On amenability of constraint satisfaction problems Michal R. Przybylek Faculty of Mathematics, Informatics and Mechanics University of Warsaw Poland [email protected] Abstract the theory) is crucial for our proof, as one can find coun- The authors of [22] showed that a constraint satisfaction terexamples to the claim in Intuitionistic Set Theories. problem (CSP) defined over rational numbers with their nat- These highly-theoretical results are of a great practical ural ordering has a solution if and only if it has a definable interest. Very many real-world decision problems of high solution. Their proof uses advanced results from topology computational complexity can be abstractly specified as the and modern model theory. The aim of this paper is threefold. classical constraint satisfaction problems (CSP): hardware (1) We give a simple purely-logical proof of their theorem verification and diagnosis: [9], [12], automated planning and and show that the advanced results from topology and model scheduling [10], [11], temporal and spatial reasoning [31], theory are not needed; (2) we introduce an intrinsic char- [7], air traffic managment [1], to name a few. Such problems acterisation of the statement “definable CSP has a solution are inherently finite. Although their computational cost is iff it has a definable solution” and investigate it ingeneral high (i.e. the problems are usualy NP-hard), they can be 2 intuitionistic set theories (3) we show that the results from solved in a finite time by a machine . This is in contrast with modern model theory are indeed needed, but for the implica- problems concerning behaviours of autonomous systems, tion reversed: we prove that “definable CSP has a solution iff where the classical variant of CSP is too restrictive. Such it has a definable solution” holds over a countable structure problems can be naturally specified as CSP with infinite sets if and only if the automorphism group of the structure is of variables (corresponding to the states of a system) and extremely amenable. infinite sets of constraints (corresponding to the transitions between the states of a system). These problems are, in gen- CCS Concepts • Theory of computation Constraint eral, undecidable — no machine can solve them in a finite and logic programming; time. In fact, depending on the choice of the specification lan- Keywords set theory with atoms, intuitionistic set theory, guage, such problems may be very high in the undecidability constraint satisfaction problem, Ramsey property, extremely hierarchy. For example, if we consider problems definable amenable group, Boolean prime ideal theorem in the First-Order theory of natural numbers, then every problem from the arithmetical hierarchy can be expressed as a definable CSP. Up until recently, we had known very little 1 Introduction about methods that can be used to solve infinite CSP. The In 1964 James D. Halpern [14] by using some combinato- first breakthought was at the begining of the century (see rial properties of the ordered Fraenkel-Mostowski model of [8] and also a survey article [6]), where researches applied set theory with atoms solved a long-standing open prob- algebraic and model-theoretic tools to analyze CSP over, so lem about independence of the Axiom of Choice from the called, infinite templates. This research inspired the Warsaw Boolean Prime Ideal Theorem. In 2015 Bartek Klin, Eryk Logical Group to investigate locally finite CSP — i.e. infinite Kopczynski, Joanna Ochremiak, and Szymon Torunczyk [22] CSP whose constraints are finite relations (see [22] and [27]). by using advanced results from topology and modern model They found that a locally finite CSP defined in the theory of theory, proved that in the ordered Fraenkel-Mostowski model rational numbers with their natural ordering can be solved of set theory with atoms an equivariant (constrained1) lo- effectively. cally finite constraint satisfaction problem has a solution if and only if it has an equivariant solution. In this paper we Example 1.1 (Finite memory machine). An important type prove that these two results are essentially the same, and, in of autonomyous systems has been defined by Kaminski, Michael fact, equivalent to many other well-known axioms/theorems and Francez [20]. The authors called these type of systems “fi- of Boolean Set Theories. The assertion of Booleaness of the nite memory machines”, or “register machines”. A finite mem- Set Theory (i.e. that the law of excluded middle holds inside ory machine is a finite automaton augmented with a finite number of registers '8 that can store natural numbers. The 1Every variable appears in at least one constraint. movement of the machine can depend on the control state, on the letter and on the content of the registers. The dependency on , ver. 1.2, 2020 2020. 2For a general reference on solving classical CSP see: [32] 1 , ver. 1.2, 2020 Michal R. Przybylek '1 B = This graph is known as infinite Kneser graph [22]. An example = < '1? '2 B '1 of a (constrained) locally finite CSP problem is the question whether a graph like G is 3-colorable. By the compactness of the first-order logic G is 3-colorable if and only if every start ( finite subgraph of G is 3-colorable. Figure 2 exhibits an ex- ample of a finite subgraph of G which cannot be colored by three colors fred, green, blueg. Theorem 19 in [22] implies that ' = 3-colourability of any graph generated by finite memory ma- = ' 2 B < 2? chines can be solved effectively. '1 B '2 Figure 1. A non-deterministic machine with two registers and a single control state (. SET PASSW ' , , : 1 2 5 1 = G , 2 5 START , , 3 1 1 4 ' < G ? , ' 2, 3 4, 5 := ? 4, 2 5, 3 , 3 4 ' : ' < G? ' < G? = AUTH AUTH AUTH G TRY 1 TRY 2 TRY 3 Figure 2. A finite counterexample to 3-colorability of infinite ' Kneser graph. = ' G ? = ? G G = ? ' the content of the registers is, however, limited — the machine GRANT can only test for equality (no formulas involving successor, AUTH addition, multiplication, etc. are allowed). Figure 1 shows a 2-register machine with a single control state (. This machine starts with a given content of the registers '1 and '2 and then at every step non-deterministically chooses a register '8 and a natural number = such that = < '8 . Value = EXIT CHNG AUTH PASSW is then stored in register '8 , whilst register '1−8 gets the value previously stored in '8 . Observe that in contrast to finite automata, the graph of Figure 3. A register machine that models access control to possible configurations in the machine from Figure 1 is infinite. some parts of the system. If we run the machine with '8 B 8, then its states + (i.e. single control state ( together with the content of the registers) will Example 1.2 . Continuing span an infinite graph G = h+, 퐸i with edges 퐸 induced by the (Access-control register machine) Example 1.1, Figure 3 presents an example of a finite memory movements of the machine: machine with one register ', whose task is to model access + = f¹(, =,<º : = 2 '1,< 2 '2g control to the red part of the system. The machine starts in 0 0 0 0 control state “SET PASW”, where it awaits for the user to pro- 퐸 = f¹¹(, =,<º, ¹(, = ,< ºº 2 + × + : ¹= = < ^ < < = º vide a password G. This password is then stored in register ', = <0 < =0 _ ¹ < ^ = ºg and the machine enters control state “START”. Inside the blue rectangle the machine can perform actions that do not require 2 On amenability of constraint satisfaction problems , ver. 1.2, 2020 authentication, whereas the actions that require authentication Tychonoff topology) is extremely amenable3 by the main theo- are presented inside the red rectangle. The red rectangle can rem in [28]. be entered by the control state “GRANT AUTH”, which can be accessed from one of three authentication states. In order to Example 1.4 (Rational numbers with finitely many con- Q t & authorise, the machine moves to control state “AUTH TRY 1”, stants). Let 0 be the structure from Example 1.3 over an @ 2 & where it gets input G from the user. If the input is the same extended signature consisting of all constants 0 for some & ⊆ & as the value previously stored in register ', then the machine finite 0 . Like in the previous example, the first order Q t & l enters control state “GRANT AUTH”. Otherwise, it moves to theory of 0 is -categorical and the topological group of ¹Q t & º control state “AUTH TRY 2” and repeats the procedure. Upon automorphisms Aut 0 is extremely amenable. second unsuccessful authorisation, the machine moves to con- Example 1.5 (Ordered vector space). Let 퐻퐹 be the free @0- trol state “AUTH TRY 3”. But if the user provides a wrong dimensional vector space over a finite field 퐹. By definition 퐻 password when the machine is in control state “AUTH TRY 3”, has a base that can be enumerated by any countable set. There- ' the register is erased (replaced with a value that is outside of fore, we can assume that there is a base hU@i@2Q enumerated the user’s alphabet) — preventing the machine to reach any of by rational numbers4 @ 2 Q. In fact we can give an explicit the control states from the red rectangle. Inside the red rectan- description of space 퐻퐹 with its standard base as follows. Let gle any action that requires authentication can be performed. & us identify 퐻퐹 with a subspace of 퐹 consisting of functions For example, the user may request the change of the password. that have finite support — i.e. functions E : & ! 퐹 with the Example of problems that we may like to ask, which can be property that the set f@ 2 & : E ¹@º < 0g is finite.