TAFA – A Tool for Admissibility in Finite Algebras Christoph R¨ othlisberger Mathematics Institute, University of Bern Sidlerstrasse 5, 3012 Bern, Switzerland [email protected]

Abstract. Checking whether a quasiequation is admissible in a finitely generated quasivariety is known to be decidable by checking validity in a suitable (finite) free algebra on finitely many generators. Nevertheless this approach is computationally unfeasible since these free algebras can be very big. TAFA is a system providing algebraic tools to search for the smallest set of algebras, according to the standard multiset well-ordering, such that a quasiequation is admissible in the quasivariety if and only if it is valid in this set of algebras.

1

Introduction

A rule is called admissible in a logic if it does not produce any new theorems when added to the logic. Admissibility is typically used to establish key properties of systems such as the completeness of a proof system, e.g., by showing that some cut-rule is admissible. Admissible rules may also be used to shorten proofs in a given system for a logic. Admissibility (often alongside equational unification) has been studied for transitive modal logics and intermediate logics in [19,9,10,13,15,7], and various kinds of proof systems to check admissibility in some of these logics have been defined in [11,14,2]. Characterizations of admissibility have also been obtained for certain many-valued logics in [6,17,16,5], but no general approach has until now been developed for the finite setting. From an algebraic perspective, a quasiequation is admissible in a finitely generated quasivariety (which provide algebraic semantics for many-valued logics) if it is valid in a suitable finite free algebra of this quasivariety. This question is decidable and there exist general methods to obtain proof systems for checking validity in finite algebras (see, e.g., [12,1,19]). However, even free algebras on a small number of generators can be very large since the “worst case” grows double-exponentially. In [18] the theoretical background is given for a general algorithm which outputs a set of algebras that can be used to check admissibility in a more efficient way. The algorithm returns the smallest (according to the standard multiset ordering) set of algebras K for a given finitely generated quasivariety Q, such that a quasiequation is admissible in Q if and only if it is valid in all the algebras of K. The main feature of the system TAFA is an implementation of this

algorithm as a tool for studying admissibility in a wide range of finite algebras. In particular, rather than generating a tableaux system for a potentially very large free algebra of a specific finite algebra or many-valued logic (which is possible by, e.g., [12]), it suffices to generate a tableaux system for the typically small output algebra(s) of TAFA. TAFA is also able to solve general algebraic problems like calculating subalgebras, different kinds of morphisms, products, congruences and their lattices and checking properties like being subdirectly irreducible. Please refer to a standard work like [4] for the details of the algebraic definitions used in this paper. TAFA is implemented in Delphi XE2 and is currently only compiled for Windows, but can easily be used on Mac and Linux using an emulator such as Wine. Many ideas concerning the data structures and basic operations are taken from the implementation of the Algebra Workbench [20]. TAFA as well as a copy of [18] are available from https://sites.google.com/site/admissibility/.

2

Basic Features of TAFA

In order to use TAFA the user has first either to define the algebras of interest in TAFA or to load some predefined (see File > Predefined algebras 1 ) or previously stored algebras (from a file). Defining a new algebra (see File > New algebra) includes giving it a name, labelling the elements and defining the operations. The user can easily rename, delete or edit algebras, their elements and operations or add some comment by either double clicking the corresponding field of the grid in the main window or using the menu Edit . The main window of TAFA contains a list showing for each algebra its name, cardinality, the names and arities of its operations, and any comments. We say that an entry of a list, e.g., an algebra in the main window, is selected if it is highlighted, and chosen if the appropriate checkbox is checked. TAFA can save the chosen algebras as a binary file (*.fab, fast, illegible), as a text file (*.fai, slower, legible) or, if the algebra is a partially ordered set with an operation “meet”, to a *.osf file which can be read by the Algebra Workbench to visualize the corresponding Hasse diagram. TAFA loads algebras from fab- or fai-files and is able to copy or remove algebras in the main window (menu File). The algebras are stored as data type TAlgebra within TAFA, which is connected to lists of the type TAlgebraUniverse and TOperationList providing further procedures and objects. Once the algebras of interest are defined in the main window, the basic operations of universal algebra listed below can be performed. 2.1

Homomorphisms

The menu item Tools > Morphisms opens a dialogue window, where the user can choose a domain A1 and a codomain A2 (of the same language) from the list of defined algebras. It is possible to choose whether to calculate all homomorphisms between A1 and A2 or only those that are surjective, injective or bijective. 1

Navigation through the menus is denoted here by Menu > Menu item.

When the button “Calculate” is pressed, TAFA lists the homomorphisms satisfying the chosen criteria. Double-clicking on an entry of the list shows the mappings from elements of A1 to elements of A2 . Using the Tools menu of this dialogue window it is also possible to add the homomorphic image as a new algebra to the main window or to save the mapping informations to a text file. 2.2

Subalgebras

The menu item Tools > Subalgebras opens a dialogue window which lists all the subalgebras of the active algebra. The subalgebras are stored as entities of TAlgebraUniverse within this dialogue window to save time (there is no need to build up the operation tables), but it is possible to add the checked subalgebras as new algebras to the main window using the menu Tools of the dialogue window. The Options menu of the dialogue window offers the possibility to (heuristically) first list the smaller and then the bigger algebras by first calculating the subalgebras generated by zero or one element, storing their sizes and then trying to combine the given generators in such a way that the subalgebras generated are potentially small. Tools > Generating subalgebra opens a dialogue window where the user can choose some elements a1 , . . . , ak of the active algebra A. TAFA then calculates the unique subalgebra of A generated by the elements a1 , . . . , ak . 2.3

Products

Having defined algebras A1 , . . . , An of the same language in the main window of TAFA, the user can calculate the direct product A1 × · · · × An using Tools > Direct product . Specifying some k ∈ N with Tools > Direct power , the direct power Ak of the selected algebra A is calculated. 2.4

Congruences

Tools > Congruences opens a dialogue window which lists the congruences Con(A) of the selected algebra A in the main window. Selecting a congruence in the list shows how the congruence is defined. The dialogue window menu Tools lets the user store the congruence lattice, given by Con(A), as a new algebra (with operations “meet” and “join”) to the main window. It is also possible to quotient the active structure with the selected congruence or to save the congruence informations to a text file. 2.5

Free Algebras

If the set K = {A1 , . . . , Ak } of algebras of the same language is chosen in TAFA, the menu item Tools > Free algebra lets the user specify a natural number n ∈ N and TAFA calculates the free algebra of K on n generators, denoted FK (n). There

is also the possibility to search for the smallest generating free algebra for K, i.e., the smallest free algebra FK (n) of K such that all A in K are homomorphic images of FK (n).

3

Validity and Admissibility

Given a set of algebras K of the same language, the quasivariety generated by K, denoted Q(K), is the smallest class of algebras closed under taking isomorphisms, subalgebras, products and ultraproducts that contains K. Let us assume that K is a finite set of finite algebras, i.e., Q(K) is finitely generated. Tm denotes the term algebra over countably infinitely many generators. We say that a quasiequation Σ ⇒ ϕ ≈ ψ, i.e., a set (possibly empty) of equations Σ implying a single equation ϕ ≈ ψ, is valid in the quasivariety Q, written Σ |=Q ϕ ≈ ψ, if for every A ∈ Q and every homomorphism h : Tm → A, whenever h(ϕ′ ) = h(ψ ′ ) for all ϕ′ ≈ ψ ′ ∈ Σ, also h(ϕ) = h(ψ). The quasiequation Σ ⇒ ϕ ≈ ψ is called admissible in the quasivariety Q if for every homomorphism σ : Tm → Tm: ∅ |=Q σ(ϕ′ ) ≈ σ(ψ ′ ) for all ϕ′ ≈ ψ ′ ∈ Σ

implies

∅ |=Q σ(ϕ) ≈ σ(ψ).

It is well known (see, e.g., [19,17,18]) that the quasiequations admissible in Q = Q(K) are the quasiequations valid in FQ (n) where n = max{|A| : A ∈ K}. Hence checking whether a given quasiequation is admissible in Q is decidable, since FQ (n) is finite if Q is a finitely generated quasivariety (see [3]). We are now interested in the “smallest” set of algebras generating the quasivariety Q(FQ (n)), i.e., the smallest set of admissibility algebras, to make checking validity faster (since the corresponding proof system will be smaller). One possibility is to make use of the Derschowitz-Manna multiset ordering ≤m defined in [8] to compare multisets of cardinalities of sets of algebras. We say that a set of finite algebras {A1 , . . . , An } is a minimal generating set for the quasivariety Q(A1 , . . . , An ) if for every set of finite algebras {B1 , . . . , Bk }: Q(A1 , . . . , An ) = Q(B1 , . . . , Bk ) implies [|A1 |, . . . , |An |] ≤m [|B1 |, . . . , |Bk |]. To obtain a minimal generating set for Q(FQ(K) (n)), we have implemented the algorithm MinGenSet(K) (see [18] for details) which returns the (unique up to isomorphism) minimal generating set for the quasivariety Q(K) generated by a finite set K of finite algebras. The algorithm MinGenSet calculates the congruence lattice of the input algebras, which takes exponential time with respect to the size of the input. Therefore we have implemented the algorithm AdmAlgs (see Figure 1), which combines the decomposing of MinGenSet with the fact that a subalgebra B of FQ(K) (n) generates the same quasivariety as FQ(K) (n), i.e., Q(B) = Q(FQ(K) (n)), if every algebra A of K is a homomorphic image of B. The idea is to reduce the sizes of the generating algebras (taking subalgebras of the free algebras) while making sure that the algebras are “complex” enough to generate the given

quasivariety (checking homomorphisms), and then to remove redundancy using MinGenSet. The algorithm Free(A, D) used in AdmAlgs calculates the smallest free algebra of D which is a prehomomorphic image of the algebra A by increasing the number of generators one a time while searching for surjective homomorphisms onto A. The algorithm SubPreHom(A, B) on the other hand returns a proper subalgebra of B which has A as a homomorphic image. If there is no such algebra, B is returned. 1: function AdmAlgs(K) 2: declare A, D : set 3: declare B, B′ : algebra 4: D ← MinGenSet(K) 5: A←∅ 6: for all A ∈ D do 7: B ← Free(A, D) 8: B′ ← SubPreHom(A, B) 9: while B′ 6= B do 10: B ← B′ 11: B′ ← SubPreHom(A, B) 12: end while 13: add B to A 14: end for 15: return MinGenSet(A) 16: end function

Fig. 1. Given a finite set K of finite algebras, return the minimal generating set of the quasivariety Q(FQ(K) (ω)).

Given a set K of algebras chosen in TAFA, the user selects the appropriate free algebra or lets the program find the smallest generating free algebra for K with Tools > Admissibility algebra. The menu Options of the dialogue window for calculating admissibility algebras then lets the user choose whether to search admissibility algebras from smaller to larger or with the usual algorithm of searching subalgebras (which is independent of the sizes). Although the latter is much quicker for small algebras, there are some cases where the heuristic method performs faster. Once the admissibility algebra is stored as a new algebra in the main window, the user can run MinGenSet (from the menu Tools) to obtain the unique smallest set of admissibility algebras for K.

4

A Case Study: 3-Element Groupoids

In [18], admissibility was studied using TAFA for various algebras and logics with up to five elements and as many operations, e.g., for the Wajsberg algebra corresponding to the 3-valued Lukasiewicz logic, for Kleene and De Morgan lattices and algebras and for Stone algebras.

In order to have a range of “arbitrary” algebras to study we have also considered the groupoids with three elements, i.e., the 3-element algebras with a binary operation. There are 3330 different groupoids up to isomorphism (out of 39 = 19683 in total) for which TAFA calculates the smallest generating free algebra. Figure 2 illustrates the distribution of the cardinalities of these free algebras. The number of generators is not always the same to produce a free algebra of a given cardinality and there are even 16 cases where three generators are needed.

Fig. 2. Cardinality of free algebras (x-axis) and number of groupoids (y-axis).

The main goal was to calculate the smallest set of admissibility algebras for all 3-element groupoids G, namely the results of MinGenSet(FG (3)). For free algebras with less than 25 elements we performed MinGenSet directly, for the larger cases we used AdmAlgs. The admissibility algebras all have fewer than 10 elements. Figure 3 lists the multisets of cardinalities of the minimal generating sets and how many times they occur. An algebra is called structurally complete if the sets of valid and admissible quasiequations coincide for this algebra, and almost structurally complete, if these sets coincide for quasiequations with unifiable premises (see [18]). These completeness-checks are accessible in TAFA via the menu Check . Performing this check to the groupoids confirmed that 654 of the investigated algebras are not structurally complete, whereas 254 of them are almost structurally complete. The remaining 2676 groupoids are structurally complete. Cards. of MinGenSet(FG(3)) Number of groupoids

2,2 16

2 3 9 2661

4 90

5 108

6 398

Fig. 3. Cardinalities of the minimal generating sets.

8 18

9 30

Acknowledgements I would like to thank George Metcalfe for his helpful comments and suggestions and Markus Sprenger for providing the source code of the Algebra Workbench. The author acknowledges support from Swiss National Science Foundation Grant 20002 129507.

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