2,857 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Logic of Non-Monotonic Interactive Proofs (Formal Theory of Temporary Knowledge Transfer)
We propose a monotonic logic of internalised non-monotonic or instant
interactive proofs (LiiP) and reconstruct an existing monotonic logic of
internalised monotonic or persistent interactive proofs (LiP) as a minimal
conservative extension of LiiP. Instant interactive proofs effect a fragile
epistemic impact in their intended communities of peer reviewers that consists
in the impermanent induction of the knowledge of their proof goal by means of
the knowledge of the proof with the interpreting reviewer: If my peer reviewer
knew my proof then she would at least then (in that instant) know that its
proof goal is true. Their impact is fragile and their induction of knowledge
impermanent in the sense of being the case possibly only at the instant of
learning the proof. This accounts for the important possibility of
internalising proofs of statements whose truth value can vary, which, as
opposed to invariant statements, cannot have persistent proofs. So instant
interactive proofs effect a temporary transfer of certain propositional
knowledge (knowable ephemeral facts) via the transmission of certain individual
knowledge (knowable non-monotonic proofs) in distributed systems of multiple
interacting agents.Comment: continuation of arXiv:1201.3667 ; published extended abstract:
DOI:10.1007/978-3-642-36039-8_16 ; related to arXiv:1208.591
The DLV System for Knowledge Representation and Reasoning
This paper presents the DLV system, which is widely considered the
state-of-the-art implementation of disjunctive logic programming, and addresses
several aspects. As for problem solving, we provide a formal definition of its
kernel language, function-free disjunctive logic programs (also known as
disjunctive datalog), extended by weak constraints, which are a powerful tool
to express optimization problems. We then illustrate the usage of DLV as a tool
for knowledge representation and reasoning, describing a new declarative
programming methodology which allows one to encode complex problems (up to
-complete problems) in a declarative fashion. On the foundational
side, we provide a detailed analysis of the computational complexity of the
language of DLV, and by deriving new complexity results we chart a complete
picture of the complexity of this language and important fragments thereof.
Furthermore, we illustrate the general architecture of the DLV system which
has been influenced by these results. As for applications, we overview
application front-ends which have been developed on top of DLV to solve
specific knowledge representation tasks, and we briefly describe the main
international projects investigating the potential of the system for industrial
exploitation. Finally, we report about thorough experimentation and
benchmarking, which has been carried out to assess the efficiency of the
system. The experimental results confirm the solidity of DLV and highlight its
potential for emerging application areas like knowledge management and
information integration.Comment: 56 pages, 9 figures, 6 table
On SAT representations of XOR constraints
We study the representation of systems S of linear equations over the
two-element field (aka xor- or parity-constraints) via conjunctive normal forms
F (boolean clause-sets). First we consider the problem of finding an
"arc-consistent" representation ("AC"), meaning that unit-clause propagation
will fix all forced assignments for all possible instantiations of the
xor-variables. Our main negative result is that there is no polysize
AC-representation in general. On the positive side we show that finding such an
AC-representation is fixed-parameter tractable (fpt) in the number of
equations. Then we turn to a stronger criterion of representation, namely
propagation completeness ("PC") --- while AC only covers the variables of S,
now all the variables in F (the variables in S plus auxiliary variables) are
considered for PC. We show that the standard translation actually yields a PC
representation for one equation, but fails so for two equations (in fact
arbitrarily badly). We show that with a more intelligent translation we can
also easily compute a translation to PC for two equations. We conjecture that
computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing
proof of the transformation from AC-representations to monotone circuits,
improved wording and literature review; 3rd v. updated literature,
strengthened treatment of monotonisation, improved discussions; 4th v. update
of literature, discussions and formulations, more details and examples;
conference v. to appear LATA 201
Reasoning About Strategies: On the Model-Checking Problem
In open systems verification, to formally check for reliability, one needs an
appropriate formalism to model the interaction between agents and express the
correctness of the system no matter how the environment behaves. An important
contribution in this context is given by modal logics for strategic ability, in
the setting of multi-agent games, such as ATL, ATL\star, and the like.
Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which
we denote here by CHP-SL, with the aim of getting a powerful framework for
reasoning explicitly about strategies. CHP-SL is obtained by using first-order
quantifications over strategies and has been investigated in the very specific
setting of two-agents turned-based games, where a non-elementary model-checking
algorithm has been provided. While CHP-SL is a very expressive logic, we claim
that it does not fully capture the strategic aspects of multi-agent systems. In
this paper, we introduce and study a more general strategy logic, denoted SL,
for reasoning about strategies in multi-agent concurrent games. We prove that
SL includes CHP-SL, while maintaining a decidable model-checking problem. In
particular, the algorithm we propose is computationally not harder than the
best one known for CHP-SL. Moreover, we prove that such a problem for SL is
NonElementarySpace-hard. This negative result has spurred us to investigate
here syntactic fragments of SL, strictly subsuming ATL\star, with the hope of
obtaining an elementary model-checking problem. Among the others, we study the
sublogics SL[NG], SL[BG], and SL[1G]. They encompass formulas in a special
prenex normal form having, respectively, nested temporal goals, Boolean
combinations of goals and, a single goal at a time. About these logics, we
prove that the model-checking problem for SL[1G] is 2ExpTime-complete, thus not
harder than the one for ATL\star
- …