Spiking neural P systems with anti-spikes(ASN P systems) are variant forms of spiking neural P systems, which are inspired by inhibitory impulses/spikes or inhibitory synapses. The typical feature of ASN P systems is when a neuron contains both spikes and anti-spikes, spikes and anti-spikes will immediately annihilate each other in a maximal way. In this paper, a restricted variant of ASN P systems, called ASN P systems without annihilating priority, is considered, where the annihilating rule is used as the standard rule, i.e., it is not obligatory to use in the neuron associated with both spikes and anti-spikes. If the annihilating rule is used in a neuron, the annihilation will consume one time unit. As a result, such systems using two categories of spiking rules(identified by(a, a) and(a, ˉa)) can achieve Turing completeness as number accepting devices. Spiking neural P systems with anti-spikes (ASN P systems) are variant forms of spiking neural P systems, which are inspired by inhibitory impulses / spikes or inhibitory synapses. The typical feature of ASN P systems is when a neuron contains both spikes and anti -spikes, spikes and anti-spikes will immediately annihilate each other in a maximal way. In this paper, a restricted variant of ASN P systems, called ASN P systems without annihilating priority, is considered, where the annihilating rule is used as the standard rule, ie, it is not obligatory to use in the neuron associated with both spikes and anti-spikes. As a result, such systems using two categories of spiking rules (identified by (a, a) and (a, ˉa)) can achieve Turing completeness as number accepting devices.