What is good about HO i ? For all i ≥ 2, HO i +1 provides two important features: • exponentially bigger auxiliary re- lations than HO i ; • nesting of relations, like in ( i + 1) -th order graphs , where each node is actually an i -th order graph, or ( i +1) -th order PERT networks , for large and complex projects, where a node may represent a PERT network itself, and the op- eration of zooming in or out al- lows navigation in depth. 32
But... The complexity of the evaluation of an HO i +1 query is exponentially higher than that of an HO i query (see above). For instance, for Existential Fourth Order Logic queries (Σ 3 1 ) the com- plexity is NTIME(2 2 ( nc ) ) � = c ∈ N While for Existential Third Or- der Logic queries (Σ 2 1 ) is NTIME(2 ( n c ) ) � = c ∈ N 33
What if...? What happens if we bound the size of the i -th order relations to be polynomial in the size of the input dbi? We could still have nesting ... 34
Besides being a requirement in some applications (like deep struc- tures where zoom operations are necessary), in many cases • nesting provides a more pow- erful language which allows sim- pler and more intuitive expressions for a query. This also happens when using pro- gramming languages with rich data structures (like OOPL): • it makes programs much sim- pler and less error-prone than us- ing the old Assembler languages of the sixties and seventies. 35
• This is convenient not only for applications to Databases in the In- dustry, but also for Theoretical re- search. • To prove that a query is in the polynomial hierarchy (PH), in many cases using higher order construc- tions in HO i,P can be much simpler than using SO (see below). • To prove that a query is in the poly-logarithmic hierarchy (PLH), in many cases using higher order constructions in HO i,plog (HO <i,plog ) can be much simpler than using SO plog (see be- low). 36
• Is nesting still relevant as to ex- pressive power ? 37
1: A General Schema of TO Formulas 38
Let σ be a relational vocabulary, which may include constant sym- bols. We define T [ σ ] as the class of TO formulas of the form: � ∃C ¯ s O ¯ s ¯ s TotalOrder( C , O ) ∧ �� � ∀ G First( G ) → α First ( G ) �� � ∧ Last( G ) → α Last ( G ) ∧ � ∀ G pred G succ Pred( G pred , G succ ) �� → ϕ ( G pred , G succ ) where 39
• C ranges over TO relations of type s = ( i 1 , . . . , i s ). ¯ • TotalOrder( C , O ), First( G ), Last( G ) and Pred( G pred , G succ ) denote fixed SO formulas. • α First ( G ) and α Last ( G ) denote arbitrary SO formulas. • ϕ ( G pred , G succ ) denotes an ar- bitrary SO formula. 40
This is a very usual, intuitive , and convenient schema in the expres- sion of natural properties of finite models. For a start, it can clearly be used to express the hypercube and for- mula-value queries as described above. 41
Additional examples are provided by the different relationships be- tween pairs of undirected graphs ( G, H ) that can be defined as or- derings of special sorts (see [Downey, Fellows, 1999]). Using the schema these relation- ships can be expressed by defining a set of possible operations that can be applied repeatedly to H , un- til a graph which is isomorphic to G is obtained. 42
In particular, the following rela- tionships fall into this category: a) G ≤ immersion H : G is an im- mersion in H ; b) G ≤ top H : G is topologically embedded or topologically contained in H ; c) G ≤ minor H : G is a minor of H ; d) G ≤ induced − minor H : G is an induced minor of H ; Interestingly, in all these cases the length of the sequence is at most linear . 43
The operations on graphs needed to define those orderings are: (E) delete an edge, (V) delete a vertex, (C) contract an edge, (T) degree 2 contraction, or sub- division removal , (L) lift an edge. 44
In particular the set of allowable operations for each of those order- ings are: { E, V, L } for ≤ immersion , { E, V, C } for ≤ minor , { E, V, T } for ≤ top , { V, C } for ≤ induced − minor . 45
[Ferrarotti, Gonz´ alez, Turull-Torres, 2017] We have the following: Theorem: Every TO formula Ψ of the above schema T can be translated into an equivalent SO formula Ψ ′ when- ever the following conditions hold. 1. The sub formulas α First , α Last and ϕ of Ψ are SO formulas. 2. There is a d ≥ 0 such that for ev- ery valuation v with v ( C ) = R , = ∃O ¯ s ¯ s ψ ( C , O ), then if A , v | |R| ≤ | dom ( A ) | d . 46
Planarity in Graphs The classical Kuratowski defini- tion of planarity , provides yet an- other example of a property that can be defined using our schema and also results in a linearly bounded sequence of structures. By Wagner’s characterization (see [Bollob´ as, 2002]) a graph is planar if and only if it contains neither K 5 nor K 3 , 3 as a minor . 47
Note that the more intuitive con- struction for planarity would be to say that there is no transformation process of linear size that arrives to a K 5 or K 3 , 3 , starting from the input graph and applying in each transition exactly one of the oper- ations in { E, V, C } above. 48
If we have the negation of a for- mula in the schema T , we can use the same translation to SO , and then add a negation in front of the SO formula. Then we have the following: Corollary: The negation ¬ Ψ of a formula Ψ of the above schema T can also be translated into an equivalent SO formula ¬ Ψ ′ whenever the two con- ditions of the previous theorem hold. 49
[Ferrarotti, Gonz´ alez, Schewe, Turull-Torres, 2018] By using the normal form for ( SO + TC 2 ) ([Imm,1999]) the following re- sult is straightforward: Theorem: The class of TO formulas of the above schema T is equivalent to the logic ( SO + TC 2 ). And, hence, equal to PSPACE. Corollary: The class of TO formulas of the schema T is closed under negation. 50
Translation to Non Det Parallel ASM [Ferrarotti, Gonz´ alez, Schewe, Turull-Torres, 2018] By using the non deterministic, parallel Abstract State Machine model ([Boerger, 2003]), it is not difficult to prove the following: Theorem: Every formula Ψ of the above schema T can be systematically translated to an equivalent non determinis- tic, parallel ASM which doesn’t use higher order formulas. 51
Note that for the sake of easily comprehensible high-level specifi- cations it is advisable to extend rigorous methods to support also higher-order logic and to investi- gate strategies for refinement to first-order. 52
Theorem provers and Non det Parallel ASM It is well known that for many cases of ASM’s, there are theorem provers which allow semi-automatic theorem proving support for many cases of ASM rules. In particular, for non determin- istic parallel ASM’s there are very interesting results. 53
[Schellhorn,Ernst,Pfhler,Bodenmller, Reif , 2018] • It is possible to compute an FO formula for each rule that im- plies clash-freedom (*) when prov- able (it is provable for many ASMs that are used in practice ). (*) for each state S a rule r yields an update set ∆( S ), i.e. a (fi- nite) set of (finite) sets of up- dates. There is a clash if there are two updates ( l, v 1 ), ( l, v 2 ) in ∆( S ) with v 1 � = v 2 . (i.e., pairs location (i.e., n -ary function symbol and an n -tuple of values), and value ) 54
• They give axioms that describe the transition relation for clash- free ASM rules as SO formu- las that can be used to verify pre/post-condition assertions, and to derive properties of ASM’s, using automated theorem provers . • They provide a Calculus for clash- free ASM rules based on sym- bolic execution for deduction , which can be used for interactive theo- rem provers , like their tool KIV . 55
[Ferrarotti, Gonz´ alez, Schewe, Turull-Torres, 2018] By using higher order logics HO i , P (see below) the following result is straightforward: Theorem: For every ASM extended with HO i , P formulas in its rules, we have an automatic refinement of the HO i , P extended ASM to an SO extended ASM. 56
Once we got the SO extended ASM, we can apply to it the na¨ ıve refine- ment strategy consisting on non- deterministically guessing the quan- tified relation variables. As na¨ ıve refinements in a stan- dard way are always possible, we believe that semi-automatic proofs could be conducted on such, though not optimal refinements. 57
QBF Solvers Alternatively, the use of QBF solvers is worth exploring . from “QBF Gallery 2014 (Com- petition)”, in the “QBF Solver Eval- uation Portal”, www.qbflib.org/index eval.php 58
“Many problems from application domains such as model checking, formal verification or synthesis are PSPACE-complete, and hence could be encoded in QBF”. “ Considerable progress has been made in QBF solving throughout the past years. However, in con- trast to SAT, QBF is not yet widely applied to practical problems in industrial settings”. 59
Once we got an SO formula φ (see below): • for every model A , there is a translation f φ ( A ) to a QBF for- mula (see [Hella, Turull-Torres, 2006a] for a translation), • we can then use a QBF solver . 60
2: Downward polynomially bounded Relations HO i , P 61
An i -th order relation R of type τ in a structure A is downward polynomially bounded ( dpb ) by d if |R| ≤ | dom ( A ) | d , and for all 2 ≤ j ≤ i − 1, all the j -th order relations that form the tuples of ( j + 1)-th order relations, are in turn dpb by d . 62
For i ≥ 3 we define HO i,P as the extension of HO i − 1 ,P , where the i - th order quantifiers restrict the car- dinality to be bounded by a poly- nomial that depends on the quan- tifier. In the alphabet of HO i,P , for ev- ery pair of positive integers d , and j , with i ≥ j ≥ 3, we have: a j -th order quantifier ∃ j,P,d and for every j -th order type τ , we have countably many j -th order vari- able symbols X j,d,τ . 63
A valuation in a structure A as- signs to each i -th order relation vari- able X j,d,τ a dpb i -th order rela- tion R of type τ in A , such that |R| ≤ | dom ( A ) | d . For any 2 < j ≤ i , the HO i,P quantifier ∃ j,P,d has the following semantics: = ∃ j,P,d X j,d,τ ϕ ( X ) A | iff there is a j -th order relation R of type τ , such that A | = ϕ ( X )[ R ] and R is dpb by d in A . 64
[Ferrarotti, Gonz´ alez, Turull-Torres, 2017] We have the following: Theorem: For all i ≥ 3, HO i , P collapses to SO . Moreover, every formula in HO i , P can be algorithmically trans- lated to an equivalent SO formula. 65
Strategy: Basically, the strategy of the trans- lation is to use a relational database with referential integrity to encode each relation variable of order ≥ 2. Let i ≥ k ≥ j ≥ 2. For each variable of order k , the db that rep- resents it consists of 2( k − 1) rela- tions. For each j -th order variable we have one relation with id’s for tu- ples of relations of order ( j − 1), and one relation for id’s of rela- tions of order ( j − 1). 66
Empty Relations We must also have in mind that the tuples of relations of any order, can have empty relations in some of its components. Then, the (SO) “ relation ” that we use to store the set of tuple iden- tifiers for a relation of type width s , is actually a set of 2 s (SO) rela- tions, one for each possible combi- nation of empty relations in such a tuple. 67
Then, for a given query, we can proceed as follows: 1. • Use an HO i,P formula, with an arbitrary order i , to express the query, 2. • translate algorithmically the HO i,P formula into an SO for- mula, 3. • evaluate the SO formula. Note that we have still (determin- istic) single exponential time com- plexity, (NP complete queries are still there !) in the third step. But we don’t have to deal with hyper exponential complexity . 68
A Note on the Different Translations The first translation (schema T of TO ) yields a more clear and in- tuitive SO formula, and the max- imum arity of the quantified SO relation variables in general seems to be much smaller . For the case of hypercube graphs the maximum arity obtained by the schema translation is 4, while for the SO formulas obtained by the HO i,P translation is 8. 69
And for the case of the Formula- Value query the maximum arity obtained by the schema translation is also 4, while for the SO formulas obtained by the HO i,P translation is 22. Note that the arity of a relation symbol in an SO formula is rele- vant for the complexity of its eval- uation (see among others [Hella, Turull-Torres,2006]). 70
Hence, and not surprisingly it makes sense to study specific schemas of TO formulas that have equiva- lent SO formulae, aiming to find more efficient translations than the general strategy used for HO i,P formulas (which had the purpose of proving equivalence , rather than looking for efficiency in the trans- lation). 71
3: Valuating Relations of Poly-logarithmic Cardinality 72
A Query in TO plog Graph Factoring [Ferrarotti, Gonz´ alez, Schewe, Turull-Torres, 2018] Roughly, let TO plog denote the fragment of TO where only valu- ations which assign TO relations of poly-logarithmic cardinality, to TO variables are considered. The SO sub-formulas in TO plog are standard SO formulas. For that matter we use typed TO variables of the form X τ, log k , mean- ing that valuations can only assign to them relations of type τ and car- dinality ≤ ( ⌈ log n ⌉ ) k . 73
The input structure is A of sig- nature σ F = � V I , E I , F I � , where ( V A I , E A I ) is a connected and loop- less undirected graph ( cu-graph ), and F A I is a TO relation which in turn consists of a set of pairs of graphs ( V A F I , E A F I ), and ( V A K , E A K ). The first graph of each pair is a cu- graph , and the second graph is a clique . 74
We define graph factoring as a decision problem. A σ F -structure A is in the class GraphFactoring iff the third-order relation F A I is a factoring of the graph ( V A I , E A I ), where the first graph of each pair in F A I is a cu-graph that is a factor of the graph ( V A I , E A I ), and the size of the corresponding clique is the exponent. 75
A straightforward consequence of the definition of graph product is that the size of any factoring cir- cuit C for a structure A is at most 2 · ⌈ log( | V A I | ) ⌉ , and the size of the TO relation F I A on any given A ∈ GraphFactoring is at most ⌈ log( | V A I | ) ⌉ . 76
� ϕ GF ≡ ∃V C E C “FactoringCircuitForG I ( V C , E C ) ∧ NodesCUgraphs( V C , E C ) ∧ RootsPrimeGraphs C ∧ RootsIn F I C � ∧ SingleOutputG I C ” where ( V C , E C ), is a TO graph of size at most 2 ·⌈ log( | V A I | ) ⌉ , whose nodes are cu-graphs, and whose edges are pairs of cu-graphs. 77
FactoringCircuitForG I ( V C , E C ) ≡ � “Digraph( V C , E C ) ∧ Acyclic( V C , E C ) ∧ Connected( V C , E C ) ∧ InDegree2 C ∧ ProductOfParents C ∧ LinearNonRoots C � ∧ NonIsomorphicRoots C ” 78
“InDegree2 C ” says that every node in the circuit has either 1 or 2 input nodes. “ProductOfParents C ” says that ev- ery node in V C is a cu-graph that is either the product of its two par- ents , or the square of its single par- ent . 79
Product( V 1 , E 1 , V 2 , E 2 , V 3 , E 3 ) ≡ �� ∃ V × E × ∀ v 1 w 1 v 2 w 2 � ( V × ( v 1 , w 1 ) ↔ ( V 1 ( v 1 ) ∧ V 2 ( w 1 ))) ∧ � E × ( v 1 , w 1 , v 2 , w 2 ) ↔ � ( v 1 = v 2 ∧ E 2 ( w 1 , w 2 )) ∨ ���� ( w 1 = w 2 ∧ E 1 ( v 1 , v 2 )) ∧ � “Isomorphic( V × , E × , V 3 , E 3 )” 80
� LinearNonRoots C ≡ ∃V C l E C l � � V C l , { int . nodes in C} ∧ “EqualTO � � E C l , E C ↾ { int . nodes in C} EqualTO ∧ LinearDigraph( V C l , E C l ) ′′ � where E C ↾ { int . nodes in C} is the restriction of the TO binary rela- tion E C to the subset of internal nodes of the set V C . 81
NumbOfProducts C ( V 0 , E 0 , V K 0 ) ≡ � ∃H “ H : V K 0 �→ Children C ( V 0 , E 0 ) � quasi injective” The quasi injectivity of the func- tion in the formula above is due to the fact that we avoid allowing multiple edges between two given nodes in the circuit C , to make the formula simpler. 82
Note that the only possible case where one single edge means that a (factor) graph is actually being used twice in the same product is at the (unique) node at depth one in the circuit. An example for this situation is the factoring circuit for an hyper- cube of order n , where the same factor graph ( K 2 ) is used n times. 83
Note: As the sizes of the valuating TO relations that make the formula ϕ GF true are poly-logarithmic , then it seems straightforward to apply the same encoding strategy as in HO i,P and translate it to an SO formula. Hence, we have the following: Corollary: TO plog = SO. 84
Though the query graph factoring can certainly be expressed in SO (for instance with a signature σ F = � V 1 I , E 2 I , V 2 F , E 3 F , V 2 K , E 3 K � ), it doesn’t seem to be easy. 85
Roughly, let SO plog denote the frag- ment of SO where only valuations which assign SO relations of poly- logarithmic cardinality, to SO vari- ables are considered. For that matter we use typed SO variables of the form X r, log k , mean- ing that valuations can only assign to them relations of arity r and car- dinality ≤ ( ⌈ log n ⌉ ) k . And let TO plog (SO plog ) denote the fragment of TO plog where only val- uations which assign SO relations of poly-logarithmic cardinality, to SO variables are considered. 86
Expected result: With the same strategy, we be- lieve that we can also prove: • TO plog (SO plog ) = SO plog . 87
[Ferrarotti, Gonz´ alez, Schewe, Turull-Torres, 2018a] On the other hand, we proved the following result: • � 1 ,plog ( b ∀ ) = NPolyLogTime. 1 • SO plog = PLH. (*) [(*) Barrington gave a characteri- zation of the class of DCL-uniform families of Boolean circuits of un- bounded fan-in, and quasi polyno- mial size (i.e., 2 (log n ) O (1) ) and con- stant depth with an equivalent logic ([Barrington, 1992]). From that re- sult the second result above follows.] 88
Where � 1 ,plog ( b ∀ ) is the existen- 1 tial fragment of SO plog where the FO ∀ is bounded to poly-logarithmic sub-domains. And PLH denotes the non deterministic Polylog-Time Hierarchy. 89
Expected result: Then, we would have also that: • TO plog (SO plog ) = PLH. This would mean that we can use a higher level language like TO plog (SO plog ) to prove that a given query is in PLH. That would make easier both the construction of the formulas and the corresponding proofs . 90
Examples in TO plog (SO plog ) : • There is an induced subgraph ( V ′ , E ′ ) of size between ⌈ log n ⌉ and ( ⌈ log n ⌉ ) c , and there is a set F of size at least ( ⌈ log n ⌉ ) 1 / 2 , of dis- joint induced subgraphs ( V ′ i , E ′ i ), s. t. the subgraphs in F are a set of prime factors of the sub- graph ( V ′ , E ′ ). • There are between ⌈ log n ⌉ and ( ⌈ log n ⌉ ) c disjoint induced sub- graphs that are cliques of sizes between ⌈ log n ⌉ and ( ⌈ log n ⌉ ) d . Note that the first query, doesn’t seem to have an easy SO plog for- mula. 91
To express it in TO plog (SO plog ) we can follow a similar strategy as for Graph-Factoring above. 92
We believe that the following queries can be also expressed in TO plog (SO plog ): • All the induced subgraphs of size between ⌈ log n ⌉ and ( ⌈ log n ⌉ ) c are prime . • There are polylog disjoint induced subgraphs of polylog size s.t. for each of them, all its prime fac- tors are disjoint induced sub- graphs of size polylog. • For every polylog size set of dis- joint induced subgraphs of poly- log size in G 1 there is a set of the same size of disjoint induced subgraphs of polylog size in G 2 , s.t. there is a bijection F : V 1 → V 2 so that the two graphs in ev- 93
ery pair in F are isomorphic . 94
So, proving that result, we would be able to use TO plog (SO plog ) logic to write probably many queries in a much simpler way than using SO plog . And still, in that way proving that the queries are in PLH. But we believe that we can do bet- ter ... 95
Expected result: Finally, we also believe that with the same strategy, we can prove: • HO i,plog (HO <i,plog ) = SO plog = PLH. 96
4: Beyond Second Order SATQBF 97
SATQBF k and SATQBF QBF k denotes the set of quanti- fied propositional formulas of the form φ ≡ ∃ ¯ x 1 ∀ ¯ x 2 . . . Q ¯ x k ( ϕ ) , where ϕ is a propositional formula over X = { x ij } 1 ≤ i ≤ k, 1 ≤ j ≤ l i , n ≥ 0, and where for 1 ≤ i ≤ k , ¯ x i = ( x i 1 , . . . , x il i ) is a tuple of l i differ- ent variables from X . 98
Note that Q is “ ∃ ” if k is odd and “ ∀ ” if k is even, and the sets X 1 , . . . , X k of variables in ¯ x 1 , . . . , ¯ x k , respectively, form a partition of X . Let QBF = � k> 0 QBF k . The semantics of the quantifiers is as follows: ∃ x ( α ( x )) ≡ α (0 /x ) ∨ α (1 /x ), and ∀ x ( α ( x )) ≡ α (0 /x ) ∧ α (1 /x ). 99
Note that, in view of the seman- tics of the quantifiers, every quanti- fied propositional formula is equiv- alent to a propositional formula , which is longer (roughly, exponen- tially longer in the number of quan- tifiers). φ is satisfiable if there is a par- tial valuation v 1 : X 1 → { T, F } , s. t. for every partial valuation v 2 : X 2 → { T, F } , there is a par- tial valuation v 3 : X 3 → { T, F } , s. t. . . . s. t. the valuation v = v 1 ∪ v 2 ∪ v 3 ∪ . . . ∪ v k makes ϕ true. 100
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