scholarly journals Modal Logic S5 Satisfiability in Answer Set Programming

2021 ◽  
Vol 21 (5) ◽  
pp. 527-542
Author(s):  
MARIO ALVIANO ◽  
SOTIRIS BATSAKIS ◽  
GEORGE BARYANNIS
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Answer Set Programming ◽  
Practical Applications ◽  
Simplified Approach ◽  
Modal Operators ◽  
Comparable Performance ◽  
Entailment Relations ◽  
Significant Attention ◽  
Answer Set

AbstractModal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead.

Representability in second-order propositional poly-modal logic

2002 ◽  
Vol 67 (3) ◽  
pp. 1039-1054 ◽  
Author(s):  
G. Aldo Antonelli ◽  
Richmond H. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Multiple Modalities ◽  
Modal Operators ◽  
Second Order Systems ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Propositional System

AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

Systems, Engineering Environments, and Competitions

AI Magazine ◽  
2016 ◽  
Vol 37 (3) ◽  
pp. 45-52 ◽  
Author(s):  
Yuliya Lierler ◽  
Marco Maratea ◽  
Francesco Ricca
Keyword(s):  
Systems Engineering ◽  
Answer Set Programming ◽  
Practical Applications ◽  
Development Tools ◽  
History Of ◽  
Answer Set

The goal of this article is threefold. First, we trace the history of the development of answer set solvers, by accounting for more than a dozen of them. Second, we discuss development tools and environments that facilitate the use of answer set programming technology in practical applications. Last, we present the evolution of the answer set programming competitions, prime venues for tracking advances in answer set solving technology.

An infinitary encoding of temporal equilibrium logic

2015 ◽  
Vol 15 (4-5) ◽  
pp. 666-680 ◽  
Author(s):  
PEDRO CABALAR ◽  
MARTÍN DIÉGUEZ ◽  
CONCEPCIÓN VIDAL
Keyword(s):  
Logic Programming ◽  
Temporal Logic ◽  
Linear Time ◽  
Answer Set Programming ◽  
Modal Operators ◽  
Temporal Logic Programming ◽  
Linear Time Temporal Logic ◽  
Answer Set

AbstractThis paper studies the relation between two recent extensions of propositional Equilibrium Logic, a well-known logical characterisation of Answer Set Programming. In particular, we show how Temporal Equilibrium Logic, which introduces modal operators as those typically handled in Linear-Time Temporal Logic (LTL), can be encoded into Infinitary Equilibrium Logic, a recent formalisation that allows the use of infinite conjunctions and disjunctions. We prove the correctness of this encoding and, as an application, we further use it to show that the semantics of the temporal logic programming formalism called TEMPLOG is subsumed by Temporal Equilibrium Logic.

scholarly journals Linear-Time Temporal Answer Set Programming

2021 ◽  
pp. 1-55
Author(s):  
FELICIDAD AGUADO ◽  
PEDRO CABALAR ◽  
MARTÍN DIÉGUEZ ◽  
GILBERTO PÉREZ ◽  
TORSTEN SCHAUB ◽  
...  
Keyword(s):  
Temporal Logic ◽  
Linear Time ◽  
Selection Criterion ◽  
Answer Set Programming ◽  
Logic Programs ◽  
Modal Operators ◽  
Linear Time Temporal Logic ◽  
Stable Models ◽  
Answer Set

Abstract In this survey, we present an overview on (Modal) Temporal Logic Programming in view of its application to Knowledge Representation and Declarative Problem Solving. The syntax of this extension of logic programs is the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). In the paper, we focus on the main recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL but involves a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). As a result, we obtain a proper extension of the stable models semantics for the general case of temporal formulas in the syntax of LTL. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between finite and infinite trace length. We also provide further useful results, such as the translation into other formalisms like Quantified Equilibrium Logic and Second-order LTL, and some techniques for computing temporal stable models based on automata constructions. In the remainder of the paper, we focus on practical aspects, defining a syntactic fragment called (modal) temporal logic programs closer to ASP, and explaining how this has been exploited in the construction of the solver telingo, a temporal extension of the well-known ASP solver clingo that uses its incremental solving capabilities.

scholarly journals A Trajectory Calculus for Qualitative Spatial Reasoning Using Answer Set Programming

2018 ◽  
Vol 18 (3-4) ◽  
pp. 355-371 ◽  
Author(s):  
GEORGE BARYANNIS ◽  
ILIAS TACHMAZIDIS ◽  
SOTIRIS BATSAKIS ◽  
GRIGORIS ANTONIOU ◽  
MARIO ALVIANO ◽  
...  
Keyword(s):  
Natural Language ◽  
Real World ◽  
Moving Objects ◽  
Spatial Reasoning ◽  
Spatial Information ◽  
Scale Up ◽  
Answer Set Programming ◽  
Qualitative Spatial Reasoning ◽  
Practical Applications ◽  
Answer Set

AbstractSpatial information is often expressed using qualitative terms such as natural language expressions instead of coordinates; reasoning over such terms has several practical applications, such as bus routes planning. Representing and reasoning on trajectories is a specific case of qualitative spatial reasoning that focuses on moving objects and their paths. In this work, we propose two versions of a trajectory calculus based on the allowed properties over trajectories, where trajectories are defined as a sequence of non-overlapping regions of a partitioned map. More specifically, if a given trajectory is allowed to start and finish at the same region, 6 base relations are defined (TC-6). If a given trajectory should have different start and finish regions but cycles are allowed within, 10 base relations are defined (TC-10). Both versions of the calculus are implemented as ASP programs; we propose several different encodings, including a generalised program capable of encoding any qualitative calculus in ASP. All proposed encodings are experimentally evaluated using a real-world dataset. Experiment results show that the best performing implementation can scale up to an input of 250 trajectories for TC-6 and 150 trajectories for TC-10 for the problem of discovering a consistent configuration, a significant improvement compared to previous ASP implementations for similar qualitative spatial and temporal calculi.

scholarly journals Generalized Target Assignment and Path Finding Using Answer Set Programming

2017 ◽  
Author(s):  
Van Nguyen ◽  
Philipp Obermeier ◽  
Tran Cao Son ◽  
Torsten Schaub ◽  
William Yeoh
Keyword(s):  
Scale Up ◽  
Answer Set Programming ◽  
Specific Information ◽  
Path Finding ◽  
Practical Applications ◽  
Target Assignment ◽  
Specific Order ◽  
Multi Agent ◽  
Unequal Number ◽  
Answer Set

In Multi-Agent Path Finding (MAPF), a team of agents needs to find collision-free paths from their starting locations to their respective targets. Combined Target Assignment and Path Finding (TAPF) extends MAPF by including the problem of assigning targets to agents as a precursor to the MAPF problem. A limitation of both models is their assumption that the number of agents and targets are equal, which is invalid in some applications such as autonomous warehouse systems. We address this limitation by generalizing TAPF to allow for (1)~unequal number of agents and tasks; (2)~tasks to have deadlines by which they must be completed; (3)~ordering of groups of tasks to be completed; and (4)~tasks that are composed of a sequence of checkpoints that must be visited in a specific order. Further, we model the problem using answer set programming (ASP) to show that customizing the desired variant of the problem is simple one only needs to choose the appropriate combination of ASP rules to enforce it. We also demonstrate experimentally that if problem specific information can be incorporated into the ASP encoding then ASP based method can be efficient and can scale up to solve practical applications.

Modal operators and functional completeness, II

1977 ◽  
Vol 42 (3) ◽  
pp. 391-399 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Kripke Semantics ◽  
Functional Completeness ◽  
Possible World ◽  
States Of Affairs ◽  
Propositional Modal Logic ◽  
Modal Operators ◽  
Fixed Frame ◽  
State Of Affairs

In the Kripke semantics for propositional modal logic, a frame W = (W, ≺) represents a set of “possible worlds” and a relation of “accessibility” between possible worlds. With respect to a fixed frame W, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a way of forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fw: (P(W))n → P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term “states of affairs” for our “possible worlds”.}In a broader sense, an n-ary connective is represented by an n-ary operatorf = {fw∣ W ∈ Fr}, where Fr is the class of all frames and each fw: (P(W))n → P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(P1,…, Pn) is true in a possible world depends only upon which modal combinations of P1,…,Pn are true in that world. (A modal combination of P1,…,Pn is the result of applying a modal connective to P1,…, Pn.) A connective C is strongly coherent if whether C(P1, …, Pn) obtains in a state of affairs depends only upon which modal combinations of P1,…, Pn obtain in that state of affairs.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

scholarly journals Open answer set programming with guarded programs

2008 ◽  
Vol 9 (4) ◽  
pp. 1-53 ◽  
Author(s):  
Stijn Heymans ◽  
Davy Van Nieuwenborgh ◽  
Dirk Vermeir
Keyword(s):  
Answer Set Programming ◽  
Answer Set
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