K . E G  V G  V G V G = { 1, …, n } Definition : A Boolean query Q is elementary if it Definition : A Boolean query Q on K is an is definable by some first-order sentence. I.e. ∀𝐻 isomorphism-closed subset of K , i.e., Q  K and for all G , H  K , 𝐻 ∈ 𝑅 ⇔ 𝐻 ⊨ 𝜚 i.e. 𝑅 = 𝜚 𝐿 G  H  ( G  Q  H  Q ) 1/23/2010 ISLA 2010 11 1/23/2010 ISLA 2010 12 2

1/23/2010 Trivial properties The theory of finite linear ordering A query over the class of finite models F which is • The sentences true in all finite linear orders: finite or co-finite: | Q | <  ; or | F – Q | <  . Δ = 𝜀: 𝐸, < ⊨ 𝜀 for all finite 𝐸 (closure under finite consequence) Fact : Every trivial query is elementary. • Every infinite model looks like: ( W any order) Proof : Q is first-order definable via the 𝜕 + 𝑋 × (𝜕 ∗ + 𝜕) + 𝜕 ∗ sentence: δ 𝐻 if Q is finite, or 𝐻∈𝑅 • All of them are elementarily equivalent , i.e. ¬δ 𝐻 if Q is co-finite. they satisfy the same first order sentences. 𝐻∉𝑅 1/23/2010 ISLA 2010 13 1/23/2010 ISLA 2010 14 Infinite models of  Parity Definition : A binary string  1   n is said to have Σ = Δ ∪ *∃𝑦 1 … 𝑦 𝑜 𝑦 𝑗 ≠ 𝑦 𝑘 : 𝑜 = 1, 2, … + even parity if  1     n = 0. 𝑗≠𝑘 Corollary : PARITY is not elementary. is complete . I.e. for every  either Σ ⊨ θ or Σ ⊨ ¬θ . Proof : a binary string of all ones, i.e.  x U ( x ), Theorem : Over  , every first-order sentence is has even parity if and only if its length is even. eventually true or eventually false (i.e. trivial). Harder : What about sparse strings? I.e. the Proof : apply compactness possibly easier problem of computing the parity of a string with very few ones, all of which are Corollary : The query EVEN consisting of even length (very) far apart. linear orders is not elementary. 1/23/2010 ISLA 2010 15 1/23/2010 ISLA 2010 16 Sparse parity  FO(<) Proof using saturated models Proof : Consider an  1 -saturated model A =  A, <, P  of . Let  be the Consider the class C of finite structures of the theory of infinite discrete linear orders (with endpoints). Observe that form  A , <, P  where P  A , and | P | « | A |. Let the substructure P =  P, <  of A is an  1 -saturated model of  , because we can relativize the types in A to P . For adjacent p and q in P , let [ p ,  = Th( C ) together with P ( min ), P ( max ), the q ] be the interval { a  A : p < a < q } and see that  [ p , q ] , <  is an  1 - saturated model of  , for the same reason. axioms for P being infinite, and that the We aim to show that any two  1 -saturated models A and A ' of  are elements of P are infinitely far apart. Clearly,  is isomorphic. Since  is complete, the respective saturated substructures P and P ' are isomorphic, say by f . To extend the isomorphism, notice finitely consistent with arbitrarily large even and that for each a  A \ P there are adjacent p and q in P such that p < a < odd parity finite models. q (since this property is a first-order sentence in Th( C )), and similarly f ( a ) is in between adjacent f ( p ) and f ( q ) in P' . Again, since  is Lemma :  is complete. complete, and because the models are saturated,  [ p , q ] , <    [ f ( p ), f ( q )] , <  . Therefore f can be extended to all of A and A' . ฀ Corollary : Sparse parity is not first-order over C . 1/23/2010 ISLA 2010 17 1/23/2010 ISLA 2010 18 3

1/23/2010 Elementary non-definability Elementary reductions Lesson : Standard techniques from model theory Say P ≤ Q if P can be first -order defined using Q. such as compactness and completeness can be Examples : acyclicity, connectivity ≤ TC used to show that a property of finite models is (  z )  E + ( z , z ) (  xy ) E + ( x , y ) not first-order definable. Exercise : parity ≤ connectivity, acyclicity. Hint : Outline of method using nonstandard models : use the historical switching circuit over ordering. Every nontrivial first-order sentence has infinite Corollary : transitive-closure  FO. models which make it true and false (not simultaneously!). Find a way to complete the Proof : PARITY is not elementary, and parity ≤ infinitary theory while preserving non-triviality. acyclicity, connectivity ≤ transitive -closure. 1/23/2010 ISLA 2010 19 1/23/2010 ISLA 2010 20 Path problems are not elementary Second-order logic 1 ) We concentrate on the purely existential ( Σ 1 Questions : What about undirected reachability? 1 ) fragments. The and purely universal ( Π 1 • REACH( a , b )  there is a path from a to b monadic fragment is restricted to quantification Exercise : REACH is not elementary. Hint : go back 1 and m Π 1 1 ). over subsets (m Σ 1 to the switching circuit and utilize the minimal Exercise : Parity over binary strings  B , <, U  . and maximal elements. Hint : Introduce a set S such that between • How about defining a connected component ? adjacent elements of S , there are exactly two elements of U (deal with endpoints separately). Given a simple graph and an identified vertex, 1 and m Π 1 1 . Write this as a formula in both m Σ 1 find all the nodes connected to it. Is this in FO? 1/23/2010 ISLA 2010 22 1/23/2010 ISLA 2010 23 1 1 Undirected acyclicity is in m Π 1 Undirected Connectivity is in m Π 1 There is no 2-regular (finite) subgraph: ∀𝐸 ∃𝑦𝐸 𝑦 ∧ ∃𝑧¬𝐸 𝑧 → ∃𝑦, 𝑧 𝐸 𝑦 ∧ ¬𝐸 𝑧 ∧ 𝐹 𝑦, 𝑧 ¬∃𝑇 ≠ ∅ ∀𝑤 ∈ 𝑇 ∃ 2 𝑣 ∈ 𝑇 𝐹(𝑣, 𝑤) Idea : there is always an edge between the two Idea : It is enough to say the relativized degrees pieces of any non-trivial partition of the graph. are at least two. This guarantees a cycle if the • Not in m Σ 1 1 . graph is finite. On the other hand, if there is a cycle then a minimal one is a 2-regular subgraph. What about “in same connected component”? 1/23/2010 ISLA 2010 24 1/23/2010 ISLA 2010 25 4