Dichotomies in CSP Karl Lieberherr inspired by the paper: Dichotomies and Duality in First-order Model Checking Problems by Barnaby Martin The 11th Mons Days of Theoretical Computer Science 2006 Irisa - Rennes 5/13/2019 CSG 260 Fall 2006

Generalized model checking Find an interpretation that satisfies a fraction t (between 0 and 1) of the constraints. The generalized model checking problem over the logic called positive existential conjunctive fragment of FOL, {and, exists}-FOL, takes as input a structure A and a sentence f in {and, exists}-FOL and asks whether there exists an interpretation for f satisfying at least the fraction t of the weighted conjuncts. This problem is equivalent to the maximum constraint satisfaction problem (MaxCSP). FOL = first-order logic 5/13/2019 CSG 260 Fall 2006

Dichotomy for {and, exists}-FOL The class MaxCSP(A) exhibits a dichotomy: For all structures A (set of binary relations) there exists an algebraic constant tA between 0 and 1 such that the set of A-formulas f in {and, exists}-FOL satisfying Fraction(f, tA) are in P and (1) Fraction(f, tA+e) is NP-complete for any e > 0. Fraction(f, t) = there exists an interpretation for f satisfying at least fraction t of the weighted constraints. There is a universal polynomial algorithm parameterized by A for case (1). This is called a P-optimal algorithm. We use the terminology: MaxCSP(A) and MinCSP(A) for the maximization and minimization version. In the minimization version we replace “at least” by “at most”. 5/13/2019 CSG 260 Fall 2006

Example A = {OneInThree} where OneInThree(x1, x2, x3) = x1+x2+x3. t {OneInThree} = 4/9. See: Lieberherr/Specker JACM 1981 and Lieberherr Journal of Algorithms 1982. http://www.ccs.neu.edu/home/lieber/p-optimal/README.html 5/13/2019 CSG 260 Fall 2006

More examples R = AllRenaming(Orn), tR = 1-1/(2**n) R = Or1 union AllRenaming(Or>=2), tR = (sqrt(5)-1)/2 = 0.618 … R = AllRenaming(Or<=n), tR = ½ for all n >= 1. 5/13/2019 CSG 260 Fall 2006

Minimization We use the terminology: MaxCSP(A) and MinCSP(A) for the maximization and minimization version. In the minimization version we replace “at least” by “at most”. We reinterpret tA as tMax,A and we introduce by analogy tMin,A. Find an A so that tMax,A is different from tMin,A. 5/13/2019 CSG 260 Fall 2006

A more general context First order predicate logic A conjunctive formula must be true, i.e., all conjuncts must be true. Drop weights. A model checking problem over a logic L takes as input a structure A and a sentence f of L and asks: A╞ f (before we had A, t╞ f), where 0<=t<=1. Parameterize over A or f. 5/13/2019 CSG 260 Fall 2006

L = FOL (first-order logic) Alphabet: G1 union G2, where G1 = {not, and, or, exists, for all, =}, G0 ={(,),R,v,0,1} R(v1,v2, … ,vn) is a formula with free variables v1,v2, … ,vn. vi=vj is a formula with free variables vi, vj if f1 and f2 are formulas, then “f1and f2”, “f1 or f2” and “not f1” are also formulae (having as free variables those free in the constituent formulae) 5/13/2019 CSG 260 Fall 2006

FOL (continued) if f contains the free variable v, then “exists v f” and “for all v f” are formulae whose free variables are exactly those of f less v. A sentence is a formula with no free variables. We currently study {and, exists}-FOL but similar questions can be asked for other subsets of FOL. 5/13/2019 CSG 260 Fall 2006

Standard Definition of Model Checking Model checking definition: Efficiently deciding whether a temporal logic formula is satisfied in a finite state machine model. 5/13/2019 CSG 260 Fall 2006

Model checking The model is usually given as a source code description in an industrial hardware description language or a special-purpose language. Such a program corresponds to a finite state machine, i.e., a directed graph consisting of nodes (or vertices) and edges. A set of atomic propositions is associated with each node, typically stating which memory elements are one. The nodes represent states of a system, the edges represent possible transitions which may alter the state, while the atomic propositions represent the basic properties that hold at a point of execution. Formally, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a model M with initial state s, decide if M,s╞ p . If M is finite, as it is in hardware, model checking reduces to a graph search. 5/13/2019 CSG 260 Fall 2006

Going full circle Symbolic algorithms avoid ever building the graph for the FSM; instead, they represent the graph implicitly using a formula in propositional logic (BDDs). More recently, SAT solvers (see Boolean satisfiability problem) are used to perform the graph search. 5/13/2019 CSG 260 Fall 2006

Model checking and traversal specifications M,s╞ p M an object graph OG, s a node in OG p a formula expressing a desired node, e.g., bypassing {X,Y} via Z bypassing R to T a strategy graph with source and target Meta level: M’,s’╞ p must hold, otherwise compile-time error message. 5/13/2019 CSG 260 Fall 2006

Modular Implementation [Kiczales / Mezini] it is textually local there is a well-defined interface that describes how it interacts with the rest of the system the interface is an abstraction of the implementation, in that it is possible to make material changes to the implementation without violating the interface an automatic mechanism enforces that every module satisfies its own interface and respects the interface of all other modules the module can be automatically composed – e.g., by a compiler – with other modules to produce a complete system 5/13/2019 CSG 260 Fall 2006

A, t╞ f 5/13/2019 CSG 260 Fall 2006