Simone Bova

A logical analysis of Mamdani-type fuzzy inference, I theoretical bases

This paper is divided into two parts. In the present Part I, our main objective is to analyse Mamdani-type fuzzy control systems in logical terms, with special emphasis on the fuzzy inference process. To that end, we provide our own inference procedure, cast in the language of standard many-valued logics. We give an ample discussion of the logical meaning of our procedure. We eventually show how to fully recover Mamdani-type fuzzy inference from the latter. In this sense, then, our proposal may be regarded as a logical interpretation of Mamdani-type fuzzy inference. In Part II of this paper, we report on the results of an experiment on the technical analysis of the financial markets based on fuzzy techniques. The core algorithm implements the inference procedure described in this first part of the paper. In Part II, we will argue that the experimental results support the claim that our present theoretical analysis provides a sound interpretation of Mamdani-type fuzzy inference.

The coherence of Łukasiewicz assessments is NP-complete

The problem of deciding whether a rational assessment of formulas of infinite-valued Lukasiewicz logic is coherent has been shown to be decidable by Mundici [1] and in PSPACE by Flaminio and Montagna [10]. We settle its computational complexity proving an NP-completeness result. We then obtain NP-completeness results for the satisfiability problem of certain many-valued probabilistic logics introduced by Flaminio and Montagna in [9].

The free n-generated BL-algebra

Abstract For each integer n ≥ 0 , we provide an explicit functional characterization of the free n -generated BL-algebra, together with an explicit construction of the corresponding normal forms.

On Compiling CNFs into Structured Deterministic DNNFs

We show that the traces of recently introduced dynamic programming algorithms for #SAT can be used to construct structured deterministic DNNF (decomposable negation normal form) representations of propositional formulas in CNF (conjunctive normal form). This allows us prove new upper bounds on the complexity of compiling CNF formulas into structured deterministic DNNFs in terms of parameters such as the treewidth and the clique-width of the incidence graph.

Finite RDP-algebras

The variety of RDP-algebras forms the algebraic semantics of RDP-logic, the many-valued propositional logic of the revised drastic product left-continuous triangular norm and its residual. We prove a Priestley duality for finite RDP-algebras, and obtain an explicit description of coproducts of finite RDP-algebras. In this light, we give a combinatorial representation of free finitely generated RDP-algebras, which we exploit to construct normal forms, strongest deductive interpolants and most general unifiers.We prove that RDP-unification is unitary, and that the tautology problem for RDP-logic is coNP-complete.

Applications of Finite Duality to Locally Finite Varieties of BL-Algebras

We are concerned with the subvariety of commutative, bounded, and integral residuated lattices, satisfying divisibility and prelinearity, namely, BL-algebras. We give an explicit combinatorial description of the category that is dual to finite BL-algebras. Building on this, we obtain detailed structural information on the locally finite subvarieties of BL-algebras that are analogous to Grigolias subvarieties of finite-valued MV-algebras. As an illustration of the power of the finite duality presented here, we give an exact recursive formula for the cardinality of free finitely generated algebras in such varieties.

The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete

Commutative, integral and bounded GBL-algebras form a subvariety of residuated lattices which provides the algebraic semantics of an interesting common fragment of intuitionistic logic and of several fuzzy logics. It is known that both the equational theory and the quasiequational theory of commutative GBL-algebras are decidable (in contrast to the noncommutative case), but their complexity has not been studied yet. In this paper, we prove that both theories are in PSPACE, and that the quasiequational theory is PSPACE-hard.

Proof search in Hájek's basic logic

We introduce a proof system for Hájeks logic <b>BL</b> based on a relational hypersequents framework. We prove that the rules of our logical calculus, called <b>RHBL</b>, are sound and invertible with respect to any valuation of <b>BL</b> into a suitable algebra, called (ω)[0,1]. Refining the notion of reduction tree that arises naturally from <b>RHBL</b>, we obtain a decision algorithm for <b>BL</b> provability whose running time upper bound is 2^{<i>O</i>(<i>n</i>)}, where <i>n</i> is the number of connectives of the input formula. Moreover, if a formula is unprovable, we exploit the constructiveness of a polynomial time algorithm for leaves validity for providing a procedure to build countermodels in (ω)[0, 1]. Finally, since the size of the reduction tree branches is <i>O</i>(<i>n</i>³), we can describe a polynomial time verification algorithm for <b>BL</b> unprovability.

A logical analysis of Mamdani-type fuzzy inference, II. An experiment on the technical analysis of financial markets

This paper is divided into two parts. In Part I, our main objective was to analyse Mamdani-type fuzzy control systems in logical terms, with special emphasis on the fuzzy inference process. To that end, we provided our own inference procedure, cast in the language of standard many-valued logics. We gave an ample discussion of the logical meaning of our procedure. We eventually showed how to fully recover Mamdani-type fuzzy inference from the latter. In this sense, then, our proposal in Part I may be regarded as a logical interpretation of Mamdani-type fuzzy inference. In the present Part II of this paper, we report on the results of an experiment on the technical analysis of the financial markets based on fuzzy techniques. The core algorithm implements the inference procedure described in the first part of the paper. The experimental results support the claim that our theoretical analysis in Part I provides a sound interpretation of Mamdani-type fuzzy inference.

Circuit Treewidth, Sentential Decision, and Query Compilation

The evaluation of a query over a probabilistic database boils down to computing the probability of a suitable Boolean function, the lineage of the query over the database. The method of query compilation approaches the task in two stages: first, the query lineage is implemented (compiled) in a circuit form where probability computation is tractable; and second, the desired probability is computed over the compiled circuit. A basic theoretical quest in query compilation is that of identifying pertinent classes of queries whose lineages admit compact representations over increasingly succinct, tractable circuit classes. Fostering previous work by Jha and Suciu (ICDT 2012) and Petke and Razgon (SAT 2013), we focus on queries whose lineages admit circuit implementations with small treewidth, and investigate their compilability within tame classes of decision diagrams. In perfect analogy with the characterization of bounded circuit pathwidth by bounded OBDD width, we show that a class of Boolean functions has bounded circuit treewidth if and only if it has bounded SDD width. Sentential decision diagrams (SDDs) are central in knowledge compilation, being essentially as tractable as OBDDs but exponentially more succinct. By incorporating constant width (linear size) SDDs and polynomial size SDDs in the picture, we refine the panorama of query compilation for unions of conjunctive queries with and without inequalities.