Parameterized aspects of team-based formalisms and logical inference

Parameterized complexity is an interesting subfield of complexity theory that has received a lot of attention in recent years. Such an analysis characterizes the complexity of (classically) intractable problems by pinpointing the computational hardness to some structural aspects of the input. In this thesis, we study the parameterized complexity of various problems from the area of team-based formalisms as well as logical inference. In the context of team-based formalism, we consider propositional dependence logic (PDL). The problems of interest are model checking (MC) and satisfiability (SAT). Peter Lohmann studied the classical complexity of these problems as a part of his Ph.D. thesis proving that both MC and SAT are NP-complete for PDL. This thesis addresses the parameterized complexity of these problems with respect to a wealth of different parameterizations. Interestingly, SAT for PDL boils down to the satisfiability of propositional logic as implied by the downwards closure of PDL-formulas. We propose an interesting satisfiability variant (mSAT) asking for a satisfiable team of size m. The problem mSAT restores the ‘team semantic’ nature of satisfiability for PDL-formulas. We propose another problem (MaxSubTeam) asking for a maximal satisfiable team if a given team does not satisfy the input formula. From the area of logical inference, we consider (logic-based) abduction and argumentation. The problem of interest in abduction (ABD) is to determine whether there is an explanation for a manifestation in a knowledge base (KB). Following Pfandler et al., we also consider two of its variants by imposing additional restrictions over the size of an explanation (ABD and ABD=). In argumentation, our focus is on the argument existence (ARG), relevance (ARG-Rel) and verification (ARG-Check) problems. The complexity of these problems have been explored already in the classical setting, and each of them is known to be complete for the second level of the polynomial hierarchy (except for ARG-Check which is DP-complete) for propositional logic. Moreover, the work by Nord and Zanuttini (resp., Creignou et al.) explores the complexity of these problems with respect to various restrictions over allowed KBs for ABD (ARG). In this thesis, we explore a two-dimensional complexity analysis for these problems. The first dimension is the restrictions over KB in Schaefer’s framework (the same direction as Nord and Zanuttini and Creignou et al.). What differentiates the work in this thesis from an existing research on these problems is that we add another dimension, the parameterization. The results obtained in this thesis are interesting for two reasons. First (from a theoretical point of view), ideas used in our reductions can help in developing further reductions and prove (in)tractability results for related problems. Second (from a practical point of view), the obtained tractability results might help an agent designing an instance of a problem come up with the one for which the problem is tractable

The variety generated by all the ordinal sums of perfect MV-chains

We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek - Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BL_Chang-algebras will be strictly connected to the one generated by Chang's MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new results concerning these last structures and their logic.Comment: This is a revised version of the previous paper: the modifications concern essentially the presentation. The scientific content is substantially unchanged. The major variations are: Definition 2.7 has been improved. Section 3.1 has been made more compact. A new reference, [Bus04], has been added. There is some minor modification in Section 3.

A Faster Tableau for CTL*

There have been several recent suggestions for tableau systems for deciding satisfiability in the practically important branching time temporal logic known as CTL*. In this paper we present a streamlined and more traditional tableau approach built upon the author's earlier theoretical work. Soundness and completeness results are proved. A prototype implementation demonstrates the significantly improved performance of the new approach on a range of test formulas. We also see that it compares favourably to state of the art, game and automata based decision procedures.Comment: In Proceedings GandALF 2013, arXiv:1307.416

Dependences in Strategy Logic

Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: First, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula, revisiting the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014]. We explain why elementary dependences, as defined by Mogavero et al., do not exactly capture the intended concept of behavioral strategies. We address this discrepancy by introducing timeline dependences, and exhibit a large fragment of SL for which model checking can be performed in 2-EXPTIME under this new semantics

Dependences in Strategy Logic

Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: First, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula, revisiting the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014]. We explain why elementary dependences, as defined by Mogavero et al., do not exactly capture the intended concept of behavioral strategies. We address this discrepancy by introducing timeline dependences, and exhibit a large fragment of SL for which model checking can be performed in 2-EXPTIME under this new semantics

Probabilistic justification logic

We present a probabilistic justification logic, PPJ⁠, as a framework for uncertain reasoning about rational belief, degrees of belief and justifications. We establish soundness and strong completeness for PPJ with respect to the class of so-called measurable Kripke-like models and show that the satisfiability problem is decidable. We discuss how PPJ provides insight into the well-known lottery paradox

Verified completeness in Henkin-style for intuitionistic propositional logic

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional logic with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature

Uncertain Reasoning in Justification Logic

This thesis studies the combination of two well known formal systems for knowledge representation: probabilistic logic and justification logic. Our aim is to design a formal framework that allows the analysis of epistemic situations with incomplete information. In order to achieve this we introduce two probabilistic justification logics, which are defined by adding probability operators to the minimal justification logic J. We prove soundness and completeness theorems for our logics and establish decidability procedures. Both our logics rely on an infinitary rule so that strong completeness can be achieved. One of the most interesting mathematical results for our logics is the fact that adding only one iteration of the probability operator to the justification logic J does not increase the computational complexity of the logic