scholarly journals Decidability and Undecidability Results for Propositional Schemata

2011 ◽  
Vol 40 ◽  
pp. 599-656 ◽  
Author(s):  
V. Aravantinos ◽  
R. Caferra ◽  
N. Peltier
Keyword(s):  
Large Class ◽  
Propositional Logic ◽  
General Class ◽  
Satisfiability Problem ◽  
Propositional Formula ◽  
Good Compromise ◽  
Proof Procedure

We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions and iterated connectives ranging over intervals parameterized by arithmetic variables. The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability.

scholarly journals Complex valued probability logics

2014 ◽  
Vol 95 (109) ◽  
pp. 73-86 ◽  
Author(s):  
Angelina Ilic-Stepic ◽  
Zoran Ognjanovic
Keyword(s):  
Propositional Logic ◽  
Linear Inequalities ◽  
Satisfiability Problem ◽  
Propositional Formula ◽  
Classical Propositional Logic ◽  
Intended Meaning ◽  
Complex Ball ◽  
Axiom Systems ◽  
Complex Valued

We present two complex valued probabilistic logics, LCOMPB and LCOMPS, which extend classical propositional logic. In LCOMPB one can express formulas of the form Bz,?? meaning that the probability of ? is in the complex ball with the center z and the radius ?, while in LCOMPS one can make statements of the form Sz,?? with the intended meaning - the probability of propositional formula ? is in the complex square with the center z and the side 2?. The corresponding strongly complete axiom systems are provided. Decidability of the logics are proved by reducing the satisfiability problem for LCOMPB (LCOMPS) to the problem of solving systems of quadratic (linear) inequalities.

scholarly journals A Theory of Satisfiability-Preserving Proofs in SAT Solving

10.29007/tc7q ◽  
2018 ◽  
Author(s):  
Adrián Rebola-Pardo ◽  
Martin Suda
Keyword(s):  
Propositional Logic ◽  
Logical Truth ◽  
Point Of View ◽  
Traditional View ◽  
Satisfiability Problem ◽  
Sat Solving ◽  
Sat Solvers ◽  
Proof Systems ◽  
Meta Level

We study the semantics of propositional interference-based proof systems such as DRAT and DPR. These are characterized by modifying a CNF formula in ways that preserve satisfiability but not necessarily logical truth. We propose an extension of propositional logic called overwrite logic with a new construct which captures the meta-level reasoning behind interferences. We analyze this new logic from the point of view of expressivity and complexity, showing that while greater expressivity is achieved, the satisfiability problem for overwrite logic is essentially as hard as SAT, and can be reduced in a way that is well-behaved for modern SAT solvers. We also show that DRAT and DPR proofs can be seen as overwrite logic proofs which preserve logical truth. This much stronger invariant than the mere satisfiability preservation maintained by the traditional view gives us better understanding on these practically important proof systems. Finally, we showcase this better understanding by finding intrinsic limitations in interference-based proof systems.

scholarly journals Finding Optimal Solutions in HTN Planning - A SAT-based Approach

2019 ◽  
Author(s):  
Gregor Behnke ◽  
Daniel Höller ◽  
Susanne Biundo
Keyword(s):  
Propositional Logic ◽  
Propositional Formula ◽  
Optimal Solutions ◽  
Hierarchical Task Network ◽  
New Approaches ◽  
Htn Planning ◽  
Overall Performance ◽  
The Relationship

Over the last years, several new approaches to Hierarchical Task Network (HTN) planning have been proposed that increased the overall performance of HTN planners. However, the focus has been on agile planning - on finding a solution as quickly as possible. Little work has been done on finding optimal plans. We show how the currently best-performing approach to HTN planning - the translation into propositional logic - can be utilised to find optimal plans. Such SAT-based planners usually bound the HTN problem to a certain depth of decomposition and then translate the problem into a propositional formula. To generate optimal plans, the length of the solution has to be bounded instead of the decomposition depth. We show the relationship between these bounds and how it can be handled algorithmically. Based on this, we propose an optimal SAT-based HTN planner and show that it performs favourably on a benchmark set.

scholarly journals Ramsey interferometry with generalized one-axis twisting echoes

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 268 ◽  
Author(s):  
Marius Schulte ◽  
Victor J. Martínez-Lahuerta ◽  
Maja S. Scharnagl ◽  
Klemens Hammerer
Keyword(s):  
Large Class ◽  
General Class ◽  
Particle Number ◽  
Sensitivity Enhancement ◽  
Local Sensitivity ◽  
Before And After ◽  
Phase Signal

We consider a large class of Ramsey interferometry protocols which are enhanced by squeezing and un-squeezing operations before and after a phase signal is imprinted on the collective spin of N particles. We report an analytical optimization for any given particle number and strengths of (un-)squeezing. These results can be applied even when experimentally relevant decoherence processes during the squeezing and un-squeezing interactions are included. Noise between the two interactions is however not considered in this work. This provides a generalized characterization of squeezing echo protocols, recovering a number of known quantum metrological protocols as local sensitivity maxima, thereby proving their optimality. We discover a single new protocol. Its sensitivity enhancement relies on a double inversion of squeezing. In the general class of echo protocols, the newly found over-un-twisting protocol is singled out due to its Heisenberg scaling even at strong collective dephasing.

scholarly journals Learning SMT(LRA) Constraints using SMT Solvers

2018 ◽  
Author(s):  
Samuel Kolb ◽  
Stefano Teso ◽  
Andrea Passerini ◽  
Luc De Raedt
Keyword(s):  
Formal Verification ◽  
Greedy Algorithm ◽  
Propositional Logic ◽  
Learning Task ◽  
Satisfiability Problem ◽  
Incremental Algorithm ◽  
Benchmark Problems ◽  
Small Subset ◽  
Smt Solvers ◽  
Recent Developments

We introduce the problem of learning SMT(LRA) constraints from data. SMT(LRA) extends propositional logic with (in)equalities between numerical variables. Many relevant formal verification problems can be cast as SMT(LRA) instances and SMT(LRA) has supported recent developments in optimization and counting for hybrid Boolean and numerical domains. We introduce SMT(LRA) learning, the task of learning SMT(LRA) formulas from examples of feasible and infeasible instances, and we contribute INCAL, an exact non-greedy algorithm for this setting. Our approach encodes the learning task itself as an SMT(LRA) satisfiability problem that can be solved directly by SMT solvers. INCAL is an incremental algorithm that achieves exact learning by looking only at a small subset of the data, leading to significant speed-ups. We empirically evaluate our approach on both synthetic instances and benchmark problems taken from the SMT-LIB benchmarks repository.

scholarly journals A Logic for Reasoning about Upper Probabilities

2002 ◽  
Vol 17 ◽  
pp. 57-81 ◽  
Author(s):  
J. Y. Halpern ◽  
R. Pucella
Keyword(s):  
Propositional Logic ◽  
Satisfiability Problem ◽  
Probability Measures ◽  
Complete Axiomatization ◽  
Np Complete

We present a propositional logic to reason about the uncertainty of events, where the uncertainty is modeled by a set of probability measures assigning an interval of probability to each event. We give a sound and complete axiomatization for the logic, and show that the satisfiability problem is NP-complete, no harder than satisfiability for propositional logic.

scholarly journals An Algorithm for β Clause Satisfiability Problem in Propositional Logic

1994 ◽  
Vol 114 (7-8) ◽  
pp. 796-804
Author(s):  
Toshio Ohyanagi ◽  
Masahito Yamamoto ◽  
Azuma Ohuchi
Keyword(s):  
Propositional Logic ◽  
Satisfiability Problem

scholarly journals General classes of impossible operations through the existence of incomparable states

2007 ◽  
Vol 7 (4) ◽  
pp. 392-400
Author(s):  
I. Chattopadhyay ◽  
D. Sarkar
Keyword(s):  
Large Class ◽  
General Class ◽  
Inner Product ◽  
Entangled States ◽  
Unitary Operators ◽  
Joint System ◽  
Bipartite State ◽  
Local Operation ◽  
Information Conservation ◽  
Hadamard Gate

In this work we show that the most general class of anti-unitary operators are nonphysical in nature through the existence of incomparable pure bipartite entangled states. It is also shown that a large class of inner-product-preserving operations defined only on the three qubits having spin-directions along x,y and z are impossible. If we perform such an operation locally on a particular pure bipartite state then it will exactly transform to another pure bipartite state that is incomparable with the original one. As subcases of the above results we find the nonphysical nature of universal exact flipping operation and existence of universal Hadamard gate. Beyond the information conservation in terms of entanglement, this work shows how an impossible local operation evolve with the joint system in a nonphysical way.

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