In the paper I sketch a theory of massively parallel proofs using cellular automata presentation of deduction. In this presentation inference rules play the role of cellular-automatic local transition functions. In this approach we completely avoid axioms as necessary notion of deduction theory and therefore we can use cyclic proofs without additional problems. As a result, a theory of massive-parallel proofs within unconventional computing is proposed for the first time.
Non-well-founded proofs including cyclic proofs have been actively studying recently (see [5] - [7], [11]). Their features consist in that in the classical theory of deduction, derivation trees, on the one hand, are finite and, on the other hand, they are without cycles, while in the non-well-founded approach they can be infinite and, at the same time, circles occur in them. Nonwell-founded proofs have different applications in computer science. In the paper I am proposing a more radical approach than other non-well-founded approaches to deduction by defining massive-parallel proofs and rejecting axioms in proof theory. This novel approach is characterized as follows:
• Deduction is considered as a transition in cellular automata, where states of cells are regarded as well-formed formulas of a logical language.
• We build up derivations without using axioms, therefore there is no sense in distinguishing logic and theory (i.e. logical and nonlogical axioms), derivable and provable formulas, etc.
• In deduction we do not obtain derivation trees and instead of the latter we find out derivation traces, i.e. a linear evolution of each singular premise.
• Some derivation traces are circular, i.e. some premises are derivable from themselves.
• Some derivation traces are infinite.
2 Proof-theoretic cellular automata
For any logical language L we can construct a proof-theoretic cellular automaton (instead of conventional deductive systems) simulating massive-parallel proofs.
Definition 1 A proof-theoretic cellular automaton is a 4-tuple A = Z d , S, N, δ , where
• d ∈ N is a number of dimensions and the members of Z d are referred as cells,
• S is a finite or infinite set of elements called the states of an automaton A, the members of Z d take their values in S, the set S is collected from well-formed formulas of a language L.
• N ⊂ Z d \ {0} d is a finite ordered set of n elements, N is said to be a neighborhood,
• δ : S n+1 → S that is δ is the inference rule of a language L, it plays the role of local transition function of an automaton A.
As we see an automaton is considered on the endless d-dimensional space of integers, i.e. on Z d . Discrete time is introduced for t = 0, 1, 2, . . . fixing each step of inferring.
For any given z ∈ Z d , its neighborhood is determined by z + N = {z + α : α ∈ N}. There are two often-used neighborhoods: (1,0), (0, -1), (0, 1)}; N M = {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}.
In the case d = 1, von Neumann and Moore neighborhoods coincide. It is easily seen that
At the moment t, the configuration of the whole system (or the global state) is given by the mapping x t : Z d → S, and the evolution is the sequence x 0 , x 1 , x 2 , . . . defined as follows: x t+1 (z) = δ(x t (z), x t (z +α 1 ), . . . , x t (z +α n )), where α 1 , . . . , α n ∈ N. Here x 0 is the initial configuration, and it fully determines the future behavior of the automaton. It is the set of all premises (not axioms).
We assume that δ is an inference rule, i.e. a mapping from the set of premises (their number cannot exceed n = |N|) to a conclusion. For any z ∈ Z d the sequence x 0 (z), x 1 (z), . . . , x t (z),. . . is called a derivation trace from a state x 0 (z). If there exists t such that x t (z) = x l (z) for all l > t, then a derivation trace is finite. It is circular/cyclic if there exists l such that x t (z) = x t+l (z) for all t.
Definition 2 In case all derivation traces of a proof-theoretic cellular automaton A are circular, this automaton A is said to be reversible.
Notice that x t+1 depends only upon x t , i.e. the previous configuration. It enables us to build the function G A : C A → C A , where C A is the set of all possible configurations of the cellular automaton A (it is the set of all mappings Z d → S, because we can take each element of this set as the initial configuration x 0 , though not every element can arise in the evolution of some other configuration). G A is called the global function of the automaton.
Example 1 (modus ponens) Consider a propositional language L that is built in the standard way with the only binary operation of implication ⊃. Let us suppose that well-formed formulas of that language are used as the set of states for a proof-theoretic cellular automaton A. Further, assume that modus ponens is a transition rule of this automaton A and it is formulated for any ϕ, ψ ∈ L as follows:
The further dynamics will depend on the neighborhood. If we assume the Moor neighborhood in the 2-dimensional space, this dynamics will be exemplified by the evolution of cell states in Fig. 1 -Fig. 3.
This example shows that first we completely avoid axioms and secondly we take premisses from the cell states of the neighborhood according to a
A with the Moor neighborhood in the 2-dimensional space, its states run over formulas set up in a propositional language L with the only binary operation ⊃, t = 0. Notice that p, q, r are propositiona
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