New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -> X as states. For such a state s and a predicate p, the validity probability s |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using L\"uders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems

Algebras of Measurements: the logical structure of Quantum Mechanics

In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute. Keywords: Quantum measurements, Measurement algebras, Quantum Logic. PACS: 02.10.-v.Comment: Submitted, 30 page

Complexity of fuzzy answer set programming under Łukasiewicz semantics

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution

(Un)decidable Problems about Reachability of Quantum Systems

We study the reachability problem of a quantum system modelled by a quantum automaton. The reachable sets are chosen to be boolean combinations of (closed) subspaces of the state space of the quantum system. Four different reachability properties are considered: eventually reachable, globally reachable, ultimately forever reachable, and infinitely often reachable. The main result of this paper is that all of the four reachability properties are undecidable in general; however, the last three become decidable if the reachable sets are boolean combinations without negation

An Algebra of Synchronous Scheduling Interfaces

In this paper we propose an algebra of synchronous scheduling interfaces which combines the expressiveness of Boolean algebra for logical and functional behaviour with the min-max-plus arithmetic for quantifying the non-functional aspects of synchronous interfaces. The interface theory arises from a realisability interpretation of intuitionistic modal logic (also known as Curry-Howard-Isomorphism or propositions-as-types principle). The resulting algebra of interface types aims to provide a general setting for specifying type-directed and compositional analyses of worst-case scheduling bounds. It covers synchronous control flow under concurrent, multi-processing or multi-threading execution and permits precise statements about exactness and coverage of the analyses supporting a variety of abstractions. The paper illustrates the expressiveness of the algebra by way of some examples taken from network flow problems, shortest-path, task scheduling and worst-case reaction times in synchronous programming.Comment: In Proceedings FIT 2010, arXiv:1101.426

Model checking Quantitative Linear Time Logic

This paper considers QLtl, a quantitative analagon of Ltl and presents algorithms for model checking QLtl over quantitative versions of Kripke structures and Markov chains

Generating Property-Directed Potential Invariants By Backward Analysis

This paper addresses the issue of lemma generation in a k-induction-based formal analysis of transition systems, in the linear real/integer arithmetic fragment. A backward analysis, powered by quantifier elimination, is used to output preimages of the negation of the proof objective, viewed as unauthorized states, or gray states. Two heuristics are proposed to take advantage of this source of information. First, a thorough exploration of the possible partitionings of the gray state space discovers new relations between state variables, representing potential invariants. Second, an inexact exploration regroups and over-approximates disjoint areas of the gray state space, also to discover new relations between state variables. k-induction is used to isolate the invariants and check if they strengthen the proof objective. These heuristics can be used on the first preimage of the backward exploration, and each time a new one is output, refining the information on the gray states. In our context of critical avionics embedded systems, we show that our approach is able to outperform other academic or commercial tools on examples of interest in our application field. The method is introduced and motivated through two main examples, one of which was provided by Rockwell Collins, in a collaborative formal verification framework.Comment: In Proceedings FTSCS 2012, arXiv:1212.657