We compare three satisfiability notions for propositional formulas in the language {not, and, or} over a fixed finite-dimensional Hilbert space H=F^d with F in {R, C}. The first is the standard Hilbert-lattice semantics on the subspace lattice L(H), where meet and join are total operations. The second is a global commuting-projector semantics, where all atoms occurring in the formula are interpreted by a single pairwise-commuting projector family. The third is a local partial-Boolean semantics, where binary connectives are defined only on commeasurable pairs and definedness is checked nodewise along the parse tree. We prove, for every fixed d >= 1, Sat_COM^d(phi) implies Sat_PBA^d(phi) implies Sat_STD^d(phi) for every formula phi. We then exhibit the explicit formula SEP-1 := (p and (q or r)) and not((p and q) or (p and r)) which is satisfiable in the standard semantics for every d >= 2, but unsatisfiable under both the global commuting and the partial-Boolean semantics. Consequently, for every d >= 2, the satisfiability classes satisfy SAT_COM^d subseteq SAT_PBA^d subset SAT_STD^d and SAT_COM^d subset SAT_STD^d, while the exact relation between SAT_COM^d and SAT_PBA^d remains open. The point of the paper is semantic comparison, not a new feasibility reduction or a generic translation theorem.
We compare three satisfiability notions for propositional formulas in the language {not, and, or} over a fixed finite-dimensional Hilbert space H=F^d with F in {R, C}. The first is the standard Hilbert-lattice semantics on the subspace lattice L(H), where meet and join are total operations. The second is a global commuting-projector semantics, where all atoms occurring in the formula are interpreted by a single pairwise-commuting projector family. The third is a local partial-Boolean semantics, where binary connectives are defined only on commeasurable pairs and definedness is checked nodewise along the parse tree. We prove, for every fixed d >= 1, Sat_COM^d(phi) implies Sat_PBA^d(phi) implies Sat_STD^d(phi) for every formula phi. We then exhibit the explicit formula SEP-1 := (p and (q or r)) and not((p and q) or (p and r)) which is satisfiable in the standard semantics for every d >= 2, but unsatisfiable under both the global commuting and the partial-Boolean semantics. Consequently, for every d >= 2, the satisfiability classes satisfy SAT_COM^d subseteq SAT_PBA^d subset SAT_STD^d and SAT_COM^d subset SAT_STD^d, while the exact relation between SAT_COM^d and SAT_PBA^d remains open. The point of the paper is semantic comparison, not a new feasibility reduction or a generic translation theorem.
Quantum propositional logic, introduced by Birkhoff and von Neumann, formalizes reasoning about experimental propositions in quantum mechanics through closed subspaces of Hilbert spaces. We provide a comprehensive complexity analysis of the satisfiability problem for quantum logic formulas over the connectives conjunction, disjunction, and negation. Distinguishing between weak satisfiability (evaluation to a nonzero subspace) and strong satisfiability (evaluation to the full space), we establish that both notions coincide with Boolean satisfiability for dimension one, are classically NP complete for dimension two, and become complete for the Blum Shub Smale class NP_R when dimension d >= 3 is fixed. Our principal contribution is twofold. First, we present two alternative constructive reductions from quantum logic satisfiability to the Existential Theory of the Reals (ETR): a realification based encoding that explicitly handles the complex to real translation through commutant characterizations, and a rank one column encoding with orthogonal splitting gadgets. Both reductions yield polynomial size systems of quadratic equations for fixed dimension, proving membership in the complexity class existsR and hence in PSPACE via Cannys algorithm. Second, we provide detailed size bounds distinguishing weak and strong satisfiability: weak satisfiability encodes with O(n D^2 + N_v D) scalar variables where n is the number of atoms, D the real dimension, and N_v the number of disjunctions, while strong satisfiability requires O(n D^2 + N_v D^2) variables due to basis vector witnesses. When dimension is part of the input, strong satisfiability becomes polynomial time equivalent to feasibility of noncommutative integer polynomial equations. These results position quantum logic satisfiability precisely between classical NP and existsR, highlighting dimension as a fundamental complexity parameter.
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the pro-positional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality [symbols omitted] applicable to the PQM lattices; and truth-functionality [symbol omitted] applicable to both the lattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional [symbol omitted] semantics for three-or-higher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum state-induced expectation-function on a PQM truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.
The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiablity for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiablity problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.
The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiability for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiability problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.
We connect both discrete and algebraic complexity theory with the satisfiability problem in certain non-Boolean lattices. Specifically, quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables: in the 1D case it coincides with Boolean propositional logic but, starting with dimension two, violates the distributive law. We introduce the weak and strong satisfiability problem for quantum logic propositional formulae. It turns out that in dimension two, both are also NP --complete. For higher-dimensional spaces ℝ d and ℂ d with d ≥ 3 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale (BSS) model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and extends the (still relatively scarce) list of problems established NP ℝ --complete with one, perhaps, closest in spirit to the classical Cook-Levin Theorem. More precisely, strong satisfiability of ∧ ∨ ∧ --terms is complete, while that of ∧ ∨--terms (i.e., those in conjunctive form) can be decided in polynomial time in dimensions d ≥ 2. The decidability of the infinite-dimensional case being still open, we proceed to investigate the case of indefinite finite dimensions. Here, weak satisfiability still belongs to NP R and strong satisfiability is still hard; the latter, in fact, turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings.
Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic terms. It turns out that in dimension two both are also NP-complete. For higher-dimensional spaces R^d and C^d with d>2 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and extends the still relatively scarce family of NP_R-complete problems with one perhaps closest in spirit to the classical Cook-Levin Theorem. Our investigations on the dimensions a term is weakly/strongly satisfiable in lead to satisfiability problems in indefinite finite and finally in infinite dimension. Here, strong satisfiability turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations